+2 maths pta question bank come book english medium

E R K HSS –ERUMIYAMPATTI Page 1 +2 STUDY MATERIALS

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COME BOOK – TEN MARKS [16 Number Of Ten Mark Questions To Be Asked For Full Test]

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 1. MATRICES AND DETERMINANTS

(ONE QUESTION FOR FULL TEST) --------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 1.1.

(3) Find the adjoint of the matrix

110

432

433

A and verify the result .A A )A adj ( )A adj (A

(6) Find the inverse of the matrix

110

432

433

A and verify that 13

AA .

(7) Show that the adjoint of

122

212

221

A is T

A3 .

(9) If

221

212

122

3

1A , prove that

TAA

1 .

EXERCISE 1.2. (3) Solve by matrix inversion method each of the following system of linear equations:

02,52752,9 zyxzyxzyx .

(4) Solve by matrix inversion method each of the following system of linear equations:

5,1353,72 zyxzyxzyx .

(5) Solve by matrix inversion method each of the following system of linear equations:

13652,43,01083 zyxyxzyx .

EXERCISE 1.4

(4) Solve the following non-homogeneous system of linear equations determinant method:

12;2;4 zyxzyxzyx


4323;02;42 zyxzyxzyx








0223

;5142

;1121

zyxzyxzyx

(10) A small seminar hall can hold 100 chairs. Three different colours ( red, blue and green) of chairs are available. The cost of a red chair is Rs. 240, cost of blue chair is Rs. 260 and the cost of a green chair is Rs. 300. The total cost of chair is Rs. 25,000. Find atleast 3 different solution of the number of chairs in each colour to be purchased.

EXERCISE 1.5. (1) Examine the consistency of the following system of equations. If it is consistent then solve the same:

(i) solve : 192;1375;25634 zyxzyxzyx

(ii) solve : 013652;043;1083 zyxzyxzyx

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(v) solve : 3333;2222;1 zyxzyxzyx

(2) Discuss the solutions of the system of equations 24,222,2 zyxzyxzyx for all

values of

(3) For what values of k, the system of equations 1,1,1 kzyxzkyxzykx

have (i) unique solution (ii) more than one solution (iii) no solution Example 1.4 :

If

312

321

111

A ,verify 3IAAAadjAadjA

Example 1.8 : Solve by matrix inversion method 2,6,932 zyxzyxzyx

Example 1.18: Solve the following non-homogeneous equations of three unknowns.

(1) 12;422;72 zyxzyxzyx

(2) 824;23;62 zyxzyxzyx

(4) 12633;8422;42 zyxzyxzyx

Example 1.19: A bag contains 3 types of coins namely Re.1, Rs. 2 and Rs. 5. There are 30 coins amounting to Rs. 100 in total. Find the number of coins in each category. Example 1.21:

Solve: 02;023;02 zyxzyxzyx

Example 1.22: Verify whether the given system of equations is consistent. If it is consistent, solve them.

02,9,52752 zyxzyxzyx

Example 1.23 :

Examine the consistency of the equations. 3247192,1333,5732 zyxzyxzyx

Example 1.24: Show that the equations 3074,1432,6 zyxzyxzyx are consistent and solve them.

Example 1.25: Verify whether the given system of equations is consistent. If it is consistent, solve them:

10222,5,5 zyxzyxzyx

Example 1.26: Investigate for what values of , the simultaneous equations zyxzyxzyx 2,1032,6

have (i) no solution (ii) a unique solution and (iii) an infinite number of solutions. Example 1.27: Solve the following homogeneous linear equations. 0,0643,052 zyxzyxzyx

Example 1.28: For what value of the equations 022,034,03 zyxzyxzyx have a

(i) trivial solution, (ii) non-trivial solution. --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 2. VECTOR ALGEBRA (TWO QUESTIONS FOR FULL TEST)

------------------------------------------------------------------------------------------------------------------------------------------------------- EXERCISE 2.2

(4) Prove that BABABA sinsincoscoscos

EXERCISE 2.4

(7) Prove that .sincoscossinsin BABABA

EXERCISE 2.5

(5) If kjckibkjia 3,52,32 Verify that cbabcacba ..

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(12) Verify dcbacdbadcba where kjia ; kib 2 ; kjic 2

; kjid 2

EXERCISE 2.7

(3) Show that the lines 31

1

1

1 zyx

and

1

1

2

1

1

2

zyxintersect and find their point of intersection.

EXERCISE 2.8.

(7) Find the vector and Cartesian equation of the plane containing the line 3

1

3

2

2

2

zyx

and parallel to the line .1

1

2

1

3

1

zyx

(8) Find the vector and Cartesian equation of the plane through the point (1,3,2) and parallel to the lines

3

3

1

2

2

1

zyx and

2

2

2

1

1

2

zyx

(9) Find the vector and Cartesian equation to the plane through the point (-1,3,2) and perpendicular to the

planes 522 zyx and .823 zyx

(10) Find the vector and Cartesian equation of the plane passing through the points A ( 1,-2,3) and B (-1,2,-1)

and is parallel to the line 4

1

3

1

2

2

zyx

(11) Find the vector and Cartesian equation of the plane through the points (1,2,3) and (2,3,1) perpendicular

to the plane .05423 zyx

(12) Find the vector and Cartesian equation of the plane containing the line 2

1

3

2

2

2

zyx and

passing through the point (-1,1,-1). (13) Find the vector and Cartesian equation of the plane passing through points with position vectors

kjikji 22,243 and .7 ki

(14) Derive the equation of the plane in the intercept form.( both in vector and cartesion form ) Example 2.16: Altitudes of a triangle are concurrent – prove by vector method. Example 2.17:

Prove that BABABA sinsincoscoscos

Example 2.29:

Prove that BABABA sincoscossinsin

Example 2.44:

Show that the lines 0

1

1

1

3

1

zyx and

3

1

02

4

zyx intersect and hence find the point of

intersection. Example 2.50: Find the vector and Cartesian equations of the plane through the point (2,-1,-3) and parallel to the lines.

4

3

2

1

3

2

zyx and .

2

2

3

1

2

1

zyx

Example 2.51: Find the vector and Cartesian equations of the plane passing through the points (-1,1,1) and (1,-1,1) and perpendicular to the plane 522 zyx

Example 2.52: Find the vector and Cartesian equations of the plane passing through the points (2,2,-1) , (3,4,2) and (7,0,

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-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 3.COMPLEX NUMBERS


EXERCISE 3.2

(8) (i) If P represents the variable complex number z. Find the locus of P, if 21

12Im

iz

z

(iii) If P represents the variable complex number z. Find the locus of P, if 1

Re 1z

z i

(v) If P represents the variable complex number z. Find the locus of P, if 23

1arg

z

z

EXERCISE 3.4

(5) If and are the roots of the equation 02222 qppxx and

py

q

tan Show that

Nnn

qyy

n

n

nn

;

sin

sin1

(6) If and are the roots of 0422

xx Prove that Nnn

innn

;3

sin21

and deduct

99

(8) If cos21

xx and show that (i) nm

x

y

y

x

m

n

n

m

cos2 (ii) nmix

y

y

x

m

n

n

m

sin2

(10) If 2sin2cos,2sin2cos ibia and 2sin2cos ic Prove that

(i) cos21

abc

abc (ii)

2cos2

222

abc

cba

EXERCISE 3.5.

(1) Find all the values of the following:

(iii) 3

2

3 i

(4) Solve: 01234

xxxx

(5) Find all the values of 4

3

2

3

2

1

i and hence prove that the product of the values is 1.

Example 3.11:

(i) If P represents the variable complex number z, find the locus of P 11

Re

iz

z

(ii) If P represents the variable complex number z, find the locus of P 31

1arg

z

z

Example 3.22:

If and are the roots of 0222

xx and ,1cot y

Show that

Nnnyy

n

nn

;

sin

sin

Example 3.23:

Solve the equation 01459

xxx

Example 3.24:


xxx

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Example 3.25:

Find all the values of 3

2

3 i --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 4. ANALYTICAL GEOMETRY (THREE QUESTIONS FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------- EXERCISE 4.1

(2) Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the following parabolas and hence sketch their graphs.

(iv) 01682

yxy (v) 031262

yxx

(5) A cable of a suspension bridge is in the form of a parabola whose span is 40 mts. The road way is 5 mts below the lowest point of the cable. If an extra support is provided across the cable 30 mts above the ground level find the length of the support if the height of the pillars are 55 mts.

EXERCISE 4.2. (6) Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram:

(ii) 068168422

yxyx

(iv) 92363291622

yxyx

(7) A kho-kho player in a practice session while running realizes that the sum of the distances from the two kho-kho pples from him is always 8m. Find the equation of the path traced by him if the distance between the poles is 6m. (8) A satellite is traveling around the earth in an elliptical orbit having the earth at a focus and of eccentricity 1/2. The shortest distance that the satellite gets to the earth is 400 kms. Find the longest distance that the satellite gets from the earth. (9) The orbit of the planet mercury around the sun is in elliptical shape with sun at a focus. The semi-major axis is of length 36 million miles and the eccentricity of the orbit is 0.206. Find (i) how close the mercury gets to sun? (ii) the greatest possible distance between mercury and sun. (10) The arch of a bridge is in the shape of a semi –ellipse having a horizontal span of 40ft and 16ft high at the centre. How high is the arch, 9ft from the right or left of the centre.

EXERCISE 4.3 (5) Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams.

(iii) 011166422

yxyx

(iv) 01866322

yxyx

EXERCISE 4.4

(5) Prove that the line 9125 yx touches the hyperbola 9922 yx and find its point of contact.

(6) Show that the line 04 yx is a tangent to the ellipse .12322 yx Find the co-ordinates of the point

of contact. EXERCISE 4.5.

(2) Find the equation of the hyperbola if (ii) its asymptotes are parallel to 0122 yx and ,082 yx (2,4) is the centre of the hyperbola and

it passes through (2,0). EXERCISE 4.6.

(3) Find the equation of the rectangular hyperbola which has for one of its asymptotes the line 052 yx

and passes through the points (6,0) and (-3,0). Example 4.7: Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectumfor the following parabolas and hence draw their graphs.

(iv) 09682

yxy

(v) 017822

yxx

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Example 4.8: The girder of a railway bridge is in the parabolic form with span 100ft. and the highest point on the arch is 10ft, above the bridge. Find the height of the bridge at 10ft, to the left or right from the midpoint of the bridge. Example 4.10: On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4mts when it is 6 mts away from the point of projection. Finally it reaches the ground 12 mts away from the starting point. Find the angle of projection. Example 4.12: Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground? Example 4.13: A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the

comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle of 3

radians with the axis of the orbit. Find (i) the equation of the comet‟s orbit (ii) how close does the comet nearer to the sun?( Take the orbit as open rightward ). Example 4.14: A cable of a suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between two towers if 1500ft, the points of support of the cable on the towers are 200ft above the road way and the lowest point on the circle is 70ft above the roadway. Find the vertical distance to the cable from a pole whose height is 122 ft. Example 4.31: Find the eccentricity, centre, foci, vertices of the following ellipses:

(iv) 044327243622

yxyx

Example 4.32: An arch is in the form of a semi-ellipse whose span is 48 feet wide. The height of the arch is 20 feet. How wide is the arch at a height of 10 feet above the base? Example 4.33: The ceiling in a hallway 20ft wide is in the shape of a semi ellipse and 18ft high at the centre. Find the height of the ceiling 4 feet from either wall if the height of the side walls is 12ft. Example 4.35: A ladder of length 15m moves with its ends always touching the vertical wall and the horizontal floor. Determine the equation of the locus of a point P on the ladder, which is 6m from the end of the ladder in contact with the floor. Example 4.56:

Find the eccentricity, centre, foci and vertices of the hyperbola 0199641816922

yxyx and also

trace the curve. Example 4.57: Find the eccentricity, centre, foci, and vertices of the following hyperbola and draw the diagram :

0164323616922

yxyx

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I (TWO QUESTIONS FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 5.6 : A boy, who is standing on a pole of height 14.7m throws a stone vertically upwards. It moves in a vertical line slightly away from the pole and falls on the ground. Its equation of motion in meters and seconds is

2

9.48.9 ttx (i) Find the time taken for upward and downward motions. (ii) Also find the maximum

height reached by the stone from the ground.

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Example 5.7 : A ladder 10m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 m/sec how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6m from the wall? Example 5.8 : A car A is travelling from west at 50 km/hr. and car B is traveling towards north at 60 km/hr. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 kilometers and car B is 0.4 kilometers from the intersection? Example 5.9 : A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. If water is being pumped into the tank at a rate of 2m3 / min , find the rate at which the water level is rising when the water is 3m deep.

EXERCISE 5.1.

(1) A missile fired from ground level rises x metres vertically upwards in t seconds and 2

2

25100 ttx .

Find (i) the initial velocity of the missile, (ii) the time when the height of the missile is a maximum (iii) the maximum height reached and (iv) the velocity with which the missile strikes the ground. (3) The distance x metres traveled by a vehicle in time t seconds after the brakes are applied is given by :

23/520 ttx .Determine (i) the speed of the vehicle (in km/hr) at the instant the brakes are applied and

(ii) the distance the car traveled before it stops. (5) The altitude of a triangle is increasing at a rate of 1 cm / min while the area of the triangle is increasing at a rate of 2 cm2 / min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2. (6) At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km / hr and ship B is sailing north at 25

km / hr. How fast is the distance between the ships changing at 4.00 p.m. (8) Two sides of a triangle have length 12 m and 15 m. The angle between them is increasing at a rate of

2° / min. How fast is the length of third side increasing when the angle between the sides of fixed length is 60° ?

(9) Gravel is being dumped from a conveyor belt at a rate of 30 ft3 / min and its coarsened such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

Example 5.13 :

Find the equations of the tangent and normal at 2

to the curve cos1,sin ayax .

Example 5.14 :

Find the equations of tangent and normal to the curve 1442

92

16 yx at ),(11

yx where 21x and

01y .

Example 5.15 :

Find the equations of the tangent and normal to the ellipse sin,cos byax at the point 4

.

Example 5.17 :

Find the angle between the curves 2xy and 22 xy at the point of intersection.

Example 5.18 :

Find the condition for the curves 12

12

1,122

ybxabyax to intersect orthogonally.

Example 5.20 : Prove that the sum of the intercepts on the co-ordinate axes of any tangent to the curve

2

0,4

sin,4

cos

ayax is equal to a.

EXERCISE 5.2.

(5) Find the equations of those tangents to the circle 5222 yx , which are parallel to the straight line

632 yx .

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(7) Let P be a point on the curve 3xy and suppose that the tangent line at P intersects the curve again at Q.

Prove that the slope at Q is four times the slope at P. (10) Show that the equation of the normal to the curve

33

sin;cos ayax at „ ‟ is 2cossincos ayx .

(11) If the curve xy 2 and kxy are orthogonal then prove that 18

2k

Example 5.34 : Evaluate : x

x

xsin

0

cotlim

Example 5.35 : Evaluate : x

x

xsin

0

lim

EXERCISE 5.6.

(11) x

x

xcos

2

tanlim

Example 5.48 (a) : Find the absolute maximum and absolute minimum values of 20,sin2 xxxxf .

Example 5.51 : Find the local minimum and maximum values of xxxxxf 234

33 .

EXERCISE 5.9 (3) Find the local maximum and minimum values of the following functions:

(iii) 246 xx (iv)

321x (v)

2sin ,0 (vi) tt cos

Example 5.52 : A farmer has 2400 feet of fencing and want to fence of a rectangular field that borders a straight river. He needs no fence along the river. What ar the dimensions of the field that has the largest area? Example 5.53 :

Find a point on the parabola xy 22 that is closest to the point (1,4)

Example 5.54 : Find the area of the largest rectangle that can be inscribed in a semi circle of radius r. Example 5.55 : The top and bottom margins of a poster are each 6 cms and the side margins are each 4cms. If the area of the printed material on the poster is fixed at 384 cms2 , find the dimension of the poster with the smallest area. Example 5.56 : Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius a is

27

8 ( volume of the sphere ).

Example 5.57 : A closed (cuboid) box with a square base is to have a volume of 2000 c.c. The material for the top and bottom of the box is to cost Rs. 3 per square cm. and the material for the sides is to cost Rs. 1.50 per square cm. If the cost of the materials is to be the least, find the dimensions of the box. Example 5.58 : A man is at a point P on a bank of a straight river, 3 km wide, and wants to reach point Q, 8 km downstream on the opposite bank, as quickly as possible. He could row his boat directly across the river to point R and then run to Q, or he could row directly to Q, or he could row to some point to between Q and R and then run to Q. If he can row at 6 km/h and run at 8 km/h where should he land to reach Q as soon as possible?

EXERCISE 5.10. (3) Show that of all the rectangles with a given area the one with smallest perimeter is a square. (4) Show that of all the rectangle with a given perimeter the one with the greatest area is a square. (5) Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r. Example 5.63 :

Discuss the curve 344 xxy with respect to concavity and points of inflection.

Example 5.64 : Find the points of inflection and determine the intervals of convexity and concavity of the Gaussion curve

2

xey

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EXERCISE 5.11. Find the intervals of concavity and the points of inflection of the following functions :

(4) 246 xxxf (5) 2sinf in ,0 (6) 432

212 xxxy

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II


EXERCISE 6.1. (3) Use differentials to find the an approximate value for the given number

(iii) 4302.102.1 y

Example 6.9 : Trace the curve 13 xy

Example 6.10 : Trace the curve 322 xy

EXERCISE 6.2 Trace the following curve :

(1) 3xy

Example 6.18 : If veuw

2 where

y

xu and ,log xyv find

x

w

and

y

w

Example 6.20 : Verify Euler‟s theorem for 22

1,

yx

yxf

Example 6.22 :

Using Euler‟s theorem, prove that uy

uy

x

ux tan

2

1

if

yx

yxu

1sin

EXERCISE 6.3.

(1) Verify xy

u

yx

u

22

for the following functions:

(ii) 22

x

y

y

xu (iii) yxu 4cos3sin (iv)

y

xu

1tan

(5) Using Euler‟s theorem prove the following :

(i) If

yx

yxu

33

1tan Prove that .2sin u

y

uy

x

ux

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 7. INTEGRAL CALCULUS (TWO QUESTIONS FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 7.25 :

Find the area between the curves ,22

xxy x-axis and the lines 2x and 4x

Example 7.26 :

Find the area between the line 1 xy and the curve .12 xy

Example 7.27 :

Find the area bounded by the curve 3xy and the line .xy

Example 7.28 :

Find the area of the region enclosed by xy 2 and 2 xy

Example 7.29 :

Find the area of the region common to the circle 1622 yx and the parabola xy 6

2 .

Example 7.30 :

Compute the area between the curve xy sin and xy cos and the lines 0x and x

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Example 7.31 :

Find the area of the region bounded by the ellipse 12

2

2

2

b

y

a

x

Example 7.32 :

Find the area of the curve 6522

xxy (i) between x = 5 and x = 6 (ii) between x = 6 and x = 7

Example 7.33 :

Find the area of the loop of the curve 223 axxay

Example 7.34 :

Find the area bounded by x-axis and an arch of the cycloid tayttax 2cos1,2sin2 .

EXERCISE 7.4.

(4) Find the area of the region bounded by the curve xxy 2

3 and the x – axis between x = -1 and x = 1.

(7) Find the area of the region bounded by the ellipse 159

22

yx

between the two latus rectums.

(8) Find the area of the region bounded by the parabola xy 42 and the line .42 yx

(9) Find the common area enclosed by the parabolas xy 942 and yx 163

2

(15) Derive the formula for the volume of a right circular cone with radius „ r „ and height „ h „. Example 7. 37 :

Find the length of the curve 324 xy between 0x and 1x

Example 7.38 :

Find the length of the curve 1

3232

a

y

a

x

Example 7.39 :

Show that the surface area of the solid obtained by revolving the arc of the curve xy sin from 0x to

x about x-axis is 21log22

Example 7.40 :

Find the surface area of the solid generated by revolving the cycloid tayttax cos1,sin about its

base ( x- axis ). EXERCISE 7.5.

(1) Find the perimeter of the circle with radius a.

(2) Find the length of the curve tayttax cos1,sin between 0t and .

(3) Find the surface area of the solid generated by revolving the arc of the parabola ,42

axy bounded by its

latus rectum about x - axis. (4) Prove that the curved surface area of a sphere of radius r intercepted between two parallel planes at a

distance a and b from the centre of the sphere is abr 2 and hence deduct the surface area of the

sphere. ab .

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 8. DIFFERENTIAL EQUATIONS (TWO QUESTIONS FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 8.7 :

Solve : 22a

dx

dyyx

Example 8.10 : Find the cubic polynomial in x which attains its maximum value 4 and minimum value 0 at x = -1 and 1 respectively.

EXERCISE 8.2.

(7) Solve the following : 12

dx

dyyx


Example 8.13 : Solve : 02 dxydyxxy

Example 8.14 : Solve : 0332323

dyyxydxxyx

Example 8.15 : Solve : 011 dyyxedxeyxyx given that ,1y where 0x

EXERCISE 8.3. Solve the following :

(5) 0322

dyxydxyx

Example 8.18 : Solve : 22121 xxxy

dx

dyx

Example 8.19 : Solve : dyxydxy 12

tan1

EXERCISE 8.4. Solve the following :

(3) 2

1

21

tan

1 y

y

y

x

dy

dx

(5) 2sin x

x

y

dx

dy

(7) dyyedyxdxy 2

sec

(9) Show that the equation of the curve whose slope at any point is equal to xy 2 and which passes

through the origin is 12 xeyx

EXERCISE 8.5. Solve the following differential equations :

(6) x

eydx

dy

dx

yd 3

2

2

223 when 0,2log yx and 0,0 yx

(10) xexyDD

2296

(11) Solve the differential equation xxyD 2sin22cos12

Example 8.34 : In a certain chemical reaction the rate of conversion of a substance at time t is proportional to the quantity of the substance still untransformed at that instant. At the end of one hour. 60 grams remain and at the end of 4 hours 21 grams. How many grams of the first substance was there initially?

Example 8.35 : A bank pays interest by continuous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year compounded continuously. Calculate the percentage increase in such an account over one year.

[ Take 0833.108

e ].

Example 8.37 : For a postmortem report, a doctor requires to know approximately the time of death of the deceased. He records the first temperature at 10.00 a.m. to be 93.4 ° F. After 2 hours he finds the temperature to be 91.4° F. If the room temperature ( which is constant) is 72° F, estimate the time of death. (Assume normal temperature of a human body to be 98.6° F).

19.4 26.6log 0.0426 2.303 log 0.00945 2.303

21.4 21.4e eand

Example 8.38 : A drug is excreted in a patients urine. The urine is monitored continuously using a catheter. A patient is

administered 10 mg of drug at time t = 0 , which is excreted at a Rate of 213t mg/h.

(i) What is the general equation for the amount of drug in the patient at time t > 0 ? (ii) When will the patient be drug free? Example 8.39 : The number of bacteria in a yeast culture grows at a rate which is proportional to the number present. If the population of a colony of yeast bacteria triples in 1 hour. Show that the number of bacteria at the end

of five hours will be 53 times of the population at initial time.

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EXERCISE 8.6. (1) Radium disappears at a rate proportional to the amount present. If 5% of the original amount disappears in 50 years, how much will remain at the end of 100 years. [ Take A0 as the initial amount ]. (2) The sum of Rs. 1000 is compounded continuously, the nominal rate of interest being four percent per

annum. In how many years will the amount be twice the original principal ? ( 6931.02log e )

(3) A cup of coffee at temperature 100° C is placed in a room whose temperature is 15° C and it cools to 60°C in 5 minutes. Find its temperature after a further interval of 5 minutes. (4) The rate at which the population of a city increases at any time is proportional to the population at that time. If there were 1,30,000 people in the city in 1960 and 1,60,000 in 1990 what population may be

anticipated in 2020? [ 52.1,2070.13

16log

42.

e

e]

(5) A radioactive substance disintegrates at a rate proportional to its mass. When its mass is 10 mgm, the rate of disintegration is 0.051 mgm per day. How long will it take for the mass to be reduced from 10

mgm to 5 mgm. ( 6931.02log e )

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 9. DISCRETE MATHEMATICS

(ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------- Example 9.18 :

Show that *,Z is an infinite abelian group where * is defined as .2* baba

Example 9.21 :

Show the set G of all matrices of the form

xx

xx where ,0 Rx is a group under matrix multiplication.

Example 9.22 :

Show that the set QbabaG ,/2 is an infinite abelian group with respect to addition.

Example 9.23 :

Let G be the set of all rational numbers except 1 and * be defined on G by abbaba * for all

., Gba Show that ( G , * ) is an infinite abelian group.

Example 9.24 :

Prove that the set of four functions 4321 ,,, ffff on the set of non-zero complex numbers

0C defined by 011

,, 4321 Czz

zfandz

zfzzfzzf forms an abelian group with

respect to the composition of functions.

Example 9.25 : Show that nn

Z , forms group.

Example 9.26 :Show that 77

,0 Z forms a group.

Example 9.27 : Show that the nth roots of unity form an abelian group of finite order with usual multiplication.

EXERCISE 9.4. (5) Show that the set G of all positive rational forms a group under the composition * defined by a

3

*ab

ba for all ., Gba

(6) Show that

0

0,

0

0,

01

10,

0

0,

0

0,

10

01

2

22

2

Where 1,1

3 form a

group with respect to matrix multiplication.

(7) Show that the set M of complex numbers z with the condition 1z forms a group with respect to the

operation of multiplication of complex numbers. (8) Show that the set G of all rational numbers except -1 forms an abelian group with respect to the

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operation * given by abbaba * for all ., Gba

(9) Show that the set 9,5,4,3,1 forms an abelian group under multiplication modulo 11.

(11) Show that the set of all matrices of the form 0,00

0

Ra

aforms an abelian group under matrix

multiplication.

(12) Show that the set ZnGn

/2 an abelian group under multiplication.

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 10. PROBABILITY DISTRIBUTIONS (ONE QUESTION FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 10.2 : A random variable X has the following probability mass function

X 0 1 2 3 4 5 6

P ( X = x )

k 3k 5k 7k 9k 11k 13k

(i) Find k.

(ii) Evaluate 635,4 XPandXPXP

(iii) What is the smallest value of x for which 1

2P X x ?

Example 10.3 : An urn contains 4 white and 3 red balls. Find the probability distribution of number of red balls in three draws one by one from the urn. (i)With replacement (ii) without replacement Example 10.10 : The total life time (in year) of 5 year old dog of a certain breed is a Random Variable whose distribution function is given by

5,

251

5,0

2xfor

x

xfor

xF Find the probability that such a five year old dog will live

(i) beyond 10 years (ii) less than 8 years (iii) anywhere between 12 to 15 years.

EXERCISE 10.1 (7) The probability density function of a random variable X is

elsewhere

xexkxf

x

,0

0,,,1

Find (i) k (ii) P ( X > 10 )

Example 10.26 : If the number of incoming buses per minute at a bus terminus is a random variable having a Poisson distribution with ,9.0 find the probability that there will be

(i) Exactly 9 incoming buses during a period of 5 minutes. (ii) Fewer than 10 incoming buses during a period of 8 minutes. (iii) At least 14 incoming buses during a period of 11 minutes.

EXERCISE 10.4.

(5) The number of accidents in a year involving taxi drivers in a city follows a Poisson distribution with mean equal to 3. Out of 1000 taxi drivers find approximately the number of drivers with

(i) no accident in a year (ii) more than 3 accidents in a year .0498.03

e

Example 10.29 : If X is normally distributed with mean 6 and standard deviation 5 find.

(i) 80 XP (ii) 106 XP

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[Area table: P(0<z<1.2) = 0.3849, P(0<z<0.4) = 0.1554, P(0<z<2) = 0.4772 ]. Example 10.30 : The mean score of 1000 students for an examination is 34 and S.D. is 16. (i) How many candidates can be expected to obtain marks between 30 and 60 assuming the normally of the distribution and (ii) determine the limit of the marks of the central 70% of the candidates. [ Area table : P(0<z<0.25) = 0.0987, P(0<z<1.63) = 0.4484 , P(0<z<1.04) = 0.3500] Example 10.31 :

Obtain k , and 2 of the normal distribution whose probability distribution function is given by

.42

2

Xkexfxx

Example 10.32 : The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with mean value 31 psi and standard deviation 0.2 psi. (i) What is the probability that the pressure for a randomly selected tyre (a) between 30.5 and 31.5 psi (b) between 30 and 32 psi (ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5 psi? [ Area table: P(0<z<2.5) = 0.4938, P(0<z<5) = 0.5000 ].

EXERCISE 10.5 (5) The mean weight of 500 male students in a certain college in 151 pounds and the standard deviation is 15 pounds. Assuming the weights are normally distributed, find how many students weigh (i) between 120 and 155 pounds (ii) more than 185 pounds. [ Area table: P(0<z<2.067) = 0.4803 , P(0<z<0.2667) = 0.1026, P(0<z<2.2667) = 0.4881]. (8) Find c , and 2

of the normal distribution whose probability function is given by

2

3.

x xf x ce X

************

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COME BOOK – SIX MARKS [16 Number Of Six Mark Questions To Be Asked For Full Test]

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT:1. MATRICES AND DETERMINANTS

(TWO QUESTIONS FOR FULL TEST) --------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 1.1

(2) Find the adjoint of the matrix

53

21A and verify the result .A A )A adj ( )A adj (A

(4) Find the inverse of each of the following matrices :

(i)

111

112

301

(ii)

121

324

731

(iii)

120

031

221

(iv)

4110

215

318

(v)

221

131

122

(5) If

37

25A and

11

12B verify that (i) 111

ABAB (ii) TTTABAB

(8) Show that the adjoint of

344

101

334

A is A itself.

(10) For

544

434

221

A show that 1 AA .

PROPERTIES: (1) State and prove reversal law for inverses of matrices. [OR]

Prove that (AB)-1 = B-1A-1, where A and B are two non-singular matrices.

EXERCISE 1.2. Solve by matrix inversion method each of the following system of linear equations:

(1) 1123,72 yxyx

(2) 02,137 yxyx

EXERCISE 1.3.

Find the rank of the following matrices:

(1)

432

323

111

(2)

484

121

6126

(3)

0312

0101

0213

(4)

0111

1032

1210

(5)

7363

2142

3121

(6)

6721

3142

4321

EXERCISE 1.4.

(1) Solve the following non-homogeneous system of linear equations by determinant method:

(iii) 18108;954 yxyx

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EXERCISE 1.5. (1) Examine the consistency of the following system of equations. If it is consistent then solve the same.

(iii) 62;1832;7 zyzyxzyx

(iv) 52687;13283;1474 zyxzyxzyx

Example 1.2: Find the adjoint of the matrix

312

321

111

A

Example 1.3: If ,41

21

A verify the result 2IAAAadjAadjA

Example 1.5:

(iv) Find the inverse of the following matrix:

121

022

113

A

Example 1.6:

If

11

21A and

21

10B verify that 111

ABAB .

Example 1.7: Solve by matrix inversion method 832,3 yxyx

Example 1.12:

Find the rank of the matrix

11715

4312

3111

Example 1.13:


323

432

111

Example 1.14:


3963

2642

1321

Example 1.15:


1012

7436

3124

Example 1.16:


2751

5121

1513

Example 1.17: Solve the following system of linear equations by determinant method.

(2) 1664;832 yxyx

Example 1.18: Solve the following non-homogeneous equations of three unknowns.

(3) 423;1;522 zyxzyxzyx

(5) 10633;8422;42 zyxzyxzyx

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-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 2. VECTOR ALGEBRA


EXERCISE 2.2 Prove by vector method: (1) If the diagonals of a parallelogram are equal then it is a rectangle. (2) The mid point of the hypotenuse of a right angled triangle is equidistant from its vertices. (3) The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of the sides.

(8) Forces of magnitudes 3 and 4 units acting in the directions kji 326 and kji 623

respectively act on a particle which is displaced from the point (2,2-1) to (4,3,1). Find the work done by the forces.

EXERCISE 2.3

(9) Let cba ,, be unit vectors such that 0.. caba and the angle between b and c is 6

. Prove

that cba 2

EXERCISE 2.4. (5) Prove by vector method that the parallelogram on the same base and between the same parallels are equal in area. (6) Prove that twice the area of a parallelogram is equal to the area of another parallelogram formed by taking as its adjacent sides the diagonals of the former parallelogram.

(8) Forces kjikjiji 2,652,72 act at a point P whose position vector is kji 234 .

Find the moment of the resultant of three forces acting at P about the point Q whose position vector is

kji 36 .

(10) Find the magnitude and direction cosines of the moment about the point (1,-2,3) of a force

kji 632 whose line of action passes through the origin.

EXERCISE 2.5. (4) Show that the points (1,3,1) , (1,1,-1) , (-1,1,1) (2,2,-1) are lying on the same plane. ( Hint : It is enough to

prove any three vectors formed by these four points are coplanar).

(7) If kjibkjia 2,532 and kjic 324 , show that cbacba

(8) Prove that cbacba iff a and c are collinear.(vector triple products is non-zero )

(10) Prove that 0... dbacdacbdcba

(11) Find dcba . if kjidkjickibkjia 2,2,2,

EXERCISE 2.6.

(4) A vector r has length 235 and direction ratios (3,4,5) , find the direction cosines and components of r .

(6) Find the vector and Cartesian equation of the line through the point (3, -4, -2) and parallel to the vector

.269 kji

(7) Find the vector and Cartesian equation of the line joining the points (1,-2,1) and (0,-2,3) EXERCISE 2.7.

(1) Find the shortest distance between the parallel lines

(i) kjitkjir 322 and kjiskjir 322

(ii) 2

3

31

1

zyx and

2

1

3

1

1

3

zyx

(2) Show that the following two lines are skew lines:

kjitkjir 2753 and kjiskjir 767

(4) Find the shortest distance between the skew lines 1

4

1

7

3

6

zyx and

4

2

2

9

3

zyx

(5) Show that (2,-1,3) , (1,-1,0) and (3,-1,6) are collinear.

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EXERCISE 2.8. (1) Find the vector and Cartesian equations of a plane which is at a distance of 18 units from the origin and

which is normal to the vector kji 872

(4) The foot of the perpendicular drawn from the origin to a plane is (8,-4,3). Find the equation of the plane.

(5) Find the equation of the plane through the point whose p.v. is kji 2 and perpendicular to the vector

.324 kji

(6) Find the vector and Cartesian equations of the plane through the point (2,-1,4) and parallel to the plane

.73124 kjir

(15) Find the Cartesian form of the following planes:

(i) ktsjtitsr 232 (ii) ktsjtsitsr 22321

EXERCISE 2.9.

(1) Find the equation of the plane which contains the two lines 4

3

3

2

2

1

zyx and 8

2

1

3

4

z

yx

(2) Can you draw a plane through the given two lines? Justify your answer.

kjitkjir 63242 and kjiskjir 832533

(3) Find the point of intersection of the line

kjiskjr 2 and xz – plane.

(4) Find the meeting point of the line kjitkjir 232 and the plane 0732 zyx

EXERCISE 2.11

(1) Find the vector equation of a sphere with centre having position vector kji 32 and radius 4 units.

Also find the equation in Cartesian form. (2) Find the vector and Cartesian equation of the sphere on the join of the points A and B having position vectors

kji 762 and kji 342 respectively as a diameter. Find also the centre and radius of the

sphere. (3) Obtain the vector and Cartesian equation of the sphere whose centre is (1, -1, 1) and radius is the same as

that of the sphere .52 kjir

(6) Show that diameter of a sphere subtends a right angle at a point on the surface by vector method. Example 2.12:

With usual notations in triangle ABC prove that bc

acbA

2cos

222

Example 2.13:

With usual notations prove (i) BcCba coscos

Example 2.14: Angle in a semi-circle is a right angle. Prove by vector method. Example 2.15: Diagonals of a rhombus are at right angles. Prove by vector methods. Example 2.24:

If kjip 743 and kjiq 326 then find .qp Verify that p and qp are perpendicular to

each other and also verify that q and qp are perpendicular to each other.

Example 2.25:

If the position vectors of three points A, B and C are respectively kjikji 54,32 and ki 7 .

Find .ACAB Interpret the result geometrically.

Example 2.26:

Prove that the area of a quadrilateral BDAC 2

1 where AC and BD are its diagonals.

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Example 2.27:

If cba ,, are the position vectors of the vertices A, B, C of a triangle ABC, then prove that the area of

triangle ABC is accbba 2

1 Deduce the condition for the points cba ,, to be collinear.

Example 2.28:

Prove that in triangle ABC with usual notations, C

c

B

b

A

a

sinsinsin by vector method.

Example 2.33:

For any three vectors cba ,, prove that cbaaccbba 2,,

Example 2.36: If kjickjibkjia 25,635,423 ,find (i) cba (ii) cba and

show that they are not equal. Example 2.37:

Let cba ,, and d be any four vectors then

(i) dcbacdbadcba

(ii) adcbbdcadcba

Example 2.38:

Prove that 2,,,, cbaaccbba

Example 2.39: Find the vector and Cartesian equations of the straight line passing through the point A with position

vector kji 43 and parallel to the vector kji 375

Example 2.40: Find the vector and Cartesian equations of the straight line passing through the points (-5,2,3) and (4,-3,6) Example 2.42: Find the shortest distance between the parallel lines

kjitjir 2 and kjiskjir 22

Example 2.43:

Show that the two lines kitjir 2 and kjisjir 2 are skew lines and find the

distance between them. Example 2.45: Find the shortest distance between the skew lines

kjijir 2 and kjikjir 2

Example 2.46: Show that the points (3,-1,-1) , (1,0,-1) and (5,-2,-1) are collinear. Example 2.48:

Find the vector and Cartesian equation of a plane which is at a distance of 8 units from the origin and which

is normal to the vector kji 223

Example 2.49: The foot of perpendicular drawn from the origin to the plane is (4,-2,-5) , find the equation of the plane. Example 2.53:

Find the equation of the plane passing through the line of intersection of the plane 1432 zyx and

4 yx and passing through the point (1,1,1)

Example 2.54: Find the equation of the plane passing through the intersection of the planes 3482 zyx and

010453 zyx and perpendicular to the plane 0423 zyx

Example 2.55:

Find the distance from the point (1,-1,2) to the plane kjtjiskjir


Example 2.56:

Find the distance between the parallel planes 3 kjir and 5 kjir

Example 2.57:

Find the equation of the plane which contains the two lines 4

3

3

2

2

1

zyx and

12

1

5

4 zyx

Example 2.58: Find the point of intersection of the line passing through the two points (1,1,-1) ; (-1,0,1) and the xy -plane.

Example 2.59:

Find the co-ordinates of the point where the line kjitkjir 43252 meets the plane

342 kjir

Example 2.62:

Find the vector and Cartesian equations of the sphere whose centre is kji 22 and radius is 3.

Example 2.63: Find the vector and Cartesian equation of the sphere whose centre is (1,2,3) and which passes through the point (5,5,3)

Example 2.64:

Find the equation of the sphere on the join of the points A and B having position vectors kji 762

and kji 342 respectively as a diameter.

Example 2.65: Find the coordinates of the centre and the radius of the sphere whose vector equation is

05010682

kjirr



EXERCISE 3.1.

(1) Express the following in the standard form iba

(ii)

i

ii

31

211

(iv)

15108

1694

23 iii

iii

(4) Find the real values of x and y for which the following equations are satisfied.

(i) iyixi 3111 (ii)

ii

iyi

i

ixi

3

32

3

21 (iii) iyixxx 2483

2

(5) For what values of x and y, the numbers yix2

3 and iyx 42

are complex conjugate of each other?

EXERCISE 3.2.

(2) Find the square root of i68

(4) Prove that the triangle formed by the points representing the complex numbers ii 42,810 and

i3111 on the Argand plane is right angled.

(5) Prove that the points representing the complex numbers iii 74,25,57 and i42 form a

parallelogram.( Plot the points and use midpoint formula).

(7) If 6

1arg

z and 3

21arg

z then prove that 1z

(8) P represents the variable complex number z. Find the locus of P, if (ii) iziz 55

(iv) 232 z .

EXERCISE 3.3.

(1) Solve the equation 02032248234

xxxx if i3 is a root.

(2) Solve the equation 01014114234

xxxx if one root is i21

(3) Solve : 010332256234

xxxx given that one of the root is i2

EXERCISE 3.4.

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(2) Simplify:

4

3

cossin

sincos

i

i

(3) If sinsinsin0coscoscos ,prove that

(i) cos33cos3cos3cos

(ii) sin33sin3sin3sin

(iii) 02cos2cos2cos

(iv) 02sin2sin2sin

(v) 2

3sinsinsincoscoscos

222222

(4) Prove that

(i) 4

cos211 2

2

nii

n

nn

(ii) 3

cos231311 n

iin

nn

(iii) 2

cos2cos2sincos1sincos11

n

iinnnn

(iv) ni

41 and 24

1

n

i are real and purely imaginary respectively

(7) If cos21

xx prove that (i) n

xx

n

ncos2

1 (ii) ni

xx

n

nsin2

1

(9) If sincos;sincos iyix prove that nmyx

yxnm

nm cos2

1

EXERCISE 3.5. (1) Find all the values of the following:

(ii) 318i

(2) If bazbaybax 22

,, show that

(i) 33baxyz

(ii) 333333 bazyx where is the complex cube root of unity.

(3) Prove that if 13 , then

(i) abccbacbacbacba 333322

(iii) 02

1

1

1

21

1

(4) Solve: (i) 044

x

Example 3.9: Find the modulus and argument of the following complex numbers:

(i) 22 i (ii) 31 i (iii) 31 i

Example 3.10:

If ,...2211 iBAibaibaiba nn prove that

(i) 22222

2

2

2

2

1

2

1 ... BAbababa nn

(ii) ZkA

Bk

a

b

a

b

a

b

n

n

,tantan....tantan11

2

21

1

11

Example 3.13:

Prove that the complex numbers iii 3333,33,33 are the vertices of an equilateral triangle in the

complex plane. Example 3.14:

Prove that the points representing the complex numbers iii 44,1,2 and i53 on the Argand plane are the

vertices of a rectangle.

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Example 3.15: Show that the points representing the complex numbers iii 33,73,97 form a right angled triangle on

the Argand diagram. Example 3.16:

Find the square root of i247

Example 3.17:


xxx ,if one of its roots is i32

Example 3.19:

Simplify:

5

4

cossin

sincos

i

i

Example 3.20:

If n is a positive integer, prove that

2sin

2cos

cossin1

cossin1nin

i

in

Example 3.21:

If n is a positive integer, prove that 6

cos2331 n

iin

nn

PROPERTIES: (1) State and prove the triangle inequality of complex numbers.

(2) For any two complex numbers ,, 21 ZZ show that

(i) 2121 ZZZZ (ii) 2121 argargarg ZZZZ

(3) For any two complex numbers ,, 21 ZZ show that

(i) 2

1

2

1

Z

Z

Z

Z (ii) 21

2

1argargarg ZZ

Z

Z

(4) Show that for any polynomial equation 0xP with real coefficients imaginary roots occur in conjugate

pairs. --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 4. ANALYTICAL GEOMETRY (ONE QUESTION FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------- EXERCISE 4.1.

(2) Find the axis, vertex ,focus equation of directrix, latus rectum, length of the latus rectum fro the following parabolas and hence sketch their graphs.

(iii) 2442

yx

(3) If a parabolic reflector is 20cm in diameter and 5cm deep, find the distance of the focus from the centre of the reflector.

(4) The focus of a parabolic mirror is at a distance of 8cm from its centre (vertex). If the mirror is 25cm deep, find the diameter of the mirror.

EXERCISE 4.2. (1) Find the equation of the ellipse if (ii) the foci are (2,-1) , (0,-1) and e = 1 / 2. (iii) the foci are 0,3 and the vertices are 0,5

(iv) the centre is (3,-4) , one of the foci is 4,33 and 2

3e

(v) the centre at the origin , the major axis is along x axis , e = 2 / 3 and passes through the point

3

5,2

(vi) the length of the semi major axis, and the latus rectum are 7 and 80 / 7 respectively, the centre is (2,5) and the major axis is parallel to y-axis.

(vii) the centre is (3,-1) , one of the foci is (6,-1) and passing through the point (8,-1).

(viii) the foci are 0,3 , and the length of the latus rectum is 32 / 5 .

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(ix) the vertices are 0,4 and 2

3e

(3) Find the locus of a point which moves so that the sum of its distances from (3,0) and (-3,0) is 9. (4) Find the equations and length of major and minor axes of

(ii) 0436109522

yxyx (iv) 092363291622

yxyx

(5) Find the equations of directrices , latus rectum and length of latus rectums of the following ellipses:

(iii) 068168422

yxyx (iv) 0234302322

yxyx

EXERCISE 4.3.

(1) Find the equatin of the hyperbola if (i) focus : (2,3) ; corresponding directrix : 2,52 eyx

(ii) Centre : (0,0) ; length of the semi-transverse axis is 5 ; e = 7 / 5 the conjugate axis is along x-axis. (iii) centre : (0,0) length of the semi – transverse axis is 6 ; e = 3, transverse axis is parallel to y-axis. (iv) centre : (1,-2) ; length of the transverse axis is 8 ; e = 5 / 4 transverse axis is parallel to x-axis. (v) centre ; (2,5) ; the distance between the directrices is 15, the distance between the foci is 20 and the

transverse axis is parallel to y – axis.

(vi) foci : 8,0 ; length of transverse axis is 12

(vii) foci : 5,3 ; e = 3

(viii) centre : (1,4) ; one of the foci (6,4) and the corresponding directrix is x = 9 / 4 . (ix) foci : (6,-1) and (-4,-1) and passing through the point (4,-1) (2) Find the equation and length of transverse and conjugate axes of the following hyperbolas:

(iii) 036369691622

yxyx

(3) Find the equations of directrices, latus rectums and length of latus rectum of the following hyperbolas.

(ii) 0832364922

yxyx

(4) Show that the locus of a point which moves so that the difference of its distances from the points (5,0) and (-

5,0) is 8 is .14416922 yx

EXERCISE 4.4. (1) Find the equations of the tangent and normal to the parabolas.

(i) xy 122 at 6,3 (ii) yx 9

2 at 1,3

(iii) 04422

yxx at 1,0 (iv) to the ellipse 63222 yx at 0,3

(v) to the hyperbola 315922 yx at 1,2

(2) Find the equations of the tangent and normal

(i) to the parabola xy 82 at

2

1t

(ii) to the ellipse 32422 yx at

4

(iii) to the ellipse 400251622 yx at

3

1t

(iv) to the hyperbola 1129

22

yx

at 6

(3) Find the equations of the tangents

(i) to the parabola xy 62 , parallel to 0523 yx

(ii) to the parabola xy 162 , perpendicular to the line 083 yx

(iii) to the ellipse 1520

22

yx

, which are perpendicular to 02 yx

(iv) to the hyperbola 64422 yx , which are parallel to 09310 yx

(4) Find the equations of the two tangents that can be drawn

(i) from the point (2,-3) to the parabola xy 42

(ii) from the point (1,3) to the ellipse 369422 yx

(iii) from the point (1,2) to the hyperbola .63222 yx

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EXERCISE 4.5. (1) Find the equation of the asymptotes to the hyperbola

(ii) 0242310822

yxyxyx

(2) Find the equation of the hyperbola if (i) the asymptotes are 0832 yx and 0123 yx and (5,3) is a point on the hyperbola.

(3) Find the angle between the asymptotes of the hyperbola

(iii) 03110165422

yxyx

EXERCISE 4.6.

(1) Find the equation of the rectangular hyperbola whose centre is

2

1,

2

1 and which passes through the

point .4

1,1

(2) Find the equation of the tangent and normal (i) at (3,4) to the rectangular hyperbolas 12xy (ii) at

4

1,2

to the rectangular hyperbola 01822 yxxy

(4) A standard rectangular hyperbola has its vertices at (5,7) and (-3,-1). Find its equation and asymptotes. (5) Find the equation of the rectangular hyperbola which has its centre at (2,1) one of its asymptotes

053 yx and which passes through the point (1,-1).

(6) Find the equations of the asymptotes of the following rectangular hyperbolas.

(ii) 01432 yxxy (iii) 0351265622

yxyxyx

(7) Prove that the tangent at any point to the rectangular hyperbola forms with the asymptotes a triangle of constant area.

Example 4.7 :

Find the axis, vertex, focus, directrix, equation of the latus rectum, length of the latus rectum for the following parabolas and hence draw their graphs.

(iii) 1822

xy

Example 4.9: The headlight of a motor vehicle is a parabolic reflector of diameter 12cm and depth 4cm. Find the position of bulb on the axis of the reflector for effective functioning of the headlight.

Example 4.11: A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 9mts. If the distance across (diameter) the top of the mirror is 160cm, how deep is the mirror at the middle?

Example 4.15: Find the equation of the ellipse whose foci are (1,0) and (-1,0) and eccentricity is 1 / 2. Example 4.16: Find the equation of the ellipse whose one of the foci is (2,0) and the corresponding directrix is x=8 and eccentricity is 1 / 2. Example 4.17: Find the equation of the ellipse with focus (-1,-3), directrix 02 yx and eccentricity 4 / 5.

Example 4.18:

Find the equation of the ellipse with foci 0,4 and vertices 0,5

Example 4.20: Find the equation of the ellipse whose centre is (1,2) , one of the foci is (1,3) and eccentricity is 1 / 2. Example 4.21:

Find the equation of the ellipse whose major axis is along x-axis, centre at the origin,passes through the point (2,1) and eccentricity 1 / 2.

Example 4.22: Find the equation of the ellipse if the major axis is parallel to y-axis, semi – major axis is 12, length of the latus rectum is 6 and the centre is (1,12).

Example 4.23: Find the equation of the ellipse given that the centre is (4,-1) , focus is (1,-1) and passing through (8,0)

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Example 4.24: Find the equation of the ellipse whose foci are (2,1) (-2,1) and length of the latus rectum is 6. Example 4.25: Find the equation of the ellipse whose vertices are (-1,4) and (-7,4) and eccentricity is 1 / 3. Example 4.26: Find the equation of the ellipse whose foci are (1,3) and (1,9) and eccentricity is 1 / 2. Example 4.27: Find the equation of a point which moves so that the sum of its distances from ( -4,0 ) and (4,0) is 10. Example 4.28: Find the equations and lengths of major and minor axes of

(iii)

116

1

9

122

yx

Example 4.29:

Find the equations of axes and length of axes of the ellipse 01236129622

yxyx

Example 4.30: Find the equations of directrices, latus rectum and length of latus rectum of the following ellipses.

(iii) 041283422

yxyx

Example 4.31: Find the eccentricity, centre, foci, vertices of the following ellipse :

(iii)

14

5

6

322

yx

Example 4.34: The orbit of the earth around the sun is elliptical in shape with sun at a focus. The semi major axis is of length 92.9 million miles and eccentricity is 0.017. Find how close the earth gets to sun and the greatest possible distance between the earth and the sun.

Example 4.36:

Find the equation of hyperbola whose directrix is 12 yx , focus (1,2) and eccentricity 3 .

Example 4.37: Find the equation of the hyperbola whose transverse axis is along x-axis. The centre is (0,0) length of semi-transverse axis is 6 and eccentricity is 3.

Example 4.38: Find the equation of the hyperbola whose transverse axis is parallel to x-axis, centre is (1,2) , length of the conjugate axis is 4 and eccentricity e = 2.

Example 4.39: Find the equation of the hyperbola whose centre is (1,2). The distance between the directrices is 20 / 3, the distance between the foci is 30 and the transverse axis is parallel to y – axis.

Example 4.41:

Find the equation of the hyperbola whose foci are 0,6 and length of the transverse axis is 8.

Example 4.42:

Find the equation of the hyperbola whose foci are 4,5 and eccentricity is 3 / 2.

Example 4.43: Find the equation of the hyperbola whose centre is (2,1) , one of the foci are (8,1) and the corresponding directrix is x =4.

Example 4.44:

Find the equation of the hyperbola whose foci are 5,0 and the length of the transverse axis is 6.

Example 4.45:

Find the equation of the hyperbola whose foci are 10,0 and passing through (2,3).

Example 4.48:

Find the equations and length of transverse and conjugate axes of the hyperbola 05616436922

yyxx .

Example 4.51: Find the equations of directrices, latus rectum and length of latus rectum of the hyperbola

05616436922

yyxx

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Example 4.52:

The foci of a hyperbola coincide with the foci of the ellipse .1925

22

yx

Determine the equation of the

hyperbola if its eccentricity is 2. Example 4.53:

Find the equation of the locus of all points such that the differences of their distances from (4,0) and (-4,0) is always equal to 2.

Example 4.54:

Find the eccentricity, centre, foci and vertices of the hyperbola 154

22

yx

and also trace the curve.

Example 4.55:

Find the eccentricity , centre , foci and vertices of the hyperbola 1186

22

xy

and also trace the curve

Example 4.58: Points A and B are 10km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.

Example 4.59:

Find the equations of the tangents to the parabola xy 52 from the point (5,13). Also find the points of

contact. Example 4.61:

Find the equation of the tangent and normal to the parabola 0222

yxx at (1,2).

Example 4.62:

Find the equations of the two tangents that can be drawn from the point (5,2) to the ellipse 147222 yx

Example 4.64:

Find the separate equations of the asymptotes of the hyperbola 0141725322

yxyxyx

Example 4.65: Find the equation of the hyperbola which passes through the point (2,3) and has the asymptotes

0734 yx and .12 yx

Example 4.66:

Find the angle between the asymptotes of the hyperbola 09612322

yxyx

Example 4.67:

Find the angle between the asymptotes to the hyperbola 0141725322

yxyxyx

Example 4.68:

Prove that the product of perpendiculars from any point on the hyperbola 12

2

2

2

b

y

a

x to its asymptotes is

constant and the value is 22

22

ba

ba

Example 4.69:

Find the equation of the standard rectangular hyperbola whose centre is

2

3,2 and which passes

through the point

3

2,1

Example 4.70:

The tangent at any point of the rectangular hyperbola 2cxy makes intercepts a, b and the normal at the

point makes intercepts p,q on the axes. Prove that 0 bqap

Example 4.71: Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact.

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-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 5.DIFFERENTIAL CALCULUS-APPLICATIONS-I


EXERCISE 5.1.

(2) A Particle of unit mass moves so that displacement after t secs is given by .42cos3 tx Find the

acceleration and kinetic energy at the end of 2 secs.

massismvmEK ,

2

1..

2

(4) Newton‟s law of cooling is given by kte

0 , where the excess of temperature at zero time is C

0

and at time t seconds is C

. Determine the rate of change of temperature after 40 s, given that

C160 and .03.0k 3201.32.1e

(7) Two sides of a triangle are 4m and 5m in length and the angle between them is increasing at a rate of 0.06 rad / sec. Find the rate at which the area of the triangle is increasing when the angle between the sides of

fixed length is .3/

Example 5.10 :

Find the equations of the tangents and normal to the curve 3xy at the point ( 1,1 ).

Example 5.11 :

Find the equations of the tangent and normal to the curve 22

xxy at the point ( 1, -2 ).

Example 5.12 :

Find the equation of the tangent at the point (a,b) to the curve 2cxy .

Example 5.16 :

Find the equation of the tangent to the parabola, xy 202 which forms an angle 45 with the x – axis.

Example 5.19 :

Show that 222ayx and 2

cxy cut orthogonally.

EXERCISE 5.2. (1) Find the equation of the tangent and normal to the curves

(i) 542

xxy at 2x (ii) ,cossin xxxy at 2

x

(iii) xy 3sin22

at 6

x (iv)

x

xy

cos

sin1 at

4

x

(2) Find the points on curve 222 yx at which the slope of the tangent is 2.

(3) Find at what points on the circle 1322 yx , the tangent is parallel to the line 732 yx

(4) At what points on the curve 014222

yxyx the tangent is parallel to (i) x – axis (ii) y – axis.

(6) Find the equation of a normal to xxy 33 that is parallel to .09182 yx

(8) Prove that the curve 14222 yx and 1126

22 yx cut each other at right angles.

(9) At what angle do the curves xay and x

by intersect ba ?

Example 5.22 : Verify Rolle‟s theorem for the following :

(v) xxexfx

0,sin

Example 5.23 : Apply Rolle‟s theorem to find points on curve ,cos1 xy where the tangent is parallel to x – axis in

2,0 .

EXERCISE 5.3.

(2) Using Rolle‟s theorem find the points on the curve 22,12

xxy where the tangent is parallel to x –

axis. Example 5.24 :

Verify Lagrange‟s law of the mean for 3xxf on 2,2

Example 5.25 : A cylindrical hole 4mm in diameter and 12 mm deep in a metal block is rebored to increase the diameter to 4.12 mm. Estimate the amount of metal removed.

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Example 5.26 :

Suppose that 30 f and 5'

xf for all values of x, how large can 2f possibly be?

Example 5.27 : It took 14 sec for a thermometer to rise from -19° C to 100°C when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at exactly 8.5 ° C / sec.

EXERCISE 5.4 (1) Verify Lagrange‟s law of mean for the following functions:

(i) 3,0,12

xxf

(ii) 2,1,1

xxf

(iii) 2,0,1223

xxxxf

(iv) 2,2,32

xxf

(v) 3,1,3523

xxxxf

(2) If 101 f and 2'

xf for 41 x how small can 4f possibly be ?

(3) At 2.00 p.m. a car‟s speedometer reads 30 miles / hr., at 2.10 pm it reads 50 miles / hr. Show that sometime between 2.00 and 2.10 the acceleration is exactly 120 miles / hr2.

Example 5.28 : Obtain the Maclaurin‟s Series for

(2) xe

1log

(3) arc xorx1

tantan

EXERCISE 5.5. (1) Obtain the Maclaurin‟s Series expansion for :

(ii) x2

cos (iii)

(iv)22

,tan

xx

Example 5.30 :

Find

x

x

x 1tan

1sin

lim1

if exists.

Example 5.31 :

2

22

sinloglim

x

x

x

Example 5.33 : Evaluate :

xxec

x

1coslim

0

Example 5.36 :

The current at time t in a coil with resistance R, inductance L and subjected to a constant electromotive force

E is given by

L

RT

eR

Ei 1 Obtain a suitable formula to be used when R is very small.

EXERCISE 5.6. Evaluate the limit for the following if exists .

(2) xx

xx

x sin

tanlim

0

(6)

x

xx

x 1

1tan2

1

lim

1

2

x1

1

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(8) x

x

x 2cot

cotlim

0

(9) .loglim2

0

xx ex

(10) 1

1

1

lim

x

x

x

(12) x

x

xlim0

(13) x

x

x1

0

coslim

Example 5.37:

Prove that the function xxxf 2cossin is not monotonic on the interval

4,0

Example 5.38:

Find the intervals in which xxxxf 20223 is increasing and decreasing.

Example 5.39 :

Prove that the function 12

xxxf is neither increasing nor decreasing in 1,0

Example 5.40 :

Discuss monotonicity of the function 2,0,sin xxxf

Example 5.41:

Determine for which values of x, the function 1,1

2

x

x

xy is strictly increasing or strictly decreasing.

Example 5.42 :

Determine for which values of x, the function 13615223

xxxxf is increasing and for which it is

decreasing. Also determine the points where the tangents to the graph of the function are parallel to the x axis.

Example 5.43:Show that 0,cossintan1

xxxxf is a strictly increasing function in the interval

4,0

EXERCISE 5.7 (3) Which of the following functions increasing or decreasing on the interval given ?

(iv) 1,211 onxxx (v)

4,0sin

onxx

(4) Prove that the following functions are not monotonic in the intervals given.

(i) 522

xx on 0,1 (ii) 2,011 onxxx

(iii) ,0sin onxx (iv)

2,0cottan

onxx

(5) Find the intervals on which f is increasing or decreasing.

(i) 220 xxxf (ii) 13

3 xxxf

(iii) 13

xxxf (iv) 2,0,sin2 xxxf

(v) xxxf cos in ,0 (vi) xxxf44

cossin in

2,0

Example 5.45 : Prove that the inequality nxxn

11 is true whenever 0x and .1n

Example 5.46: Prove that

2,0,tansin

xxxx

EXERCISE 5.8. (1) Prove the following inequalities :

(i) 0,2

1cos

2

xx

x

(ii) 0,

6sin

3

xx

xx

(iii) xx 1

tan for all 0x (iv) xx 1log for all 0x .

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Example 5.48 :

Find the absolute maximum and minimum values of the function. 42

1,13

23 xxxxf

Example 5.49 :

Discuss the curve 344 xxy with respect to local extrema.

Example 5.50 :

Locate the extreme point on the curve xxy 632 and determine its nature by examine the sign of the

gradient on either side. EXERCISE 5.9.

(1) Find the critical numbers and stationary points of each of the following functions .

(iii) 2544 xxxf (iv)

1

1

2

xx

xxf

(vi) 2sin2

f in ,0 (vii) sinf in 2,0

(2) Find the absolute maximum and absolute minimum values of f on the given interval:

(i) 3,0,222

xxxf (ii) 1,4,212

xxxf

(iii) 5,3,1123

xxxf (iv) 2,1,92

xxf

(v) 2,1,1

x

xxf

(vi)

3,0,cossin

xxxf

(vii) ,,cos2 xxxf

(3) Find the local maximum and minimum values of the following functions:

(i) xx 3 (ii) xxx 452

23

EXERCISE 5.10. (1) Find the numbers whose sum is 100 and whose product is a maximum. (2) Find two positive numbers whose product is 100 and whose sum is minimum.

(6) Resistance to motion F, of a moving vehicle is given by .1005

xx

F Determine the minimum value of

resistance. Example 5.62 :

Determine where the curve 133

xxy is can cave upward and where it is concave downward. Also find the

inflection points. Example 5.65 :

Determine the points of inflection if any, of the function 233

xxy

Example 5.66 :

Test for points of inflection of the curve 2,0,sin xxy

EXERCISE 5.11. Find the intervals of concavity and the points of inflection of the following functions:

(1) 311 xxf (3) xxxxf 452

23


(ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------- Example 6.2 :

Compute the values of y and dy if 1223

xxxxfy where x changes (i) from 2 to 2.05 and (ii) from

2 to 2.01. Example 6.3 :

Use differentials to find an approximate value for .653

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Example 6.5 :

The time of swing T of a pendulum is given by kT where k is a constant. Determine the percentage

error in the time of swing if the length of the pendulum / changes from 32.1 cm to 32.0 cm. Example 6.6 :

A circular template has a radius of 10 cm 02.0 . Determine the possible error in calculating the area of the

templates. Find also the percentage error. Example 6.7.

Show that the percentage error in the nth root of a number is approximately n

1 times the percentage in the

number. EXERCISE 6.1.

(3) Use differentials to find an approximate value for the given number

(i) 1.36 (ii) 1.10

1 (iv) 6

97.1

(4) The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error in computing (i) the volume of the cube and (ii) the surface area of cube.

(5) The radius of a circular disc is given as 24cm with a maximum error in measurement of 0.02cm. (i) Use differentials to estimate the maximum error in the calculated area of the disc. (ii) Compute the relative error? Example 6.15:

If zyxu tantantanlog , prove that 22sin

x

ux

Example 6.16:

If xzzyyxU then show that 0 zyx UUU

Example 6.17:

Suppose that 2

xyez where tx 2 and ty 1 then find

dt

dz

Example 6.19:

If 22 zyxw and .;sin;cos tztytx Find

dt

dw.

Example 6.21:

If u is a homogenous function of x and y of degree n , prove that y

un

y

uy

yx

ux

1

2

22

EXERCISE 6.3.

(1) Verify xy

u

yx

u

22

for the following function:

(i) 223 yxyxu

(3) Using chain rule find dt

dw for each of the following:

(i) xyew where 32

, tytx

(ii) 22log yxw where tt

eyex

,

(iii) 22

yx

xw

where tytx sin,cos

(iv) zxyw where tztytx ,sin,cos

(4) (i) Find r

w

and

w if 22

log yxw where sin,cos ryrx

(ii) Find u

w

and

v

w

if 22

yxw where uvyvux 2,22

(iii) Find u

w

and

v

w

if xyw

1sin

where ., vuyvux

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(5) Using Euler‟s theorem prove the following:

(ii)

y

xxyu sin

2 show that .3uy

uy

x

ux

(iii) If u is a homogeneous function of x and y of degree n, prove that x

un

yx

uy

x

ux

1

2

2

2

(iv) If byaxzeV

and z is a homogenous function of degree n in x and y prove that

.Vnbyaxy

Vy

x

Vx

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 7. INTEGRAL CALCULUS

(ONE QUESTIONS FOR FULL TEST) --------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 7.1. Evaluate the following problems using second fundamental theorem:

(3) 1

0

249 dxx (4) dxxx 2sinsin2

4

0

2

(6)

2

02

cos9

sin

x

dxx

(7)

2

12

65 xx

dx (8)

dx

x

x

1

0 2

31

1

sin (9)

2

0

cos2sin

dxxx (10) dxexx

1

0

2

Example 7.7 : Evaluate :

2

2

sin

dxxx

Example 7.8 : Evaluate

2

2

2sin

dxx

Example 7.9: Evaluate

dx

xfxf

xf

2

0 cossin

sin

Example 7.10: Evaluate dxxx

n

1

0

1

Example 7.11: Evaluate dxx

2

0

tanlog

Example 7.12: Evaluate

3

6 cot1

x

dx

EXERCISE 7.2. Evaluate the following Problems using properties of integration.

(3) dxxx cossin

2

0

3

(4)

2

2

3cos

dxx (5)

(7) dxx

1

0

11

log (8)

3

0 3 xx

dxx (9) dxxx

101

0

1 (10)

3

6tan1

x

dx

Example 7.13: Evaluate: dxx5

sin

Example 7.14: Evaluate dxx6

sin

Example 7.15: Evaluate (iii) dxx

4sin

2

0

9

(iv) dxx3cos

6

0

7

Example 7.16: Evaluate dxxx2

2

0

4cossin

Example 7.17: Evaluate (i) dxexx23

(ii) dxexx

1

0

4

2

2

2cossin

dxxx

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EXERCISE 7.3.

(1) Evaluate : (i) dxx4

sin (ii) dxx5

cos

(3) Evaluate : (i) dxx2cos

4

0

8

(ii) dxx3sin

6

0

7

(4) Evaluate : (i) dxexx

1

0

2

Example 7.18: Find the area of the region bounded by the line 3,1,0623 xxyx and x-axis.

Example 7.19: Find the area of the region bounded by the line 4,1,01553 xxyx and x-axis.

Example 7.20:

Find the area of the region bounded by the line 3,2,452

xxxxy and the x-axis.

Example 7.21: Find the area of the region bounded by the line 5,3,12 yyxy and y - axis.

Example 7.22: Find the area of the region bounded by the line 3,1,42 yyxy and y - axis.

Example 7.23:

(i) Evaluate the integral dxx 5

1

3

(ii) Find the area of the region bounded by the line xy 3 , x = 1 and x = 5

Example 7.24:

Find the area bounded by the curve xy 2sin between the ordinates x = 0 , x = and x – axis.

Example 7.35: Find the volume of the solid that results when the ellipse

012

2

2

2

bab

y

a

x is revolved about the minor axis.

Example 7.36:

Find the volume of the solid generated when the region enclosed by 2, yxy and 0x is revolved

about the y – axis. EXERCISE 7.4.

(1) Find the area of the region bounded by the line 1 yx and

(i) x - axis , x = 2 and x = 4 (ii) x - axis , x = -2 and x = 0 (2) Find the area of the region bounded by the line 0122 yx and

(i) y - axis , y = 2 and y = 5 (ii) y - axis , y = -1 and y = -3

(3) Find the area of the region bounded by the line 5 xy and the x-axis between the ordinates x = 3 and x = 7

(5) Find the area of the region bounded by ,362

yx y - axis , y = 2 and y = 4.

(6) Find the area included between the parabola axy 42 and its latus rectum.

(10) Find the area of the circle whose radius is a. Find the volume of the solid that result when the region enclosed by the given curves:

(11) 0,2,1,12

yxxxy is revolved about the x – axis.

(12) 22

2 axxay is revolved about x - axis, a>0.

(13) 1,0,3

yxxy is revolved about the y - axis.

(14) 12

2

2

2

b

y

a

x is revolved about major axis .0 ba

(16) The area of the region bounded by the curve ,1xy x - axis, 1x and x . Find the volume of the solid

generated by revolving the area mentioned about x - axis.

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-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 8. DIFFERENTIAL EQUATIONS

(ONE QUESTION FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------- Example 8.2: Form the differential equation from the following equations.

(iii) 122 ByAx

EXERCISE 8.1. (2) Form the differential equations by eliminating arbitrary constants given in brackets against each

(ix) BABxAeyx

,3cos2

(3) Find the differential equation of the family of straight lines m

amxy when (i) m is the parameter ;

(ii) a is the parameter ; (iii) a , m both are parameters. (4) Find the differential equation that will represent the family of all circles having centres on the x-axis and the

radius is unity. Example 8.3:

Solve : xyyxdx

dy 1

Example 8.4:

Solve: 0sec1tan32

dyyedxyexx

Example 8.5:

Solve: 01

1 2

1

2

2

x

y

dx

dy

Example 8.6:

Solve: 012

dyx

ydxye

x

Example 8.8:

Solve: dxexydyxx

45

4

Example 8.9:

Solve: ,022

dyxydxyx if it passes through the origin.

Example 8.11: The normal lines to a given curve at each point (x, y) on the curve pass through the point (2,0). The curve passes through the point (2, 3). Formulate the differential equation representing the problem and hence find the equation of the curve.

EXERCISE 8.2. Solve the following:

(1) 0sec5sin2sec2

dxyxdyx (2) 0costan2

dxyedyxx

(3) 02222

dxxyydyyxx (4) 02

dyedxyxx

(5) 0897522

dxyydyxx (6) yxdx

dy sin

(8) dxedyxdxyxy

if it cuts the y-axis.

Example 8.12:

Solve : x

y

x

y

dx

dytan

Example 8.16:

Solve: dxyxydxxdy22


(1) 2

2

x

y

x

y

dx

dy

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(2)

yxx

yxy

dx

dy

3

2

(3) dxxydyyx 22

(4) xyydx

dyx 2

22 given that y = 1, when x = 1.

(6) Find the equation of the curve passing through (1,0) and which has slope x

y1 at ., yx

Example 8.17: Solve: xxydx

dycos2cot

Example 8.20: Solve: 2

11 xeydx

dyx

x

Example 8.21: Solve: xxydx

dysintan2


(1) xydx

dy (2)

222

1

1

1

4

x

yx

x

dx

dy (4) xxy

dx

dyx cos21

2

(6) xxydx

dy (8) 2

adx

dyxy

Example 8.25: Solve: xeyDD

221213


2286


3296

Example 8.28: Solve: x

eyDD 2

1

2252

Example 8.29: Solve: xyD 2sin42

Example 8.30: Solve: xyDD 3cos1342

Example 8.31: Solve: xyD 3sin92

Example 8.32: Solve: xyDD 232

Example 8.33: Solve: 2214 xyDD

EXERCISE 8.5. Solve the following differential equations:

(1) xeyDD

22127 (2) x

eyDD32

134

(3) 4491472

xeyDD (4) xx

eeyDD 5121322

(5) 012

yD when x = 0 , y = 2 and when 2,2

yx

(7) 2243 xyDD (8) xxyDD cossin32

2

(9) xyD 3sin92

(12) xyD22

cos5

(13) xyDD 2sin322

(14) 323143

xeyDD

Example 8.36 : The temperature T of a cooling object drops at a rate proportional to the difference T - S, where S is constant

temperature of surrounding medium. If initially CT 150 , find the temperature of the cooling object at any

time t.

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(TWO QUESTIONS FOR FULL TEST) -------------------------------------------------------------------------------------------------------------------------------------------------------- Example 9.4 : Construct the truth table for the following statements :

(iii) qqp ~ (iv) qP ~~~

Example 9.5:

Construct the truth table for rqp ~

Example 9.6 :

Construct the truth table for rqp

EXERCISE 9.2. Construct the truth tables for the following statements:

(7) qpqp ~

(9) rqp

(10) rqp

Example 9.7 : Show that qPqp ~~~

Example 9.10 : (i) Show that pqP ~~ is a tautology.

(ii) Show that qpq ~ is a contradiction.

Example 9.11 :

Use the truth table to determine whether the statement qpqP ~~ is a tautology.

EXERCISE 9.3 (1) Use the truth table to establish which of the following statements are tautologies and which are contradictions

.

(i) pqP ~

(ii) qpqp ~

(iii) qpqp ~~

(iv) qpq ~

(v) pqpp ~~

(2) Show that qpqp ~

(3) Show that pqqpqp

(4) Show that pqqpqp ~~

(5) Show that qpqp ~~~

(6) Show that qp and pq are not equivalent.

(7) Show that qpqp is a tautology.

Example 9.12 :

Prove that ,Z is an infinite abelian group.

Example 9.13 :

Show that .,0R is an infinite abelian group. Here denotes usual multiplication.

Example 9.14 : Show that the cube roots of unity forms a finite abelian group under multiplication. Example 9.15 : Prove that the set of all 4th roots of unity forms an abelian group under multiplication. Example 9.16 :

Prove that ,C is an infinite abelian group.

Example 9.17 : Show that the set of all non-zero complex numbers is an abelian group under the usual multiplication of complex numbers.

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Example 9.19: Show that the set of all 2 X 2 non-singular matrices forms a non-abelian infinite group under matrix multiplication, (where the entries belong to R). Example 9.20 :

Show that

10

01,

10

01,

10

01,

10

01 form an abelian group, under multiplication of matrices.

PROPERTIES: (1) State and prove cancellation laws on groups. (2) State and prove reversal law on inverse of a group. --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 10. PROBABILITY DISTRIBUTIONS (TWO QUESTIONS FOR FULL TEST)

-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 10.1 : Find the probability mass function, and the cumulative distribution function for getting „ 3 „ s when two dice are thrown. Example 10.4 : A continuous random variable X follows the probability law,

elsewhere

xxkxxf

,0

10,110

Find k.

Example 10.5 :

A continuous random variable X has p.d.f. 23xxf , 10 x , Find a and b such that.

(i) aXPaXP and (ii) 05.0 bXP

Example 10.6 :

If the probability density function of a random variable is given by

elsewhere

xxkxf

,0

10,12

Find (i) k (ii) the distribution function of the random variable . Example 10.8 :

If

elsewhere

exx

A

xf

,0

1,3

is a probability density function of a continuous random variable X ,

find exp

EXERCISE 10.1 (1) Find the probability distribution of the number of sixes in throwing three dice once. (2) Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the

probability distribution of the number of queens. (3) Two bad oranges are accidentally mixed with ten good ones. Three oranges are drawn at random without

replacement from this lot. Obtain the probability distribution for the number of bad oranges. (4) A discrete random variable X has the following probability distributions.

X

0

1

2

3

4

5

6

7

8

P(x)

a

3a

5a

7a

9a

11a

13a

15a

17a

(i) Find the value of a (ii) Find 3xP (iii) Find 73 xP

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(6) For the p.d.f

elsewhere

xxcxxf

,0

10,13

find (i) the constant C (ii)

2

1xP

(8) For the distribution function given by

11

10

00

2

x

xx

x

xF

Find the density function. Also evaluate (i) 75.05.0 XP (ii) 5.0XP (iii) 75.0XP

(10) A random variable X has a probability density function

elsewhere

xkxf

,0

20,

Find (i) k (ii)

20

XP (iii)

2

3

2

XP

Example 10.11 : Two unbiased dice are thrown together at random. Find the expected value of the total number of points shown up.

Example 10.12 : The probability of success of an event is p and that of failure is q. Find the expected number of trials to get a first success.

Example10.13 : An urn contains 4 white and 3 red balls. Find the probability distribution of the number of red balls in three draws when a ball is drawn at random with replacement. Also find its mean and variance.

Example 10.14 : A game is played with a single fair die. A player wins Rs. 20 if a 2 turns up, Rs. 40 if a turns up, loses Rs. 30 if a 6 turns up. While he neither wins nor loses if any other face turns up. Find the expected sum of money he can win.

Example 10.15 : In a continuous distribution the p.d.f of X is

otherwise

xxxxf

,0

20,24

3

Find the mean and the variance of the distribution.

Example 10.16:

Find the mean and variance of the distribution

elsewhere

xexf

x

,0

0,33

EXERCISE 10.2 (1) A die is tossed twice. A success is getting an odd number on a toss. Find the mean and the variance of the

probability distribution of the number of successes. (3) In an entrance examination a student has to answer all the 120 questions. Each question has four options

and only one option is correct. A student gets 1 mark for a correct answer and loses half mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the answer to each question at random

(4) Two cards are drawn with replacement from a well shuffled deck of 52 cards. Find the mean and variance for the number of aces.

(5) In a gambling game a man wins Rs. 10 if he gets all heads or all tails and loses Rs. 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.

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(6) The probability distribution of a random variable X is given below:

If XXY 22 find the mean and variance of Y.

(7) Find the Mean and Variance for the following probability density functions:

(i)

otherwise

xxf

,0

1212,24

1

(ii)

otherwise

xifexf

x

,0

0,

(iii)

otherwise

xifexxf

x

,0

0,

Example 10.17 :

Let X be a binomially distributed variable with mean 2 and standard deviation .

3

2Find the corresponding

probability function. Example 10.18 :

A pair of dice is thrown 10 times. If getting a doublet is considered a success find the probability of (i) 4 success (ii) No success.

EXERCISE 10.3 (4) Four coins are tossed simultaneously. What is the probability of getting (a) exactly 2 heads (b) at least two

heads (c) at most two heads. (5) The overall percentage of passes in a certain examination is 80. If 6 candidates appear in the examination

what is the probability that at least 5 pass the examination. Example 10.23 : If a publisher of non-technical books takes a great pain to ensure that his books are free of typological

errors, so that the probability of any given page containing atleast one such error is 0.005 and errors are independent from page to page (i) what is the probability that one of Rs. 400 page novels will contain

exactly one page with error. (ii) atmost three pages with errors. .819.0;1363.02.02

ee

Example 10.24 : Suppose that the probability of suffering a side effect from a certain vaccine is 0.005. If 1000 persons are inoculated. Find approximately the probability that (i) atmost 1 person suffer. (ii) 4,5 or 6 persons

suffer. .0067.05

e

Example 10.25 :

In a Poisson distribution if 32 XPXP find 5XP .050.03

egiven

EXERCISE 10.4 (3) 20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10

bolts chosen at random exactly 2 will be defective using (i) Binomial distribution (ii) Poisson distribution.

.1353.02

e

(4) Alpha particles are emitted by a radioactive source at an average rate of 5 in a 20 minutes interval. Using Poisson distribution find the probability that there will be (i) 2 emission (ii) at least 2 emission in a particular

20 minutes interval. .0067.05

e

EXERCISE 10.5 (3) Suppose that the amount of cosmic radiation to which a person is exposed when flying by jet across the

United States is a random variable having a normal distribution with a mean of 4.35 m rem and a standard deviation of 0.59 m rem. What is the probability that a person will be exposed to more than 5.20 m rem of cosmic radiation of such a flight.

[Area table : P(0<z<1.44) = 0.4251].

X 0 1 2 3

P ( X =x ) 0.1 0.3 0.5 0.1

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(4) The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If 5000 pairs are issued, how many pairs would be expected to need replacement within 12 months.

[ Area table : P(0<z<2) = 0.4772]. (6) If the height of 300 students are normally distributed with mean 64.5 inches and standard deviation 3.3

inches. Find the height below which 99% of the student lie. [Area table : P(0<z<2.33) = 0.49]. (7) Marks in an aptitude test given to 800 students of a school was found to be normally distributed. 10% of the students scored below 40 marks and 10% of the students scored above 90 marks. Find the number of students scored between 40 and 90.

****************

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COME BOOK – THREE MARKS (For two subdivisions – Each 3 Mark )

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT:1. MATRICES AND DETERMINANTS

-------------------------------------------------------------------------------------------------------------------------------------------------------- EXERCISE 1.1

(1) Find the adjoint of the following matrices:

(i)

42

13 (ii)

342

050

321

(iii)

121

213

352

EXERCISE 1.4 (1) Solve the following non-homogeneous system of linear equations by determinant method: (i) 43;523 yxyx (ii) 1264;532 yxyx

Example 1.1: Find the adjoint of the matrix

dc

baA

Example 1.5: Find the inverse of the following matrices:

(i)

41

21 (ii)

24

12 (iii)

cossin

sincos

Example 1.11: Find the rank of the matrix

115

642

321

Example 1.17: Solve the following system of linear equations by determinant method. (1) 732;3 yxyx (3) 733;2 xyyx

Example 1.20: Solve: 022;02;02 zyxzyxzyx


-------------------------------------------------------------------------------------------------------------------------------------------------------- EXERCISE 2.1

(2) If kjia 2 and kjib 23 find baba 23

(5) Find the angles which the vector kji 2 makes with the coordinate axes.

(6) Show that the vector kji is equally inclined with the coordinate axes.

(7) If a

and b

are unit vectors inclined at an angle , then prove that

(i) ba

2

1

2cos

(ii)

ba

ba

2

tan

(8) If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is .3

(12) Show that the vectors kjikji 53,23 and kji 42 form a right angled triangle.

(13) Show that the points whose position vectors jikjikji ,542,34 form a right angle.

EXERCISE 2.2

(6) A force magnitude 5 units acting parallel to kji 22 displaces the point of application from (1,2,3) to

(5,3,7). Find the work done.

(7) The constant forces kjikji 2,652 and ji 72 act on a particle which is displaced from

position kji 234 to position .36 kji Find the work done.

EXERCISE 2.3

(3) Find the unit vectors perpendicular to the plane containing the vectors kji 2 and kji 2

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(4) Find the vectors whose length 5 and which are perpendicular to the vectors kjia 43 and

kjib 256

(7) If kjia 23 and kib 3 then find ba . Verify that a and b are perpendicular to ba

separately.

(8) For any three vectors cba ,, show that 0 bacacbcba

(10) If dcba and ,dbca show that da and cb parallel.

EXERCISE 2.4 (1) Find the area of parallelogram ABCD whose vertices are A (-5,2,5) , B (-3,6,7) , C (4,-1,5) and D (2,-5,3)

(2) Find the area of the parallelogram whose diagonals are represented by kji 632 and kji 263

(3) Find the area of the parallelogram determined by the sides kji 32 and kji 23

(4) Find the area of the triangle whose vertices are (3,-1,2) , (1,-1,-3) and (4,-3,1)

(9) Show that torque about the point A (3,-1,3) of a force kji 24 trough the point B (5,2,4) is

.82 kji

EXERCISE 2.5

(1) Show that vectors cba ,, are coplanar if and only if accbba ,, are coplanar.

(2) The volume of a parallelepiped whose edges are represented by kjikjki 152,3,12 is

546. Find the value of .

(3) Prove that abccba if and only if cba ,, are mutually perpendicular.

(6) Prove that 0 bacacbcba

(9) For any vector a prove that akakjajiai 2

EXERCISE 2.6 (2) (i) Can a vector have direction angles .60,45,30

(ii) Can a vector have direction angles ?120,60,45

(3) What are the d.c.s of the vector equally inclined to the axes? (5) Find the direction cosines of the line joining (2,-3,1) and (3,1,-2). (8) Find the angle between the following lines.

6

4

3

1

2

1

zyx and

2

4

2

21

zyx

(9) Find the angle between the lines.

kjijir 2475

kikir 432

EXERCISE 2.7. (6) If the points ( , 0,3 ) , (1,3,-1) and (-5,-3,7) are collinear then find .

EXERCISE 2.10 (1) Find the angle between the following planes: (i) 92 zyx and 72 zyx

(ii) 1432 zyx and 4 yx

(iii) 73 kjir and 1024 kjir

(2) Show that the following planes are at right angles.

152 kjir and 33 kjir

(3) The planes 1032 kjir and 53 kjir are perpendicular. Find .

(4) Find the angle between the line 2

3

1

1

3

2

zyxand the plane 0543 zyx

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(5) Find the angle between the line kjikjir 23 and the plane .1 jir

EXERCISE 2.11

(4) If A (-1,4,-3) is one end of a diameter AB of the sphere 015223222

zyxzyx , then find the

coordinates of B. (5) Find the centre and radius of each the following spheres.

(iii) 5284222

zyxzyx (iv) 0116242

kjirr

Example 2.6:

For any vector r prove that kkrjjriirr

Example 2.8:

For any two vectors a and b prove that

2222

2 bababa

Example 2.9:

If a

and b

are unit vectors inclined at an angle , then prove that ba

2

1

2sin

Example 2.10:

If ,0 cba 5,3 ba and ,7c find the angle between a and b .

Example 2.11:

Show that the vectors kji 2 , kji 53 , kji 443 form the sides of a right angled triangle.

Example 2.20:

If ba , are any two vectors, then prove that 2222

bababa

Example 2.21:

Find the vectors of magnitude 6 which are perpendicular to both the vectors kji 34 and

kji 22

Example 2.22:

If 5,13 ba and 60 ba then find ba

Example 2.23:

Find the angle between the vectors kji

2 and kji 2 by using cross product.

Example 2.30:

Show that the area of a parallelogram having diagonals kji 23 and kji 43 is 35 .

Example 2.31:

A force given by kji 423 is applied at the point (1,-1,2) . Find the moment of the force about the point

(2,-1,3). Example 2.32:

If the edges kjickjibkjia 357,375,573 meet at a vertex, find the volume

of the parallelopiped. Example 2.34:

If 0.,0.,0. cxbxax and 0x then show that cba ,, are coplanar.

Example 2.35:

If cbacba then prove that 0 bac

Example 2.41:

Find the angle between the lines kjitkjir 2223 and kjiskjr 62325

Example 2.47:

Find the value of if the points (3,2,-4) , (9,8,-10) and (,4,-6) are collinear. Example 2.60: Find the angle between 42 zyx and 42 zyx

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Example 2.61:

Find the angle between the line kjikjir 222 and the plane

0623 kjir --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 3.COMPLEX NUMBERS --------------------------------------------------------------------------------------------------------------------------------------------------------

EXERCISE 3.1 (1) Express the following in the standard form a + ib

(i)

21

32

i

i

(2) Find the real and imaginary parts of the following complex numbers:

(i) i1

1 (ii)

i

i

34

52

(3) Find the least positive integer n such that 11

1

n

i

i

EXERCISE 3.2

(1) If iyxniiii 1...31211 show that 2221...10.5.2 yxn

(3) If 1,02z find z.

(6) Express the following complex numbers in polar form.

(i) i322 (ii) 31 i (iii) i1 (iv) i1

EXERCISE 3.4

(1) Simplify :

612

57

5sin5cos4sin4cos

3sin3cos2sin2cos

ii

ii

EXERCISE 3.5 (1) Find all the values of the following:

(i) 3

1

i

(3) Prove that if ,13 then (ii) 1

2

31

2

3155

ii

Example 3.4:

Express the following in the standard form of a + ib (iv) i

i

43

55

Example 3.5:

Find the real and imaginary parts of the complex number 12

31920

i

iiz

Example 3.6:

If iziz 23,2 21 and iz2

3

2

13

find the conjugate of (i)

21 zz (ii) 4

3z

Example 3.8:

Find the modulus or the absolute value of

i

ii

43

2131

Example 3.12:

Graphically prove that 321321

zzzzzz

Example 3.18:

Simplify :

86

33

sincos4sin4cos

3sin3cos2sin2cos

ii

ii

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-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 4. ANALYTICAL GEOMETRY

-------------------------------------------------------------------------------------------------------------------------------------------------------- EXERCISE 4.1

(1) Find the equation of the parabola if (i) Focus : (2,-3) ; directrix : 032 y

(ii) Focus : (-1,3) ; directrix : 332 yx

(iii) Vertex : (0,0) ; focus : (0,-4) (iv) Vertex : (1,4) ; focus : ( -2,4) (v) Vertex : (1,2) ; latus rectum : y = 5 (vi) Vertex : (1,4) ; open leftward and passing through the point (-2,10) (vii) Vertex : (3,-2) ; open downward and the length of the latus rectum is 8. (viii) Vertex : (3,-1) ; open rightward ; the distance between the latus rectum and the directrix is 4. (ix) Vertex : (2,3) ; open upeard ; and passing through the point (6,4).

EXERCISE 4.2 (1) Find the equation of the ellipse if

(i) one of the foci is (0,-1) , the corresponding directrix is 0163 x and 5

3e

Example 4.1: Find the equation of the following parabola with indicated focus and directrix. (i) 0;0, aaxa (ii) 032;2,1 yx (iii) 02;3,2 y

Example 4.2: Find the equation of the parabola if (i) the vertex is (0,0) and the focus is (-a,0) , a > 0 (ii) the vertex is (4,1) and the focus is (4,-3) Example 4.3: Find the equation of the parabola whose vertex is (1,2) and the equation of the latus rectum is x = 3. Example 4.4:

Find the equation of the parabola if the curve is open rightward, vertex is (2,1) and passing through point (6,5)

Example 4.5: Find the equation of the parabola if the curve is open upward, vertex is (-1,-2) and the length of the latus rectum is 4. Example 4.6:

Find the equation of the parabola if the curve is open leftward, vertex is (2,0) and the distance between the latus rectum and directrix is 2

Example 4.40: Find the equation of the hyperbola whose transverse axis is parallel to y - axis , centre (0,0) , length of semi-conjugate axis is 4 and eccentricity is 2.

Example 4.60:

Find the equation of the tangent at t = 1 to the parabola xy 122 .

Example 4.63:

Find the equation of chord of contact of tangents from the point (2,4) to the ellipse 205222 yx


-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 5.2:

The luminous intensity I candelas of a lamp at varying voltage V is given by : .10424

VI

Determine the

voltage at which the light is increasing at a rate of 0.6 candelas per volt. Example 5.21: Using Rolle‟s theorem find the value (s) of c.

(i) 11,12

xxxf

(ii) .,, babxaxbaxxf

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(iii) 32

1,3452

23 xxxxxf

Example 5.22: Verify Rolle‟s theorem for the following:

(i) 10333

xxxxf

(ii) xxxf 0,tan

(iii) 11, xxxf

(iv) xxxf 0,sin2

(vi) 20,21 xxxxxf

EXERCISE 5.3.

(1) Verify Rolle‟s theorem for the following functions:

(i) xxxf 0,sin (ii) 10,2

xxxf

(iii) 20,1 xxxf (iv) 2

3

2

3,94

3 xxxxf

Example 5.28: Obtain the Maclaurin‟s Series for

1) x

e

EXERCISE 5.5 (1) Obtain the Maclaurin‟s Series expansion for:

(i) xe

2

Example 5.32:

Evaluate: x

x e

x2

lim

EXERCISE 5.6 Evaluate the limit for the following if exists.

(1) x

x

x 2

sinlim

2

(3)

x

x

x

1

0

sinlim

(4) 2

2lim

2

x

xnn

x

(5)

x

x

x 1

2sin

lim

(7) x

xe

x

loglim

EXERCISE 5.7

(1) Prove that xe is strictly increasing function on R.

(2) Prove that log x is strictly increasing function on ,0

(3) Which of the following functions are increasing or decreasing on the interval given?

(i) 12x on 2,0 (ii) xx 32

2 on

2

1,

2

1 (iii) x

e on 1,0

Example 5.44:

Prove that xex

1 for all .0x

Example 5.47:

Find the critical numbers of xx 453

EXERCISE 5.9 (1) Find the critical numbers and stationary points of each of the following functions.

(i) 232 xxxf (ii) 13

3 xxxf

Example 5.59:

Determine the domain of concavity (convexity) of the curve .22

xy

Example 5.60:

Determine the domain of convexity of the function xey

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Example 5.61:

Test the curve 4xy for points of inflection.

EXERCISE 5.11 Find the intervals of concavity and the points of inflection of the following functions:

(2) xxxf 2


-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 6.4:

The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of atmost 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?

Example 6.8: Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 1%

EXERCISE 6.1 (2) Find the differential dy and evaluate dy for the given values of x and dx.

(i) 2

1,5,1

2 dxxxy

(ii) 1.0,2,1334

dxxxxxy

(iii) 05.0,1,532

dxxxy

(iv) 02.0,0,1 dxxxy

(v) 05.06

,cos dxxxy

EXERCISE 6.3

(2) (i) If 22

yxu , show that .uy

uy

x

ux

(ii) If x

ye

y

xeu x

y

y

x

cossin , show that .0

y

uy

x

ux

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 7. INTEGRAL CALCULUS -------------------------------------------------------------------------------------------------------------------------------------------------------- Example 7.1:

Evaluate dxx

x

2

02

cos1

sin

Example 7.2:

Evaluate dxexx

1

0

Example 7.3:

Evaluate a

dxxa

0

22

Example 7.4:

Evaluate:

2

0

2cos

dxxex

EXERCISE 7.1 Evaluate the following problems using second fundamental theorem:

(1)

2

0

2sin

dxx (2)

2

0

3cos

dxx

(5)

1

0 24 x

dx (11)

2

0

3cos

dxxex (12)

2

0

sin

dxxex

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Example 7.5:

Evaluate

4

4

23sin

dxxx

Example 7.6:

Evaluate dxx

x

1

1 3

3log

EXERCISE 7.2 Evaluate the following problems using properties of integration.

(1) dxxx4

1

1

cossin

(2)

4

4

33cos

dxxx

(6)

4

4

2sin

dxxx

Example 7.15:

Evaluate (i) dxx

2

0

7sin

(ii) dxx

2

0

8cos

EXERCISE 7.3

(2) Evaluate : (i)

2

0

6sin

dxx (ii)

2

0

9cos

dxx


-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 8.2: Form the differential equation from the equations.

(i) BxAeyx

2 (ii) xBxAey

x3sin3cos (iv) axay 4

2

EXERCISE 8.1 (2) Form the differential equations by eliminating arbitrary constants given in brackets against each

(i) axay 42

(ii) bacbxaxy ,2

(iv) bab

y

a

x,1

2

2

2

2

(v) BABeAeyxx

,52

(vi) BAeBxAyx

,3

(vii) DCxDxCeyx

,2sin2cos3

Example 8.20:

Solve : 0652

yDD

Example 8.21:

Solve: 0962

yDD

Example 8.22:

Solve: 012

yDD


-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 9.4: Construct the truth table for the following statements: (i) qp ~~ (ii) qp ~~

EXERCISE 9.2 Construct the truth tables for the following statements: (1) qp ~ (2) qp ~~ (3) qp ~

(4) pqp ~ (5) qqp ~ (6) qp ~~

(8) qqp ~

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PROPERTIES: (1) Prove that identity element of a group is unique. (2) Prove that inverse element of an element of a group is unique.

(3) Show that ,11

Gaaa a group.

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 10. PROBABILITY DISTRIBUTIONS

-------------------------------------------------------------------------------------------------------------------------------------------------------- Example 10.7:

If

xxxF

1tan

2

1

is a distribution function of a continuous variable X , find 10 xP

Example 10.9:

For the probability density function

0,0

0,22

x

xexf

x

, find F(2)

EXERCISE 10.1 (5) Verify that following are probability density functions.

(a)

elsewhere

xx

xf

,0

30,9

2

(b)

xx

xf ,1

11

2

(9) A continuous random variable x has the p.d.f defined by

elsewhere

xcexf

ax

,0

0, Find the value of c if a > 0.

EXERCISE 10.2

(2) Find the expected value of the number on a die when thrown. Example 10.19: In a Binomial distribution if n = 5 and 223 XPXP find p.

Example 10.20: If the sum of mean and variance of a Binomial Distribution is 4.8 for 5 trials find the distribution. Example 10.21: The difference between the mean and the variance of a Binomial distribution is 1 and the difference between their squares is 11. Find n.

EXERCISE 10.3 (1) The mean of a binomial distribution is 6 and its standard deviation is 3. Is this statement true or false?

Comment. (2) A die is thrown 120 times ad getting 1 or 5 is considered a success. Find the mean and variance of the

number of successes. (3) If on an average 1 ship out of 10 do not arrive safely to ports. Find the mean and the standard deviation of

ships returning safely out of a total of 500 ships. (6) In a hurdle race a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5 / 6. What

is the probability that he will knock down less than 2 hurdles. Example 10.22: Prove that the total probability is one.

EXERCISE 10.4

(1) Let X have a Poisson distribution with mean 4. Find (i) 3XP (ii) 52 XP .0183.04

e

(2) If the probability of a defective fuse from a manufacturing unit is 2% in a box of 200 fuses find the probability that

(i) exactly 4 fuses are defective (ii) more than 3 fuses are defective .0183.04

e

Example 10.27: Let Z be a standard normal variate. Calculate the following probabilities. (i) 2.10 ZP (ii) 02.1 ZP

(iii) Area to the right of Z = 1.3 (iv) Area to the left of Z = 1.5 (v) 5.22.1 ZP (vi) 5.02.1 ZP (vii) 5.25.1 ZP

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Example 10.28: Let Z be a standard normal variate. Find the value of c in the following problems. (i) 05.0 cZP (ii) 94.0 cZcP

(iii) 05.0 cZP (iv) 31.00 ZcP

EXERCISE 10.5 (1) If X is a normal variate with mean 80 and standard deviation 10, compute the following probabilities by

standardizing. (i) 100XP (ii) 80XP

(iii) 10065 XP (iv) XP 70

(v) 9585 XP

(2) If Z is a standard normal variate, find the value of c for the following (i) 25.00 cZP (ii) 40.0 cZcP (iii) 85.0 cZP

*********

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COME BOOK – ONE MARK [10 Number Of One Mark Questions to be Asked For Full Test]



1. The rank of the matrix

21

42 is

1.1 2.2 3.0 4. 8


12

17 is

1. 9 2. 2 3. 1 4. 5 3. If A and B are matrices conformable to multiplication then (AB)T is 1. ATBT 2. BTAT 3. AB 4. BA 4. (AT)-1 is equal to 1. A-1 2. AT 3. A 4. (A-1)T 5. If (A) = r , then which of the following is correct?

1. all the minors of order r which do not vanish. 2. A has atleast one minor of r which does not vanish and all higher order minors vanish. 3. A has atleast one (r+1) order minor which vanishes. 4. all (r+1) and higher order minors should not vanish. 6. Which of the following is not elementary transformation? 1. Ri Rj 2. Ri 2Ri+Rj 3. Ci Cj+Ci 4. Ri Ri+Cj

7. Equivalent matrices are obtained by 1. taking inverses 2. taking transposes 3. taking adjoints 4. taking finite number of elementary transformations 8. In echelon form, which of the following is incorrect?

1. Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. 2. The first non-zero entry in each non-zero row is 1. 3. The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row. 4. Two rows can have same number of zeros before the first non-zero entry

9. If 0 then the system is

1. consistent and has unique solution 2. consistent and has infinitely many solutions 3. inconsistent 4. either consistent or inconsistent

10. In the system of 3 linear equations with three unknowns, if 0 and one of zyx

or ,, is non-zero then

the system is 1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation.

11. In the system of 3 linear equations with three unknowns, if 0,0,0,0 zyx

and atleast

one 2x2 minor of 0 then the system is

1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation.

12. In the system of 3 linear equations with three unknowns, if 0 and all 2x2 minors of 0 and atleast

one 2x2 minor of zyx

oror is non-zero then the system is

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1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation.

13. In the system of 3 linear equations with three unknowns, if 0 and all 2x2 minors of zyx

,,, are

zeros and atleast one non-zero element is in then the system is 1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation. 14. Every homogeneous system(linear) 1. is always consistent 2. has only trivial solution 3. has infinitely many solution 4. need not be consistent

15. If BAA ,)( then the system is

1. consistent and has infinitely many solution 2. consistent and has a unique solution 3. consistent 4. inconsistent

16. If BAA ,)( = the number of unknowns then the system is


17. BAA ,)( then the system is


18. In the system of 3 linear equations with three unknowns, BAA ,)( = 1, then the system

1.has unique solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent

19. In the homogeneous system with three unknowns, )( A = number of unknowns then the system has

1. only trivial solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent 20. In the system of three linear equations with three unknowns, in the non-homogeneous system

BAA ,)( = 2, then the system

1. has unique solution 2. reduces to two equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent

21. In the homogeneous system )( A < the number of unknowns then the system has

1. only trivial solution 2. trivial solution and infinitely many non-trivial solutions 3. only non-trivial solutions 4. no solution 22. Cramer‟s rule is applicable only (with three unknowns) when

1. 0 2. 0 3. 0,0 x

4. 0zyx

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23. Which of the following statement is correct regarding homogeneous system? 1. always inconsistent 2. has only trivial solution 3. has only non-trivial solutions

4. has only trivial solution only if rank of the coefficient matrix is equal to the number of unknowns



1. The value of ba when kjia

2 and kjib

744 is

1. 19 2. 3 3. -19 4. 14

2. The value of ba when kja

2 and kib

2 is

1. 2 2. -2 3. 3 4. 4

3. The value of ba when kja

2 and kjib

232 is

1. 7 2. -7 3. 5 4.6

4. If kjim

2 and kji

294 are perpendicular, then m is

1.-4 2. 8 3. 4 4. 12

5. If kji

295 and kjim

2 are perpendicular, then m is

1.16

5 2. -

16

5 3.

5

16 4. -

5

16

6. If a

and b

are two vectors such that ,4a

3b

and ba = 6, then the angle between a

and b

is

1. 6

2. -

6

3.

3

4.

3

7. The angle between the vectors kji

623 and kji

84 is

1.

63

34cos

1 2.

63

34sin

1 3.

63

34sin

1 4.

63

34cos

1

8. The angle between the vectors ji

and kj

is

1. 3

2.

3

2 3. -

3

4.

3

2

9. The projection of the vector kji

47 on kji

362 is

1. 8

7 2.

66

8 3.

7

8 4.

8

66

10. ba when kjia

22 and kjib

236 is

1. 4 2. -4 3. 3 4. 5

11. If the vectors kji

2 and kji

2 are perpendicular to each other, then is

1. 3

2 2.

3

2 3.

2

3 4.

2

3

12. If the vectors a

= kji

923 and b

= kjmi

3 are perpendicular, then „m‟ is

1. -15 2. 15 3. 30 4. -30

13. If the vectors a

= kji

923 and b

= kjmi

3 are parallel, then „m‟ is

1. 2

3 2.

3

2 3.

2

3 4.

3

2

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14. If cba

,, are three mutually perpendicular unit vectors, then cba

1. 3 2. 9 3. 33 4. 3

15. If ba

60, ba

40 and b

46, then a

is

1. 22 2. 21 3. 18 4. 11

16. Let u

, v

and w

be vector such that 0

wvu . If 3u

, 4v

and 5w

, then uwwvvu

is

1. 25 2. -25 3. 5 4. 5

17. The projection of ji

on z-axis is

1. 0 2. 1 3. -1 4. 2

18. The projection of kji

22 on kji

52 is

1. 30

10 2.

30

10 3.

3

1 4.

30

10


3 on kji

24 is

1. 21

9 2. -

21

9 3.

21

81 4. -

21

81

20. The work done in moving a particle from the point A with position vector kji

762 to the point B, with

position vector kji

53 by a force F

kji

3 is

1. 25 2. 26 3. 27 4. 28

21. The work done by the force F

kjia

in moving the point of application from (1, 1, 1) to (2, 2, 2)

along a straight line is given to be 5 units. The value of a is 1. -3 2. 3 3. 8 4. -8

22. If a

=3, b

4 and ba

=9, then ba

is

1.3 7 2.63 3.69 4. 69

23. The angle between the two vectors a

and if ba

= ba

is

1.4

2.

3

3.

6

4.

2

24. If a

=2, b

=7 and kjiba

623 then the angle between a

and b

is

1.4

2.

3

3.

6

4.

2

25. The direction cosines of a vector whose direction ratios are 2, 3, -6 are

1.

7

6,

7

3,

7

2 2.

49

6,

49

3,

49

2 3.

7

6,

7

3,

7

2 4.

7

6,

7

3,

7

2

26. The unit normal vectors to the plane 522 zyx are

1. kji

22 2. kji

223

1 3. kji

22

3

1 4. kji

22

3

1

27. The length of the perpendicular from the origin to the plane 261243. kjir

is

1. 26 2. 169

26 3. 2 4.

2

1

28. The distance from the origin to the plane 752. kjir

is

1. 30

7 2.

7

30 3.

7

30 4.

30

7

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29. Chord AB is a diameter of the sphere 1862 kjir

with coordinate of A as

(3, 2, -2). Then the coordinates of B is 1. (1, 0, 10) 2. (-1, 0, -10) 3. (-1, 0, 10) 4. (1, 0, -10)

30. The centre and radius of the sphere )42( kjir

5 are

1. (2, -1, 4) and 5 2. (2, 1, 4) and 5 3. (-2, 1, 4) and 6 4. (2, 1, -4) and 5

31. The centre and radius of the sphere )43(2 kjir

4 are

1.

2,

2

1,

2

3, 4 2.

2,

2

1,

2

3and 2 3.

2,

2

1,

2

3, 6 4.

2,

2

1,

2

3and 5

32. The vector equation of a plane passing through a point whose position vector is a

and perpendicular to a

vector n

is

1. nanr

2. nanr

3. nanr

4. nanr

33. The vector equation of a plane whose distance from the origin is p and perpendicular to a unit vector n̂ is

1. pnr

. 2. qnr ˆ.

3. pnr

4. pnr ˆ.

34. The non-parametric vector equation of a plane passing through a point whose position vector is a

and

parallel to u

and v

is

1. 0,, vuar

2. 0vur

3. 0 vuar

4. 0vua

35. The non-parametric vector equation of a plane passing through the points whose position vectors are a

.

b

and parallel to v

is

1. 0 vabar

2. 0 vabr

3. 0vba

4. 0bar

36. The non-parametric vector equation of a plane passing through three non-collinear points whose position

vectors are cba

,, is

1. 0 acabar

2. 0bar

3. 0cbr

4. 0cba

37. The vector equation of a plane passing through the line of intersection of the planes 11

. qnr

and

22

. qnr

is

1. 0..2211

qnrqnr

2. 2121

.. qqnrnr

3. 2121qqnrnr

4. 2121

qqnrnr

38. The angle between the line btar

and the plane qnr

is connected by the relation

1. q

na

cos 2.

nb

nb

cos 3. n

ba

sin 4. nb

nb

sin

39. The vector equation of a sphere whose centre is origin and radius „a‟ is

1. ar

2. acr

3. ar

4. ar



1. The complex number form of 35 is

1. i 35 2. - i 35 3. i 35 4. 35i

2. The complex number form of 3- 7 is

1. -3+i 7 2. 3- i 7 3. 3- i7 4. 3+ i7

3. Real and imaginary parts of 34 i are

1. 3,4 2. 3,4 3. 3 , 4 4. 3 , 4

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4. Real and imaginary parts of i2

3are

1. 2

3,0 2. 0,

2

3 3. 3,2 4. 3, 2

5. The complex conjugate of 72 i is

1. 72 i 2. 72 i 3. 72 i 4. 72 i

6. The complex conjugate of 94 i is

1. 94 i 2. 94 i 3. 4 – i9 4 .– 4 – i9

7. The complex conjugate of 5 is

1. 5 2. 5 3. 5i 4. 5i

8. The standard form )( iba of )7(23 ii is

1. i4 2. - i4 3. i4 4. i44

9. If )72()68( iiiba then the values of „a‟ and „b‟ are

1. 15,8 2. 15,8 3. 9,15 4. 8,15

10. If p+iq = (2-3i)(4+2i), then q is 1. 14 2. -14 3. -8 4. 8 11. The conjugate of (2+i)(3-2i) is 1. 8-i 2. -8-i 3. -8+i 4. 8+i 12. The real and imaginary parts of (2+i)(3-2i) are 1. -1, 8 2. -8, 1 3. 8, -1 4. -8, -1 13. The modulus values of -2+2i and 2-3i are

1. 5,5 2. 13,52 3. 13,22 4. - 4, 1

14. The modulus values of -3-2i and 4+3i are

1. 5,5 2. 7,5 3. 1,6 4. 5,13

15. The cube roots of unity are

1. in G.P. with common ratio 2. in G.P. with common difference 2

3. In A.P. with common difference 4. in A.P. with common difference 2

16. The arguments of nth roots of a complex number differ by

1. n

2 2.

n

3.

n

3 4.

n

4

17. Which of the following statements is correct? 1. negative complex numbers exist 2. order relation does not exist in real numbers

3. order relation exist in complex numbers 4. )1( i > )23( i is meaningless

18. Which of the following are correct?

a. zz )(Re b. zzmI )( c. zz d. nnzz

1. (a), (b) 2. (b), (c) 3. (b), (c) and (d) 4. (a), (c) and (d)

19. The values of zz is

1. )(Re2 z 2. )(Re z 3. )(Im z 4. )(Im2 z

20. The value of zz is

1. )(Im2 z 2. )(Im2 zi 3. )( zmI 4. )(Im zi


1. || z 2. 2|| z 3. ||2 z 4. 2

||2 z

22. If |||| 21zzzz then the locus of z is

1. a circle with centre at the origin

2. a circle with centre at 1

z

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3. a straight line passing through the origin

4. is a perpendicular bisector of the line joining 1

z and 2

z

23. If is a cube roots of unity, then

1. 12 2. 01 3. 01

2 4. 01

2

24. The principal value of zarg lies in the interval

1.

2,0

2. ],( 3. ,0 4. ]0,(

25. If 1

z and2

z are any two complex numbers then which one of the following is false?

1. )(Re)(Re)(Re2121

zzzz 2. )()()(Im2121

zIzIzzmm

3. 2121

argarg)(arg zzzz 4. ||||||2121

zzzz

26. The fourth roots of unity are

1. ii 1,1 2. ii 1, 3. i ,1 4. 1,1

27. The fourth roots of unity form the vertices of 1. an equilateral triangle 2. a square 3. a hexagon 4. a rectangle 28. Cube roots of unity are

1. 2

31,1

i 2.

2

31,

ii 3.

2

31,1

i 4.

2

31,

ii

29. The number of values of q

p

i sincos where p and q are non-zero integers prime to each other, is

1. p 2. q 3. p+q 4.( p-q)

30. The value of iiee

is

1. cos2 2. cos 3. sin2 4. sin


is

1. sin 2. sin2 3. sini 4. sin2 i

32. Geometrical interpretation of z is

1. reflection of z on real axis 2. reflection of z on imaginary axis 3. rotation of z about origin

4. rotation of z about origin through 2/ in clockwise direction

33. If ibazibaz 21

, then 21

zz lies on

1. real axis 2. imaginary axis 3. the line y= x 4. the line y = -x 34. Which one of the following is incorrect?

1. ninin

sincos)sin(cos

2. ninin

sincos)sin(cos

3. ninin

cossin)cos(sin

4.

sincossincos

1i

i

35. Polynomial equation P(x)=0 admits conjugate pairs of imaginary roots only if the coefficients are 1. imaginary 2. complex 3. real 4. either real or complex

36. Identify the correct statement 1. Sum of the moduli of two complex numbers is equal to their modulus of the sum 2. Modulus of the product of the complex numbers is equal to the sum of their moduli 3. Arguments of the product of two complex numbers is the product of their arguments. 4. Arguments of the product of two complex numbers is equal to sum of their arguments.

37. Which of the following is not true?

1. 2121

zzzz 2. 2121

zzzz 3. 2

)(Rezz

z

4. i

zzz

2)(Im

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38. If 1

z and 2

z are complex numbers then which of the following is meaningful?

1. 21

zz 2. 21

zz 3. 21

zz 4. 21

zz

39. Which of the following is incorrect?

1. ||)(Re zz 2. ||)(Im zz 3. 2|| zzz 4. ||)(Re zz


1. ||||||2121

zzzz 2. ||||||2121

zzzz

3. ||||||2121

zzzz 4. ||||||2121

zzzz


1. z is the mirror image of z on the real axis

2. The polar form of z is ,r

3. –z is the point symmetrical to z about the origin

4. The polar form of -z is ,r

42. Which of the following is incorrect? 1. Multiplying a complex number by i is equivalent to rotating the number counter clockwise about the

origin through an angle 90 .

2. Multiplying a complex number by - i is equivalent to rotating the number clockwise about the origin

through an angle 90.

3. Dividing a complex number by i is equivalent to rotating the number counter clockwise about the

origin through an angle 90 .

4. Dividing a complex number by i is equivalent to rotating the number clockwise about the origin

through an angle 90 .

43. Which of the following is incorrect regarding nth roots of unity? 1. The number of distinct roots is n

2. the roots are in G.P. with common ratio n

cis2

3. the arguments are in A.P. with common difference n

2.

4. product of the roots is 0 and the sum of the roots is 1 . 44. Which of the following are true?

1. If n is a positive integer then ninin

sincossincos

2. If n is a negative integer then ninin

sincossincos

3. If n is a fraction then nin sincos is one of the values of ni )sincos( .


sincossincos

1. (i), (ii), (iii), (iv) 2. (i), (iii), (iv) 3. (i), (iv) 4. (i) only

45. If )('),(),(),0,0(221

ZBZBZAO are the complex numbers in a argand plane then which of the

following are correct?

(i) In the parallelogram OACB, represents 21

ZZ

(ii) In the argand plane E represents 21ZZ where OE = OA.OB and OE makes an angle

arg( 1z )+arg(

2z ) with positive real axis.

(iii) In the argand parallelogram DAOB ' , D represents 21

ZZ

(iv) In the argand plane F represents 2

1

Z

Z where

OB

OAOF and OF makes an angle arg(

1z )-arg(

2z )

with positive real axis. 1. (i), (ii), (iii), (iv) 2. (i), (iii), (iv) 3. (i), (iv) 4. (i) only

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46. If Z = 0, then the arg(Z) is

1. 0 2. 3. 2

4. Indeterminate

-------------------------------------------------------------------------------------------------------------------------------------------------------- UNIT: 4. ANALYTICAL GEOMETRY

(One Question For Full Test) --------------------------------------------------------------------------------------------------------------------------------------------------------

1. The axis of the parabola xy 42 is

1. 0x 2. 0y 3. 1x 4. 1y

2. The vertex of the parabola xy 42 is

1. )0,1( 2. )1,0( 3. )0,0( 4. )1,0(

3. The focus of the parabola xy 42 is

1. )1,0( 2. )1,1( 3. )0,0( 4. )0,1(

4. The directrix of the parabola xy 42 is

1. 1y 2. 1x 3. 1y 4. 1x

5. The equation of the latus rectum of xy 42 is

1. 1x 2. 1y 3. 4x 4. 1y

6. The length of the latus rectum of xy 42 is

1. 2 2. 3 3. 1 4. 4

7. The axis of the parabola yx 42

is

1. 1y 2. 0x 3. 0y 4. 1x

8. The vertex of the parabola yx 42

is

1. )1,0( 2. )1,0( 3. )0,1( 4. )0,0(

9. The focus of the parabola yx 42

is

1. )0,0( 2. )1,0( 3. )1,0( 4. )0,1(

10. The directrix of the parabola yx 42

is

1. 1x 2. 0x 3. 1y 4. 0y

11. The equation of the latus rectum of yx 42

is

1. 1x 2. 1y 3. 1x 4. 1y

12. The length of the latus rectum of yx 42

is

1. 1 2. 2 3. 3 4. 4

13. The axis of the parabola xy 82

is

1. 0x 2. 2x 3. 2y 4. 0y

14. The vertex of the parabola xy 82

is

1. )0,0( 2. )0,2( 3. )2,0( 4. )2,2(

15. The focus of the parabola xy 82

is

1. )2,0( 2. )2,0( 3. )0,2( 4. )0,2(

16. The equation of the directrix of the parabola xy 82

is

1. 02 y 2. 02 x 3. 02 y 4. 02 x

17. The equation of the latus rectum of xy 82

is

1. 02 y 2. 02 y 3. 02 x 4. 02 x

18. The length of the latus rectum of xy 82

is

1. 8 2. 6 3. 4 4. - 8

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19. The axis of the parabola yx 202 is

1. 5y 2. 5x 3. 0x 4. 0y

20. The vertex of the parabola yx 202 is

1. )5,0( 2. )0,0( 3. )0,5( 4. )5,0(

21. The focus of the parabola yx 202 is

1. )0,0( 2. )0,5( 3. )5,0( 4. )0,5(

22. The equation of the directrix of the parabola yx 202 is

1. 05 y 2. 05 x 3. 05 x 4. 05 y

23. The equation of the latus rectum of the parabola yx 202 is

1. 05 x 2. 05 y 3. 05 y 4. 05 x

24. The length of the latus rectum of the parabola yx 202 is

1. 20 2. 10 3. 5 4. 4 25. If the centre of the ellipse is (2, 3) one of the foci is (3, 3) then the other focus is

1. )3,1( 2. )3,1( 3. )3,1( 4. )3,1(

26. The equation of the major and minor axes of 149

22

yx

are

1. 2,3 yx 2. 2,3 yx 3. 0,0 yx 4. 0,0 xy

27. The equation of the major and minor axes of 123422 yx are

1. 2,3 yx 2. 0,0 yx 3. 2,3 yx 4. 0,0 xy

28. The length of the minor and major axes of 149

22

yx

are

1. 4,6 2. 2,3 3. 6,4 4. 3,2

29. The length of the major and minor axes of 123422 yx are

1. 32,4 2. 3,2 3. 4,32 4. 2,3

30. The equation of the directrices 1916

22

yx

are

1. 7

4y 2.

7

16x 3.

7

16x 4.

7

16y

31. The equation of the directrices 25x2+9y2 = 225 are

1. 25

4x 2.

4

25x 3.

25

4y 4.

4

25y

32. The equation of the latus rectum of 1916

22

yx

are

1. 7y 2. 7x 3. 7x 4. 7y

33. The equation of the latus rectum of 22592522 yx are

1. 5y 2. 5x 3. 4y 4. 4x

34. The length of the latus rectum of 1916

22

yx

is

1. 2

9 2.

9

2 3.

16

9 4.

9

16

35. The length of the latus rectum of 22592522 yx is

1. 5

9 2.

5

18 3.

9

25 4.

18

5

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36. The eccentricity of the ellipse 1925

22

yx

is

1. 5

1 2.

5

3 3.

5

2 4.

5

4

37. The eccentricity of the ellipse 194

22

yx

is

1. 3

5 2.

5

3 3.

5

3 4.

3

2

38. The eccentricity of the ellipse 16x2+25y2= 400 is

1. 5

4 2.

5

3 3.

4

3 4.

5

2

39. The centre of the ellipse 1925

22

yx

is

1. )0,0( 2. )0,5( 3. )5,3( 4. )5,0(

40. The centre of the ellipse 194

22

yx

is

1. )3,0( 2. )3,2( 3. )0,0( 4. )0,3(

41. The foci of the ellipse 1925

22

yx

are

1. )5,0( 2. )4,0( 3. )0,5( 4. )0,4(


22

yx

are

1. 0,5 2. 5,0 3. 5,0 4. 0,5

43. The foci of the ellipse 16x2+25y2= 400 are

1. 0,3 2. 3,0 3. 5,0 4. 0,5

44. The vertices of the ellipse 1925

22

yx

are

1. )5,0( 2. )3,0( 3. )0,5( 4. )0,3(

45. The vertices of the ellipse 194

22

yx

are

1. )3,0( 2. )0,2( 3. )0,3( 4. )2,0(

46. The vertices of the ellipse 16x2+25y2= 400 are

1. )4,0( 2. )0,5( 3. )0,4( 4. )5,0(

47. If the centre of the ellipse is (4, -2) and one of the foci is (4, 2), then the other focus is

1. )6,4( 2. )4,6( 3. )6,4( 4. )4,6(

48. The equations of transverse and conjugate axes of the hyperbola 149

22

yx

are

1. 3;2 yx 2. 0;0 xy 3. 2;3 yx 4. 0,0 yx

49. The equations of transverse and conjugate axes of the hyperbola 14491622 xy are

1. 0;0 xy 2. 4;3 yx 3. 0;0 yx 4. 4;3 xy

50. The equations of transverse and conjugate axes of the hyperbola 36002514422 yx are

1. 0;0 xy 2. 5;12 yx 3. 0;0 yx 4. 12;5 yx

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51. The equations of transverse and conjugate axes of the hyperbola 8y2-2x2 = 16 are

1. 2;22 yx 2. 22;2 yx 3. x = 0 ; y = 0 4. y = 0 ; x = 0

52. The equation of the directrices of the hyperbola 149

22

yx

are

1. y = 13

9 2. x =

9

13 3. y =

9

13 4. x =

13

9

53. The equation of the directrices of the hyperbola 14491622 xy are

1. 9

5x 2.

5

9y 3.

5

9x 4.

9

5y

54. The equation of the latus rectum of the hyperbola 149

22

yx

are

1. 13y 2. 13y 3. 13x 4. 13x

55. The equation of the latus rectum of the hyperbola 14491622 xy are

1. 5y 2. 5x 3. 5y 4. 5x

56. The length of the latus rectum of the hyperbola 149

22

yx

is

1. 3

4 2.

3

8 3.

2

3 4.

4

9

57. The eccentricity of the hyperbola 1259

22

xy

is

1. 3

34 2.

3

5 3.

3

34 4.

5

34

58. The centre of the hyperbola 400162522 yx are

1. )4,0( 2. )5,0( 3. )5,4( 4. )0,0(

59. The foci of the hyperbola 1259

22

xy

are

1. )34,0( 2. )0,34( 3. )34,0( 4. )0,34(

60. The vertices of the hyperbola 400162522 yx are

1. )4,0( 2. )0,4( 3. )5,0( 4. )0,5(

61. The equation of the tangent at (3, -6) to the parabola xy 122 is

1. 03 yx 2. 03 yx 3. 03 yx 4. 03 yx

62. The equation of the tangent at (-3, 1) to the parabola yx 92 is

1. 0323 yx 2. 0332 yx 3. 0332 yx 4. 0323 yx

63. The equation of chord of contact of tangents from the point (-3, 1) to the parabola xy 82 is

1. 0124 yx 2. 0124 yx 3. 0124 xy 4. 0124 xy

64. The equation of chord of contact of tangents from the point (2, 4) to the ellipse 205222 yx is

1. 055 yx 2. 055 yx 3. 055 yx 4. 055 yy

65. The equation of chord of contact of tangents from the point (5, 3) to the hyperbola 246422 yx is

1. 012109 yx 2. 012910 yx 3. 012109 yx 4. 012910 yx

66. The combined equation of the asymptotes to the hyperbola 900253622 yx is

1. 0362522 xx 2. 02536

22 yx 3. 02536

22 yx 4. 03625

22 yx

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67. The angle between the asymptotes of the hyperbola 2782422 yx is

1. 3

2.

3

2

3

or 3.

3

2 4.

3

2

68. The point of contact of the tangent cxmy and the parabola axy 42 is

1.

m

a

m

a 2,

2 2.

m

a

m

a,

2

2 3.

2

2,

m

a

m

a 4.

m

a

m

a 2,

2

69. The point of contact of the tangent cxmy and the ellipse 12

2

2

2

b

y

a

x is

1.

c

ma

c

b22

, 2.

c

b

c

ma22

, 3.

c

b

c

ma22

, 4.

c

b

c

ma22

,

70. The point of contact of the tangent cxmy and the hyperbola 12

2

2

2

b

y

a

x is

1.

c

b

c

ma22

, 2.

c

b

c

ma22

, 3.

c

b

c

ma22

, 4.

c

b

c

am22

,

71. The true statements of the following are a. Two tangents and 3 normals can be drawn to a parabola from a point. b. Two tangents and 4 normals can be drawn to an ellipse from a point. c. Two tangents and 4 normals can be drawn to an hyperbola from a point. d. Two tangents and 4 normals can be drawn to an rectangular hyperbola from a point. 1. a, b, c and d 2. a, b only 3. c, d only 4. a, b and c

72. If ''1

t , ''2

t are the extremities of any focal chord of a parabola axy 42 then ;

21tt is

1. 1 2. 0 3. 1 4. 2

1

73. The normal at ''1

t on the parabola axy 42 meets the parabola at ''

2t then

1

1

2

tt is

1. 2

t 2. 2

t 3. 21

tt 4. 2

1

t

74. The condition that the line 0 nmylx may be a normal to the ellipse 12

2

2

2

b

y

a

x is

1. 02223

nmalmal 2. 2

222

2

2

2

2)(

n

ba

m

b

l

a

3. 2

222

2

2

2

2)(

n

ba

m

b

l

a 4.

2

222

2

2

2

2)(

n

ba

m

b

l

a

75. The condition that the line 0 nmylx may be a normal to the hyperbola 12

2

2

2

b

y

a

x is

1. 02223

nmalmal 2. 2

222

2

2

2

2)(

n

ba

m

b

l

a

3. 2

222

2

2

2

2)(

n

ba

m

b

l

a 4.

2

222

2

2

2

2)(

n

ba

m

b

l

a

76. The condition that the line 0 nmylx may be a normal to the parabola axy 42 is

1. 02223

nmalmal 2. 2

222

2

2

2

2)(

n

ba

m

b

l

a

3. 2

222

2

2

2

2)(

n

ba

m

b

l

a 4.

2

222

2

2

2

2)(

n

ba

m

b

l

a

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77. The chord of contact of tangents from any point on the directrix of the parabola axy 42 passes through

its 1. vertex 2. focus 3. directrix 4. latus rectum

78. The chord of contact of tangents from any point on the directrix of the ellipse 12

2

2

2

b

y

a

xpasses through


79. The chord of contact of tangents from any point on the directrix of the hyperbola 12

2

2

2

b

y

a

x passes

through its 1. vertex 2. focus 3. directrix 4. latus rectum

80. The point of intersection of tangents at ''1

t and ''2

t to the parabola axy 42 is

1. )),((2121

tattta 2. ))(,(2121

ttatta 3. )2,(2

atat 4. ))(,(2121

ttatta

81. If the normal to the R.H. 2cxy at ''

1t meets the curve again at ''

2t then

2

3

1tt

1. 1 2. 0 3. -1 4. -2

82. The locus of the point of intersection of perpendicular tangents to the parabola axy 42 is

1. latus rectum 2. Directrix 3. tangent at the vertex 4. axis of the parabola

83. The locus of the foot of perpendicular from the focus on any tangent to the ellipse 12

2

2

2

b

y

a

xis

1. 2222bayx 2. 222

ayx 3. 2222bayx 4. 0x

84. The locus of the foot of perpendicular from the focus on any tangent to the hyperbola 12

2

2

2

b

y

a

x is

1. 2222bayx 2. 222

ayx 3. 2222bayx 4. 0x

85. The locus of the foot of perpendicular from the focus on any tangent to the parabola axy 42 is

1. 2222bayx 2. 222

ayx 3. 2222bayx 4. 0x

86. The locus of the point of intersection of perpendicular tangents to the ellipse 12

2

2

2

b

y

a

xis

1. 2222bayx 2. 222

ayx 3. 2222bayx 4. 0x

87. The locus of the point of intersection of perpendicular tangents to the hyperbola 12

2

2

2

b

y

a

x is

1. 2222bayx 2. 222

ayx 3. 2222bayx 4. 0x

88. The condition that the line 0 nmylx may be a tangent to the parabola axy 42 is

1. 22222nmbla 2. ln

2am 3. 22222

nmbla 4. 224 nlmc

89. The condition that the line 0 nmylx may be a tangent to the ellipse 12

2

2

2

b

y

a

xis

1. 22222nmbla 2. ln

2am 3. 22222

nmbla 4. 224 nlmc

90. The condition that the line 0 nmylx may be a tangent to the hyperbola 12

2

2

2

b

y

a

x is

1. 22222nmbla 2. ln

2am 3. 22222

nmbla 4. 224 nlmc

91. The condition that the line 0 nmylx may be a tangent to the rectangular hyperbola 2cxy is

1. 22222nmbla 2. nlam

2 3. 22222nmbla 4. 22

4 nlmc

92. The foot of a perpendicular from a focus of the hyperbola on an asymptote lies on the 1. centre 2. corresponding directrix 3. vertex 4. L.R.

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(One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Let „h' be the height of the tank. Then the rate of change of pressure „p‟ of the tank with respect to height

is

1. dt

dh 2.

dt

dp 3.

dp

dh 4.

dh

dp

2. If the temperature C

of the certain metal rod of „ l ‟ metres is given by 20000004.000005.01 l

then the rate of change of „ l ‟in m/

C when the temperature is 100 C

is

1. 0.00013 m/

C 2. 0.00023 m/

C 3. 0.00026 m/

C 4. 0.00033 m/

C

3. The following graph gives the functional relationship between distance and time of a moving car in m/sec. The speed of the car is

1. 2. smx

t/ 3. sm

dt

dx/ 4. sm

dx

dt/

4. The distance-time relationship of a moving body is given by y = F(t), then the acceleration of the body is the

1. gradient of the velocity/time graph 2. gradient of the distance/time graph 3. gradient of the acceleration/time graph 4. gradient of the velocity/distance graph 5. The distance travelled by a car in „t‟ seconds is given by x = 3t3-2t2+4t-1. Then the initial velocity and initial

acceleration respectively are

1. (-4m/s, 4m/2

s ) 2. (4m/s, -4m/2

s ) 3. (0, 0) 4. (18.25m/s, 23m/2

s )

6. The angular displacement of a fly wheel in radians is given by 3229 tt . The time when the angular

acceleration zero is 1. 2.5s 2. 3.5s 3. 1.5s 4. 4.5s

7. Food pockets were dropped from an helicopter during the flood and distance fallen in „t‟ seconds is given by

2

2

1gty (g = 9.8 m/

2s ).Then the speed of the food pocket after it has fallen for „2‟ seconds is

1. 19.6 m/sec 2. 9.8 m/sec 3. -19.6 m/sec 4. -9.8 m/sec

8. An object dropped from the sky follows the law of motion 2

2

1gtx (g = 9.8 m/

2s ). The acceleration of the

object when t = 2 is 1. -9.8 m/sec2 2. 9.8 m/sec2 3. 19.6 m/sec2 4. -19.6 m/sec2 9. A missile fired from ground level rises x metres vertically upwards in „t‟ seconds and x = t(100-12.5t). Then the maximum height reached by the missile is

1. 100m 2. 150 m 3. 250 m 4. 200 m

10. A continuous graph y = f (x) is such that )(' xf as 1xx at

11, yx . Then )( xfy has a

1. vertical tangent y = 1x 2. horizontal tangent x = 1

x

3. vertical tangent x = 1x 4. horizontal tangent y =

1y

11. The curve )( xfy and )( xgy cut orthogonally if at the point of intersection

1. slope of f(x) = slope of g(x) 2. slope of f(x) + slope of g(x) = 0 3.. slope of f(x) / slope of g(x) = -1 4. [slope of f(x)][slope of g(x)] = -1

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12. The law of the mean can also be put in the form

1. 10),(')()( hahfafhaf 2. 10),(')()( hahfafhaf

3. 10),(')()( hahfafhaf 4. 10),(')()( hahfafhaf

13. spitaloHl 'ˆ' rule cannot be applied to 3

1

x

x as 0x because f(x) = x+1 and g(x) = x+3are

1. not continuous 2. not differentiable

3. not in the indeterminate form as 0x 4. in the indeterminate form as 0x

14. If bxgax

)(lim and f is continuous at x = b then

1.

)())(( limlim xgfxfgaxax

2.

)())(( limlim xgfxgfaxax

3.

)())(( limlim xfgxgfaxax

4.

)())(( limlim xgfxgfaxax

15. x

x

x tanlim

0

is

1. 1 2. -1 3. 0 4.

16. f is a real valued function defined on an interval RI (R being the set of real numbers) increases on I.

Then

1. )()(21

xfxf whenever Ixxxx 2121

,, 2. )()(21


,,

3. )()(21


,, 4. )()(21


,,

17. If a real valued differentiable function y = f (x) defined on an open interval I is increasing then

1. 0dx

dy 2. 0

dx

dy 3. 0

dx

dy 4. 0

dx

dy

18. f is differentiable function defined on an interval I with positive derivative. Then f is 1. increasing on I 2. decreasing on I 3. strictly increasing on I 4. strictly decreasing on I 19. The function f (x) = x3 is 1. increasing 2. decreasing 3. strictly decreasing 4. strictly increasing 20. If the gradient of a curve changes from positive just before P to negative just after then “P” is a 1. minimum point 2. maximum point 3. inflection point 4. discontinuous point 21. The function f (x) = x2 has 1. a maximum value at x=0 2. minimum value at x=0 3. finite no. of maximum values 4. infinite no. of maximum values 22. The function f(x) = x3 has 1. absolute maximum 2. absolute minimum 3. local maximum 4. no extrema

23. If f has a local extremum at a and if )(' af exists then

1. )(' af <0 2. )(' af >0 3. )(' af =0 4. )(" af =0

24. In the following figure, the curve y = f(x) is

1. concave upward 2. convex upward 3. changes from concavity to convexity 4. changes from convexity and concavity

25. The point that separates the convex part of a continuous curve from the concave part is 1. the maximum point 2. the minimum point 3. the inflection point 4. critical point

26. f is a twice differentiable function on an interval I and if 0)('' xf for all x in the domain I of f , then f is

1. concave upward 2.convex upward 3. increasing 4. decreasing

Y

X

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27. 0

xx is a root of even order for the equation )(' xf = 0, then 0

xx is a

1. maximum point 2. minimum point 3. inflection point 4. critical point

28. If 0

x is the x-coordinate of the point of inflection of a curve y = f(x), then (second derivative exists)

1. 0)(0xf 2. 0)('

0xf 3. 0)(''

0xf 4. 0)(''

0xf

29. The statement “If f is continuous on a closed interval [a, b], then f attains an absolute maximum value

)(cf and an absolute minimum value )(df at some number c and d in [a, b] ” is

1. The extreme value theorem 2. Fermat‟s theorem 3. Law of mean 4. Rolle‟s theorem

30. The statement: “ If f has a local extremum (minimum or maximum) at c and if )(' cf exists then 0)(' cf

” is 1. the extreme value theorem 2. Fermat‟s theorem 3. Law of mean 4. Rolle‟s theorem 31. Identify the false statement: 1. all the stationary numbers are critical numbers 2. at the stationary point the first derivative is zero 3. at critical numbers the first derivative need not exist 4. all the critical numbers are stationary numbers 32. Identify the correct statement: a. a continuous function has local maximum then it has absolute maximum b. a continuous function has local minimum then it has absolute minimum c. a continuous function has absolute maximum then it has local maximum d. a continuous function has absolute minimum then it has local minimum 1. a and b 2. a and c 3. c and d 4. a, c and d 33. Identify the correct statements: a. Every constant function is an increasing function b. Every constant function is a decreasing function c. Every identity function is an increasing function

d. Every identity function is a decreasing function 1. a, b and c 2. a and c 3. c and d 4. a, c and d

34. Which of the following statement is incorrect? 1. Initial velocity means velocity at t = 0 2. Initial acceleration means acceleration at t = 0 3. If the motion is upward, at the maximum height, the velocity is not zero 4. If the motion is horizontal, v = 0 when the particle comes to rest

35. Which of the following statements are correct (1

m and 2

m are slopes of two lines)

a. If the two lines are perpendicular then 1

m2

m = -1

b. If 1

m2

m = -1, then two lines are perpendicular

c. If 1

m = 2

m , then the two lines are parallel

d. If 2

1

1

mm

then the two lines are perpendicular

1. b, c and d 2. a, b and d 3. c and b 4. a and b 36. One of the conditions of Rolle‟s theorem is

1. f is defined and continuous on (a, b) 2. f is differentiable on [a, b]

3. )()( bfaf 4. f is differentiable on ],( ba

37. If „a‟ and „b‟ are two roots of a polynomial 0)( xf , then Rolle‟s theorem says that there exists atleast

1.one root between a and b for 0)(' xf

2. two roots between a and b for 0)(' xf

3. one root between a and b for 0)('' xf

4. two roots between a and b for 0)('' xf

38. A real valued function which is continuous on [a, b] and differentiable on (a, b) then there exists atleast one c in

1. [a, b] such that 0)(' cf 2. (a, b) such that 0)(' cf

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3. (a, b) such that 0)()(

ab

afbf 4. (a, b) such that )('

)()(cf

ab

afbf

39. In the law of mean, the value '' satisfies the condition

1. 0 2. 0 3. 1 4. 10

40. Which of the statements are correct? a. Rolle‟s theorem is a particular case of Lagranges law of mean b. Lagranges law of mean is a particular case of generalized law of mean(Cauchy) c. Lagranges law of mean is a particular case of Rolle‟s theorem d. Generalised law of mean is a particular case of Lagrange‟s law of mean (Cauchy) 1. b, c 2. c, d 3. a, b 4. a, d --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (*** Question For Full Test)

-------------------------------------------------------------------------------------------------------------------------------------------------------- 1. For the function y=x3+2x2 the value of dy when x=2 and dx = 0.1 is 1. 1 2. 2 3. 3 4. 4

2. If yxyxyxU22234

33 then x

u

is

1. xyxyx 66423 2. 224

363 xyyxx 3. 223664 xyyxx 4. xyyxx 364

223

3. If ),( yxfu then with usual notations, yxxy

uu if

1. u is continuous 2. x

u is continuous 3. y

u is continuous 4. yx

uuu ,, are continuous

4. If ),( yxfu is a differentiable function of x and y ; x and y are differentiable functions of „t‟ then

1. t

y

y

f

t

x

x

f

dt

du

.. 2.

t

y

y

t

dt

dx

x

f

dt

du

..

3. dt

dy

y

f

dt

dx

x

f

dt

du..

4.

t

y

y

f

t

x

x

f

t

u

..

5. If ),( yxf is a homogeneous functions of degree n then y

fy

x

fx

=

1. f 2. nf 3. fnn )1( 4. fnn )1(

6. If yxyxyxyxu22234

33),( then yx

u

2

is

1. xxy 612 2. xxy 612 3. xyx 6122

4. xxy 6122


33),( then

xy

u2

1. xxy 612 2. xxy 612 3. xyx 6122

4. xxy 6122


33),( then

2

2

x

u

1/ 222

363 xyxy 2. 2

66 xy 3. xyx 6122

4. yyx 661222


33),( then

2

2

y

u

1. 2

66 xy 2. xxy 612 3. xyx 6122

4. 222

363 xyxy

10. The differential on y of the function 4xy is

1. 4/3

4

1 x 2. dxx

4/3

4

1 3. dxx

4/3 4. 0

11. The differential of y if 5xy is

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1. 4

5 x 2. dxx4

5 3. dxx5

5 4. 5

5 x

12. The differential of y if 124 xxy is

1. dxxx 2

1

3)24(

2

1

2. dxxxxx )24()1(2

1 32

1

24

3. 2

1

3)24(

2

1

xx 4. )24()1(2

1 32

1

24xxxx

13. The differential of y if 32

2

x

xy is

1. dxx

2)32(

7

2. dx

x2

)32(

1

3. dx

x2

)32(

7

4.

2)32(

7

x

14. The differential of y if y = sin2x is

1. .2cos2 x 2. dxx.2cos2 3. dxx.2cos2 4. dxx.2cos

15. The differential of xx tan is

1. )tansec(22

xxx 2. dxxxx )tansec(2

3. xdxx2

sec 4. dxxxx )tansec(2


33),( then y

u

is

1. 22363 xxyy 2. 222

363 xxyy 3. 222363 xyxy 4. 2222

363 xyxy

17. The curve )1(222

xxy is defined only for

1. 2x and 2x 2. 1x and 1x 3. 1x and 1x 4. 1x and 1x


xxy is symmetrical about

1. x-axis only 2. y-axis only 3. x and y axes only 4. x,y axes and the origin


xxy has

1. only one loop between x=0 and x=1 2. two loops between x=-1 and x=0 3. two loops between x=-1 and 0; 0 and 1 4. no loop


xxy has

1. an asymptote x = -1 2. an asymptote x= 1 3. two asymptotes x =1 and x = -1 4. no asymptote

21. The curve )6()2(22

xxxy exists for

1. 62 x 2. 62 x 3. 62 x 4. 62 x

22. The x-intercept of the curve )6()2(22

xxxy is

1. 0 2. 6, 0 3. 2 4. 2

23. An asymptote to the curve )6()2(22

xxxy is

1. 2x 2. 2x 3. 6x 4. 6x

24. The curve )6()2(22

xxxy has

1. only one loop between x = 0 and x = 6 2. two loops between x = 0 and x = 6 3. only one loop between x = -2 and x = 6 4. two loops between x= -2 and x=6

25. The curve )1(22

xxy is defined only for

1. 1x 2. 1x 3. 1x 4. 1x

26. The curve )1(22

xxy is symmetrical about

1. y-axis only 2. x-axis only 3. both the axes 4. origin only

27. The curve )1(22

xxy has

1. an asymptote y =0 2. an asymptote x = 1 3. an asymptote y = 1 4. no asymptote

28. The curve )1(22

xxy has

1. only one loop between x = -1 and x = 0 2. only one loop between x= 0 and x=1 3. two loops between x = -1 and x = 1

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4. no loop

29. The curve 22)()( bxaxy , 0, ba and ba does not exist for

1. ax 2. bx 3. axb 4. ax

30. The curve 22)()( bxaxy is symmetrical about

1. origin only 2. y-axis only 3. x-axis only 4. both x and y axis

31. The curve 22)()( bxaxy , has 0, ba and ba

1. an asymptote x = a 2. an asymptote x = b 3. an asymptote y = a 4. no asymptotes

32. The curve 0,,)()(22

babxaxy and ba has

1. a loop between x = a and x = b 2. two loops between x = a and x = b 3. two loops between x=0 and x= a 4. no loop

33. The curve )1()1(22

xxxy is defined for

1. 11 x 2. 11 x 3. 11 x 4. 11 x

34. The curve )1()1(22

xxxy is symmetrical about

1. both the axes 2. origin only 3. y-axis only 4. x-axis only

35. The asymptote to the curve )1()1(22

xxxy is

1. 1x 2. 1y 3. 1y 4. 1x

36. The curve )1()1(22

xxxy has

1. a loop between x = -1 and x = 1 2. a loop between x = -1 and x = 0 3. a loop between x = 0 and x = 1 4. no loop

37. The curve )(22222

xaxya is defined for

1. ax and ax 2. ax and ax

3. ax and ax 4. ax and ax

38. The curve )(22222

xaxya is symmetrical about

1. x-axis only 2. y-axis only 3. both the axes 4. both the axes and origin

39. The curve )(22222

xaxya has

1. an asymptote x = a 2. an asymptote x = -a 3. an asymptote x = 0 4. no asymptote

40. The curve )(22222

xaxya has

1. a loop between x = a and x = -a 2. two loops between x = -a and x = 0; x = 0 and x = a 3. two loops between x = 0 and x = a 4. no loop

41. The curve 22)2()1( xxy is not defined for

1. 1x 2. 2x 3. 2x 4. 1x

42. The curve 22)2()1( xxy is symmetrical about

1. both x and y axes 2. x-axis only 3. y-axis only 4. both the axes and origin

43. The curve 22)2()1( xxy has

1. an asymptote x=1 2. an asymptote x=2 3. two asymptotes x=1 and x=2 4. no asymptote

44. The curve 22)2()1( xxy has

1. two loops between x= 0 and x=2 2. one loop between x= 0 and x=1 3. one loop between x=1 and x= 2 4. no loop --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 7. INTEGRAL CALCULUS (One Question For Full Test)

--------------------------------------------------------------------------------------------------------------------------------------------------------

1. If xdxIn

nsin , then

nI

1. 2

1 1cossin

1

n

nI

n

nxx

n 2.

2

1 1cossin

1

n

nI

n

nxx

n

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3. 2

1 1cossin

1

n

nI

n

nxx

n 4.

n

nI

n

nxx

n

1cossin

1 1

2. dxxf

a

)(

2

0

= ,)(2

0

dxxf

a

if

1. f(2a-x) = f(x) 2. f(a-x) = f(x) 3. f(x) = -f(x) 4. f(-x) = f(x)

3. dxxf

a

)(

2

0

= 0 , if

1. f(2a-x) = f(x) 2. f(2a-x) = -f(x) 3. f(x) = -f(x) 4. f(-x) = f(x)

4. If f(x) is an odd function then dxxf

a

a

)(

is

1. dxxf

a

)(2

0

2. dxxf

a

)(

0

3.0 4.

a

dxxaf

0

)(

5. dxxf

a

)(

0

+ dxxaf

a

)2(

0

=

1. dxxf

a

)(

0

2. 2 dxxf

a

)(

0

3. dxxf

a

)(

2

0

4. dxxaf

a

)(

2

0

6. If f(x) is even then dxxf

a

a

)(

is

1. 0 2. 2 dxxf

a

)(

0

3. dxxf

a

)(

0

4. -2 dxxf

a

)(

0

7. dxxf

a

)(

0

is

1. dxaxf

a

)(

0

2. dxxaf

a

)(

0

3. dxxaf

a

)2(

0

4. dxaxf

a

)2(

0

8. dxxf

b

a

)( is

1.

a

dxxf

0

)(2 2.

b

a

dxxaf )( 3.

b

a

dxxbf )( 4.

b

a

dxxbaf )(

9. If n is a positive integer , then dxexaxn

0

=

1. n

a

n 2.

na

n 1 3.

1

1

na

n 4.

1

na

n

10. If n is odd, then xdxn

cos

2

0

is

1. 25

4

3

2

1

n

n

n

n

n

n 2.

22

1

4

5

2

31

n

n

n

n

n

n

3. 12

3

5

4

3

2

1

n

n

n

n

n

n 4. 1

3

2

4

5

2

31

n

n

n

n

n

n

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11. If n is even, then xdxn

sin

2

0

is

1. 25

4

3

2

1

n

n

n

n

n

n 2.

22

1

4

5

2

31

n

n

n

n

n

n

3. 12

3

5

4

3

2

1

n

n

n

n

n

n 4. 1

3

2

4

5

2

31

n

n

n

n

n

n

12. If n is even, then xdxn

cos

2

0

is

1. 25

4

3

2

1

n

n

n

n

n

n 2.

22

1

4

5

2

31

n

n

n

n

n

n

3. 12

3

5

4

3

2

1

n

n

n

n

n

n 4. 1

3

2

4

5

2

31

n

n

n

n

n

n

13. If n is odd, then

2/

0

sin

dxxn

1. 25

4

3

2

1

n

n

n

n

n

n 2.

22

1

4

5

2

31

n

n

n

n

n

n

3. 12

3

5

4

3

2

1

n

n

n

n

n

n 4. 1

3

2

4

5

2

31

n

n

n

n

n

n

14. dxxf

b

a

)( is

1. - dxxf

b

a

)( 2. dxxf

a

b

)( 3. - dxxf

a

)(

0

4. 2 dxxf

b

)(

0

15. The area bounded by the curve x = g(y) to the right of y-axis and the two lines y=c and y=d is given by

1. xdx

d

c

2. xdy

a

c

3. ydy

d

c

4. xdy

d

c

16. The area bounded by the curve x = f(y), y-axis and the lines y=c and y=d is rotated about y-axis. Then the volume of the solid is

1. dyx

d

c

2

2. dxx

d

c

2

3. dxy

d

c

2

4. dyy

d

c

2

17. The area bounded by the curve x = f(y) to the left of y-axis and between the lines y=c and y=d is

1. xdy

d

c

2. - xdy

d

c

3. ydx

d

c

4. - ydx

d

c

18. The arc length of the curve y = f(x) from x=a to x=b is

1.

b

a

dxdx

dy2

1 2.

d

c

dxdy

dx2

1 3. 2

b

a

dxdx

dyy

2

1 4) dxdy

dxy

b

a

2

12

19. The surface area obtained by revolving the area bounded by the curve y= f(x), the two ordinates x=a, x=b

and x-axis, about x-axis is

1.

b

a

dxdx

dy2

1 2.

d

c

dxdy

dx2

1 3. 2

b

a

dxdx

dyy

2

1 4) dxdy

dxy

b

a

2

12

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20. dxexx45

0

is

1. 6

4

6 2.

54

6 3.

64

5 4.

54

5

21. dxxemx 7

0

is

1. m

m

7

2.

7

7

m

3.

17

m

m 4.

8

7

m

22. dxex

x

26

0

is

1. 7

2

6 2.

62

6 3. 62

6 4. 62

7

23. If xdxIn

ncos , then

nI

1. 2

1 1sincos

1

n

nI

n

nxx

n 2.

2

1 1sincos

n

nI

n

nxx

3. 2

1 1sincos

1

n

nI

n

nxx

n 4.

2

1 1sincos

1

n

nI

n

nxx

n


(One Question For Full Test) --------------------------------------------------------------------------------------------------------------------------------------------------------

1. The order and degree of the differential equation 7

53

2

2

3

3

y

dx

dy

dx

yd

dx

yd are

1. 3, 1 2. 1, 3 3. 3, 5 4. 2, 3

2. The order and degree of the differential equation dy

dxx

dx

dyy 34 are

1. 2, 1 2. 1, 2 3. 1, 1 4. 2,2


32

2

2

4

dx

dy

dx

yd are

1. 2, 1 2. 1, 2 3. 2, 4 4. 4,2

4. The order and degree of the differential equation 22')'1( yy are

1. 2, 1 2. 1, 2 3. 2, 2 4. 1, 1

5. The order and degree of the differential equation 2xy

dx

dy are

1. 1, 1 2. 1, 2 3. 2, 1 4. 0, 1

6. The order and degree of the differential equation xyy 2

' are

1. 2, 1 2. 1, 1 3. 1, 0 4. 0, 1

7. The order and degree of the differential equation 0'3"32 yyy are

1. 2, 2 2. 2, 1 3. 1, 2 4. 3, 1

8. The order and degree of the differential equation dx

dyyx

dx

yd

2

2

are

1. 2, 1 2. 1, 2 3. 2, 2

1 4. 2,2

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3

3

3

2

2

dx

yd

dx

dyy

dx

yd are

1. 2, 3 2. 3, 3 3. 3, 2 4. 2,2


2

3)'(" yyy are

1. 2, 3 2. 3, 3 3. 3, 2 4. 2,2

11. The order and degree of the differential equation 22)"()"(' yxyy are

1. 1, 1 2. 1, 2 3. 2, 1 4. 2,2

12. The order and degree of the differential equation 22)"()"(' yxxyy are

1. 2, 2 2. 2, 1 3. 1, 2 4. 1, 1


2

xdy

dxx

dx

dy

are

1. 2, 2 2. 2, 1 3. 1, 2 4. 1, 3 14. The order and degree of the differential equation sinx(dx+dy) = cosx(dx-dy) are 1. 1, 1 2. 0, 0 3. 1, 2 4. 2, 1 15. The differential equation corresponding to xy = c2 where c is an arbitrary constant, is

1. 0" xxy 2. 0" y 3. 0' yxy 4. 0" xxy

16. In finding the differential equation corresponding to y = emx where m is the arbitrary constant, then m is

1. 'y

y 2.

y

y ' 3. 'y 4. y

17. The solution of a linear differential equation QPxdy

dx where P and Q are functions of y is

1. cQdxFIFIy .. 2. cQdyFIFIx ..

3. cQdyFIFIy .. 4. cQdxFIFIx ..

18. The solution of a linear differential equation QPydx

dy where P and Q are functions of x is

1. cQdxFIFIy .. 2. cQdyFIFIx ..

3. cQdyFIFIy .. 4. cQdxFIFIx ..

19. Identify the incorrect statement. 1. The order of a differential equation is the order of the highest order derivative occurring in it.

2. The degree of the differential equation is the degree of the highest order derivative which occurs in it. (The derivatives are free from radicals and fractions).

3. ),(

),(

2

1

yxf

yxf

dx

dy is the first order first degree homogeneous differential equation.

4. x

exydx

dy is a linear differential equation in x.


(One Question For Full Test) -------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Which of the following are statements? (i) Chennai is the capital of TamilNadu. (ii) The earth is a planet. (iii) Rose is a flower. (iv) Every triangle is an isosceles triangle. 1. all 2. (i) and (ii) 3. (ii) and (iii) 4. (iv) only

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2. Which of the following are not statements? (i) Three plus four is eight. (ii) The sun is a planet. (iii) Switch on the light. (iv) Where are you going? 1. (i) and (ii) 2. (ii) and (iii) 3. (iii) and (iv) 4. (iv) only 3. The truth values of the following statements are (i) Ooty is in Tamilnadu and 3+4 = 8 (ii) Ooty is in Tamilnadu and 3+4 = 7 (iii) Ooty is in Kerala and 3+4 = 7 (iv) Ooty is in Kerala and 3+4 = 8 1. FTFF 2. FFFT 3. TTFF 4.TFTF 4. The truth values of the following statements are

(i) Chennai is in India or 2 is an integer.

(ii) Chennai is in India or 2 is an irrational number.

(iii) Chennai is in China or 2 is an integer.

(iv) Chennai is in China or 2 is an irrational number.

1. TFTF 2. TFFT 3. FTFT 4. TTFT 5. Which of the following are not statements? (i) All natural numbers are integers. (ii) A square has five sides. (iii) The sky is blue. (iv) How are you? 1. (iv) only 2. (i) and (iv) 3. (i), (ii), (iii) 4. (iii) and (iv) 6. Which of the following are statements? (i) 7+2<10. (ii) The set of rational numbers is finite. (iii) How beautiful you are! (iv) Wish you all success. 1. (iii), (iv) 2. (i), (ii) 3. (i), (iii) 4. (ii), (iv) 7. The truth values of the following statements are (i) All the sides of a rhombus are equal in length.

(ii) 1+ 19 is an irrational number.

(iii) Milk is white. (iv) The number 30 has four prime factors. 1. TTTF 2. TTTT 3. TFTF 4. FTTT 8. The truth values of the following statements are (i) Paris is in France. (ii) sinx is an even function. (iii) Every square matrix is non-singular. (iv) Jupiter is a planet. 1. TFFT 2. FTFT 3. FTTF 4.FFTT 9. Let p be “Kamala is going to school” and q be “There are twenty students in the class”.“Kamala is not

going to school or there are twenty students in the class” stands for

1. qp 2. qp 3. p 4. qp

10. If p stands for the statement “Sita likes reading” and q for the statement “Sita likes playing”.“Sita likes neither reading nor playing” stands for

1. qp 2. qp 3. qp 4. qp

11. If p is true and q is unknown then

1. p is true 2. )( pp is false 3. )( pp is true 4. qp is true

12. If p is true and q is false then which of the following is not true? 1. qp is false 2. qp is true 3. qp is false 4. qp is true

13. Which of the following is not true? 1. Negation of a negation of a statement is the statement itself. 2. If the last column of its truth table contains only T then it is tautology. 3. If the last column of its truth table contains only F then it is contradiction. 4. If p and q are any two statements then qp is a tautology.

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14. Which of the following are binary operations on R? a. a*b = min{a, b} b. a*b = max{a, b} c. a*b = a d. a*b = b 1. all 2. a, b and c 3. b,c and d 4. c, d 15. „+‟ is not a binary operation on 1. N 2. Z 3. C 4. Q-{0} 16. „-‟ is a binary operation on 1. N 2. Q-{0} 3. R-{0} 4. Z 17. „ ‟ is a binary operation on 1. N 2. R 3. Z 4. C-{0}

18. In congruence modulo5, },25/{ zkkxzx represents

1. 0 2. 5 3. 7 4. 2 .

19. 11512 is

1. 55 2. 12 3. 7 4. 11

20. 738

is

1. 10 2. 8 3. 5 4. 2

21. In the group (G, .), G = {1, -1, i, -i} the order of -1 is 1. -1 2. 1 3. 2 4. 0 22. In the group (G, .), G = {1, -1, i, -i} the order of -i is 1. 2 2. 0 3. 4 4. 3

23. In the group (G, .), G = },,1{2

, is cube root of unity, then O(2

) is

1. 2 2. 1 3. 4 4. 3

24. In the group ),(44

Z , order of [0] is

1.1 2. 3. can‟t be determined 4. 0


Z , )]3[(O is

1.4 2. 3 3. 2 4. 1

26. In ,S , Syxxyx ,, then „ ‟ is

1. only associative 2. only commutative 3. associative and commutative 4. neither associative nor commutative 27. In (N, *), x*y = max{x, y}, x, y N, then (N, *) is 1. only closed 2. only semi group 3. only monoid 4. a group 28. The set of positive even integers, with usual multiplication forms 1. a finite group 2. only a semi group 3. only a monoid 4. an infinite group 29. The set of positive even integers, with usual addition forms 1. a finite group 2. only a semi group 3. only a monoid 4. an infinite group

30. In the group ),0(55Z , ]3[O is

1. 5 2. 3 3. 4 4. 2 31. In the group (G, .), G = {1, -1, i, -i} the order of 1 is 1. 2 2. 0 3. 4 4. 1 32. In the group (G, .), G = {1, -1, i, -i} the order of i is 1. 2 2. 0 3. 4 4. 3

33. In the group (G, .), G = },,1{2

, is cube root of unity, then O( ) is

1. 2 2. 1 3. 4 4. 3

34. In the group (G, .), G = },,1{2

, is cube root of unity, then O(1) is

1. 2 2. 1 3. 4 4. 3


Z , order of O( [1]) is

1.1 2. 3. cannot be determined 4. 4


Z ,order of O([2]) is

1.1 2. 2 3.cannot be determined 4. 0


1. 5 2. 3 3. 4 4. 2

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1. 5 2. 3 3. 4 4. 2


1. 1 2. 2 3. 3 4. 4 --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 10. PROBABILITY DISTRIBUTIONS (One Question For Full Test)

-------------------------------------------------------------------------------------------------------------------------------------------------------- 1. A discrete random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of value 2. A continuous random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of values 3. If X is a discrete random variable then P(X a) = 1. P(X<a) 2. 1 – P(X a) 3. 1 – P(X<a) 4. 0 4. If X is a continuous random variable then P(X a) = 1. P(X<a) 2. 1 – P(X>a) 3. P(X>a) 4. 1-P(X a-1) 5 If X is a continuous random variable then P(a<X<b) = 1. P(a X b) 2. P(a<X b) 3. P(a X<b) 4. all the three above 6. A continuous random variable X has probability density function „f(x)‟ then

1. 1)(0 xf 2. 0)( xf 3. 1)( xf 4. 1)(0 xf

7. A discrete random variable X has probability mass function p(x), then

1. 1)(0 xp 2. 0)( xp 3. 1)( xp 4. 1)(0 xp

8. Mean and variance of binomial distribution are

1. nq, npq 2. np, npq 3. np, np 4. np, npq

9. Which of the following is or are correct regarding normal distribution curve ? a. symmetrical about the line X = (mean)

b. Mean = median = mode c. unimodal d. Points of inflection are at X =

1. a, b only 2. b,d only 3. a.b,c only 4. all 10. For a standard normal distribution the mean and variance are

1. 2, 2. , 3. 0, 1 4. 1, 1

11. The probability density function of the standard normal variate Z is )( z =

1.

2

2

1

2

1 z

e

2.

2

2

1 z

e

3.

2

2

1

2

1 z

e

4.

2

2

1

2

1 z

e

12. If X is a discrete random variable then which of the following is correct ?

1. 1)(0 xF 2. 1)(0)( FandF

3. )(][1

nnn

xFxFxXP 4. F(x) is a constant function

13. If X is a continuous random variable then which of the following is incorrect ?

1. )()(' xfxF 2. 0)(1)( FandF

3. )()(][ aFbFbxaP 4. )()(][ aFbFbxaP


(i) E(aX+b) = aE(X)+b (ii) 2

122)'('

(iii) iancevar2 (iv) var(aX+b) = a2var(X)

1. all 2. (i), (ii), (iii) 3. (ii), (iii) 4. (i), (iv)

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15. Which of the following is not true regarding the normal distribution? 1. skewness is zero. 2. mean = median = mode

3. the points of inflection are at X =

4. maximum height of the curve is 2

1

*****************

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COME BOOK ONE MARK KEY

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 1 2 2 4 2 4 4 4 1 2

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 3 2 4 1 3 2 4 3 1 2

Q.NO 21 22 23 ANS 2 1 4

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 1 1 1 3 3 4 4 4 3 1

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 3 1 2 4 1 2 1 1 1 4

Q.NO 21 22 23 24 25 26 27 28 29 30

ANS 2 1 1 3 1 4 3 1 4 1

Q.NO 31 32 33 34 35 36 37 38 39

ANS 2 1 4 1 1 1 1 4 3

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 1 2 2 1 3 1 1 2 4 3

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 4 3 3 4 1 1 4 4 1 2

Q.NO 21 22 23 24 25 26 27 28 29 30

ANS 2 4 3 2 3 3 2 1 2 1

Q.NO 31 32 33 34 35 36 37 38 39 40

ANS 4 1 1 3 3 4 4 4 4 4

Q.NO 41 42 43 44 45 46

ANS 4 3 4 2 1 4

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 2 3 4 2 1 4 2 4 2 3

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 2 4 4 1 3 2 4 1 3 2

Q.NO 21 22 23 24 25 26 27 28 29 30

ANS 3 4 2 1 1 4 2 3 1 2

Q.NO 31 32 33 34 35 36 37 38 39 40

ANS 4 2 3 1 2 4 1 2 1 3

Q.NO 41 42 43 44 45 46 47 48 49 50

ANS 4 2 1 3 1 2 3 2 3 1

Q.NO 51 52 53 54 55 56 57 58 59 60

ANS 3 4 2 4 1 2 3 4 1 2

Q.NO 61 62 63 64 65 66 67 68 69 70

ANS 4 3 1 3 4 2 3 1 2 3

Q.NO 71 72 73 74 75 76 77 78 79 80

ANS 4 1 1 3 4 1 2 2 2 2

Q.NO 81 82 83 84 85 86 87 88 89 90

ANS 3 2 2 2 4 3 1 2 1 3

Q.NO 91 92

ANS 4 2

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Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 4 1 1 1 2 3 1 2 4 3

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 4 2 3 2 1 1 2 3 4 2

Q.NO 21 22 23 24 25 26 27 28 29 30

ANS 2 4 3 1 3 1 3 3 1 2

Q.NO 31 32 33 34 35 36 37 38 39 40

ANS 4 1 1 3 1 3 1 4 4 3

Q.NO 41

ANS 1

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 2 1 4 3 2 1 1 4 1 2

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 2 2 3 2 4 3 2 4 3 4

Q.NO 21 22 23 24 25 26 27 28 29 30

ANS 1 2 2 1 1 2 4 2 3 3

Q.NO 31 32 33 34 35 36 37 38 39 40

ANS 4 4 2 4 4 3 1 4 4 2

Q.NO 41 42 43 44

ANS 4 2 4 3

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 1 1 2 3 3 2 2 4 4 4

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 2 2 4 2 4 1 2 1 3 3

Q.NO 21 22 23

ANS 4 4 4

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 1 2 3 4 1 2 2 4 2 1

Q.NO 11 12 13 14 15 16 17 18 19

ANS 3 1 4 1 3 2 2 1 4

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 1 3 1 4 1 2 1 1 4 1

Q.NO 11 12 13 14 15 16 17 18 19 20

ANS 4 4 4 1 4 4 4 4 3 4

Q.NO 21 22 23 24 25 26 27 28 29 30

ANS 3 3 4 1 1 1 3 2 2 3

Q.NO 31 32 33 34 35 36 37 38 39

ANS 4 3 4 2 4 2 3 4 1

Q.NO 1 2 3 4 5 6 7 8 9 10

ANS 4 2 3 3 4 2 1 4 4 3

Q.NO 11 12 13 14 15

ANS 4 3 4 1 4

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BOOK BACK – ONE MARK [30 Number Of One Mark Questions to be Asked For Full Test]


(THREE QUESTIONS FOR FULL TEST) --------------------------------------------------------------------------------------------------------------------------------------------------------


844

422

211

is

1)1 2) 2 3)3 4)4

2. The rank of the diagonal matrix

0

4

0

2

1

is

1) 0 2)2 3)3 4)5 3. If A = [2 0 1], then rank of AAT is 1)1 2)2 3)3 4)0

4. If A =

3

2

1

then the rank of AAT is

1)3 2)0 3)1 4)2

5. If the rank of the matrix

01

10

01

is 2, then is

1)1 2)2 3)3 4) any real number 6. If A is a scalar matrix with scalar k≠0, of order 3, then A-1 is

1) Ik

2

1 2) I

k3

1 3) I

k

1 4) kI

7. If the matrix

541

31

231

k has an inverse then the values of k

1) k is any real number 2) k = -4 3) k≠ -4 4) k≠4

8. If A=

43

12,then (adj A)A =

1)

5

10

05

1

2)

10

01 3)

50

05 4)

50

05

9. If A is a square matrix of order n then isAadj

1)2

A 2) n

A 3) 1n

A 4) A

10. The inverse of the matrix

001

010

100

is

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1)

100

010

001

2)

001

010

100

3)

001

010

100

4)

100

010

001

11. If A is a matrix of order 3, then det(kA) 1) k3 det (A) 2) k2 det (A) 3) k det (A) 4) det (A) 12. If I is unit matrix of order n, where k≠0 is a constant, then adj(kI)= 1) kn(adj I) 2) k(adj I) 3) k2(adj I)) 4) kn-1(adj I) 13. If A and B are any two matrices such that AB=O and A is non-singular, then 1) B = О 2) B is singular 3) B is non-singular 4) B = A

14. If A =

50

00 , then A12 is

1)

600

00 2)

1250

00 3)

00

00 4)

10

01

15. Inverse of

25

13 is

1)

35

12 2)

31

52 3)

35

13 4)

21

53

16. In a system of 3 linear non-homogeneous equation with three unknowns, if andx

y 0 , z = 0 then the system has 1) unique solution 2) two solutions 3) infinitely many solutions 4) no solution 17. The system of equations ax + y + z = 0 ; x +by + z = 0 ; x + y + cz = 0 has a non-trivial solution then

cba 1

1

1

1

1

1

1)1 2)2 3)-1 4)0

18. If aex + bey = c ; pex +qey = d and 1 = qp

ba; 2 =

qd

bc; 3 =

dp

ca then the value of (x ,y) is

1)

1

3

1

2, 2)

1

3

1

2log,log 3)

2

1

3

1log,log

4)

3

1

2

1log,log

19. If the equation -2x + y + z = l ; x - 2y +z = m ; x + y -2z = n such that l+m+n = 0, then the system has 1) a non-zero unique solution 2) trivial solution 3) infinitely many solution 4) No solution --------------------------------------------------------------------------------------------------------------------------------------------------------

2. VECTOR ALGEBRA (FOUR QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------

20. If a

is a non-zero vector and m is a non-zero scalar then m a

is a unit vector if

1) m = ±1 2)a = m 3) a = m

1 4) a =1

21. If a

and b

are two unit vectors and is the angle between them, then ( ba

) is a unit vector if

1) =3

2) =

4

3) =

2

4) =

3

2

22. If a

and b

include an angle 1200 and their magnitude are 2 and 3 then ba

. is equal to

1) 3 2) - 3 3)2 4) - 2

3

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23. If )()()( bacacbcbau

, then

1) u is a unit vector 2) cbau

3) 0

u 4) 0

u

24. If 0

cba , 5,4,3 cba

, then the angle between a

and b

is

1)6

2)

3

2 3)

3

5 4)

2

25. The vectors kcjbiaandkji

432 are perpendicular when

1) a = 2 , b = 3 , c = -4 2) a = 4 , b = 4 , c = 5 3) a = 4 , b = 4 , c = -5 4) a = -2 , b = 3 , c = 4

26. The area of the parallelogram having a diagonal kji

3 and a side kji

43 is

1) 310 2) 306 3) 302

3 4) 303

27. If baba

then

1) a

is parallel to b

2) a

is perependicular to b

3) ba

4) a

and b

are unit vectors

28. If qp

, and qp

are vectors of magnitude then the magnitude of qp

is

1) 2 2) 3 3) 2 4) 1

29. If yxbacacbcba

)()()( then

1) 0

x 2) 0

y

3) x

and y

are parallel 4) 0

x or 0

y or x

and y

are parallel

30. If kjiQSkjiPR

23,2 then the area of the quadrilateral PQRS is

1) 35 2) 310 3) 2

35 4)

2

3

31. The projection of OP on a unit vector OQ equals thrice the area of parallelogram OPRQ.

Then POQ is

1) tan-1 3

1 2) cos-1

10

3 3) sin-1

10

3 4) sin-1

3

1

32. If the projection of a

on b

and projection of b

on a

are equal then the angle between a

+ b

and

a

- b

is

1)2

2)

3

3)

4

4.

3

2

33. If cbacba

)()( for non-coplanar vectors cba

,, then

1) a

parallel to b

2) b

parallel to c

3) c

parallel to a

4) 0

cba

34. If a line makes 450, 600 with positive direction of axes x and y then the angle it makes with the z-axis is 1) 300 2) 900 3) 450 4) 600

35. If accbba

,, = 64, then cba

,, is

1) 32 2) 8 3) 128 4) 0

36. If accbba

,, = 8 then cba

,, is

1) 4 2) 16 3) 32 4) -4

37. The value of ikkjji

,, is equal to

1)0 2)1 3) 2 4) 4

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38. The shortest distance of the point (2 , 10 , 1) from the plane ⃗⃗ [ kji

43 ]= 262 is

1) 262 2) 26 3)2 4) 26

1

39. The vector )()( dcba

is

1) perpendicular to cba

,, and d

2) parallel to the vectors ba

and dc

3) parallel to the line of intersection of the plane containing a

and b

and the plane containing c

and d

4) perpendicular to the line of intersection of the plane containing banda

and the plane containing

dandc

40. If cba

,, are a right handed triad of mutually perpendicular vectors of magnitude a , b , c then the

value of ],,[ cba

is

1) a2b2c2 2) 0 3) abc2

1 4) abc

41. If cba

,, are non-coplanar and ][][ accbbaaccbba

then ],,[ cba

is

1) 2 2) 3 3) 1 4) 0

42. jtisr

is the equation of

1) a straight line joining the points i

and j

2) xoy plane

3) yoz plane 4) zox plane

43. If the magnitude of moment about the point kj

of a force kjai

acting through the point ji

is 8 then the value of a is

1) 1 2) 2 3) 3 4) 4

44. The equation of the line parallel to 3

52

5

3

1

3

zyx and passing through the point (1 , 3, 5) in

vector form is

1) )53()35( kjitkjir

2) )35()53( kjitkjir

3) )53()2

35( kjitkjir

4) )

2

35()53( kjitkjir

45. The point of intersection of the line )723()( kjitkir

and the plane 8).( kjir

is

1) (8 , 6 , 22) 2) (-8 , -6 , -22) 3) (4 , 3 , 11) 4) (-4 , -3 , -11) 46. The equation of the plane passing through the point (2 , 1 , -1) and the line of intersection of the planes

;0)3.( kjir

and 0)2.( kjr

is

1) x + 4y -z = 0 2) x + 9y +11z = 0 3) 2x + y - z +5 =0 4) 2x -y +z = 0


kji

acting on a particle, if the particle is displaced from A (3,3,3)

to the point B(4,4,4) is 1) 2 units 2) 3 units 3) 4 units 4) 7 units

48. If kjia

32 and kjib

23 then a unit vector perpendicular to a

and b is

1) 3

kji

2)

3

kji

3)

3

2kji

4)

3

kji

49. The point of intersection of the lines 8

4

4

4

6

6

zyx and

2

3

4

2

2

1

zyx is

1) ( 0 , 0 , -4) 2) (1 , 0 , 0) 3)(0 , 2 , 0) 4) (1 , 2 ,0)

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50. The point of intersection of the lines kjitkjir

2()32( ) and

)32()532( kjiskjir

is

1) (2 , 1 , 1) 2) (1 , 2 , 1) 3) (1 , 1 ,2) 4) (1 , 1 ,1)

51. The shortest distance between the lines 4

3

3

2

2

1

zyx and

5

5

4

4

3

2

zyx is

1) 3

2 2)

6

1 3)

3

2 4)

62

1

52. The shortest distance between the lines 3

5

2

1

4

3

zyxand

3

3

2

2

4

1

zyx is

1) 3 2) 2 3) 1 4) 0

53. The following two lines are 11

1

2

1 zyx

and

2

1

5

1

3

2

zyx

1) parallel 2) intersecting 3) skew 4) perpendicular 54. The centre and radius of the sphere given by x2+y 2 +z 2 - 6x + 8y - 10z +1 = 0 is 1) (-3, 4, -5), 49 2) (-6, 8, -10), 1 3)(3, -4, 5),7 4) ( 6, -8, 10 ), 7 --------------------------------------------------------------------------------------------------------------------------------------------------------

3. COMPLEX NUMBERS (THREE QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------

55. The value of

100

2

31

i +

100

2

31

iis

1) 2 2) 0 3) -1 4) 1

56. The modulus and amplitude of the complex number

3

43

i

e are respectively

1) e 9, 2

2) e 9,

2

3) e 6,

4

3 4) e 9,

4

3

57. If )4()5( nim is the complex conjugate of (2m +3) + i (3n - 2), then ( n, m ) are

1)

8,

2

1 2)

8,

2

1 3)

8,

2

1 4)

8,

2

1

58. If x 2 + y 2 = 1, then the value of iyx

iyx

1

1 is

1) x –iy 2) 2x 3) -2iy 4) x + iy

59. The modulus of the complex number 2 + 3i is

1) 3 2) 13 3) 7 4) 7

60. If A + iB = (a1 + ib1) (a2 + ib2) (a3 + ib3 ), then A2 + B2 is 1) a1

2 + b12 + a2

2 + b22 + a3

2 + b32 2) (a1 + a2 + a3)

2 + (b1 + b2 + b3) 2

3) (a12 + b1

2) (a22 + b2

2) (a3 2 + b3

2) 4) (a12 +a2

2+a32) (b1

2 + b22+b3

2) 61. If a = 3+i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and-az are 1) Vertices of a right angled triangle 2) Vertices of an equilateral triangle 3) Vertices of an isosceles triangle 4) Collinear

62. The points 4321

,,, zzzz in the complex plane are the vertices of a parallelogram taken in order if and only

if

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1) 3241

zzzz 2) 4231

zzzz

3) 4321

zzzz 4) 4321

zzzz

63. If z represents a complex number then arg (z) + arg ( z ) is

1) 4

2)

2

3) 0 4)

4

64. If the amplitude of a complex number is 2

, then the number is

1) purely imaginary 2) purely real 3) 0 4) neither real nor imaginary

65. If the point represented by the complex number iz is rotated about the origin through the angle 2

in the

counter clockwise direction then the complex number representing the new position is 1) iz 2) –iz 3) -z 4) z 66. The polar form of the complex number (i 25)3 is

1) cos 2

+ i sin

2

2) cos + i sin 3) cos - i sin 4) cos

2

- i sin

2

67. If P represents the variable complex number z and if zz 212 , then the locus of P is

1) the straight line x = 4

1 2) the straight line y =

4

1

3) the straight line z = 2

1 4) the circle x2 + y2 - 4x - 1 = 0

68.

i

i

e

e

1

1 =

1) cos + i sin 2) cos - i sin 3) sin - i cos 4) sin + i cos

69. If n

z = cos3

n + i sin

3

n then

6321... zzzz is

1) 1 2) -1 3) i 4) –i 70. If z lies in the third quadrant then z lies in the 1) first quadrant 2) second quadrant 3) third quadrant 4) fourth quadrant

71. If x = cos + i sin then the value of n

n

xx

1 is

1) 2 cos n 2) 2i sin n 3) 2 sin n 4) 2 i cos n

72. If a = cos - i sin , b = cos - i sin , c = cos - i sin then( a 2 c2 - b 2 ) / abc is

1) cos2 ( )(2sin) i 2) -2 cos ( )

3) -2 i sin ( ) 4) 2 cos ( )

73. If Z1 = 4 + 5i , Z2 = -3 + 2i then 2

1

Z

Zis

1) i13

22

13

2 2) - i

13

22

13

2 3) i

13

23

13

2

4) i

13

22

13

2

74. The value of i + i22 + i23 + i24 + i25 is 1) i 2) -i 3) 1 4) -1 75. The conjugate of i13 + i14 + i15 + i16 is 1) 1 2) -1 3) 0 4) –i 76. If -i +2 is one root of the equation ax2 - bx + c = 0 then the other root is 1) -i -2 2) i - 2 3) 2 +i 4) 2i + i

77. The quadratic equation whose roots are i 7 is

1) x 2 + 7 = 0 2) x2 - 7 = 0 3) x2 + x +7 = 0 4) x2 - x -7 = 0

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78. The equation having 4 - 3i and 4 + 3i as roots is 1) x2 +8x + 25 = 0 2) x2 + 8x - 25 = 0 3) x2 - 8x + 25 = 0 4) x2 - 8x - 25 = 0

79. If i

i

1

1 is a root of the equation ax2 + bx +1 = 0 ,where a, b are real then ( a, b) is

1) ( 1, 1) 2) ( 1, -1) 3) ( 0,1) 4) ( 1, 0 ) 80. If –i+3 is a root of x2 - 6x + k = 0, then the value of k is

1) 5 2) 5 3) 10 4) 10

81. If is a cube root of unity then the value of (1 - + 2

)4+ (1 + -2

)4 is

1) 0 2) 32 3) -16 4) -32 82. If is the nth root of unity then

1) 1 +2

+4

+ …. = + 3

+ 5

+ …. 2) n

= 0

3) n

= 1 4) = 1n

83. If is the cube root of unity then the value of (1- ) ( 1- 2

) ( 1-4

) ( 1 - 8

) is

1) 9 2) -9 3) 16 4) 32 --------------------------------------------------------------------------------------------------------------------------------------------------------

4. ANALYTICAL GEOMETRY (THREE QUESTIONS FOR FULL TEST)

------------------------------------------------------------------------------------------------------------------------------------------------------ 84. The axis of the parabola y2 - 2y +8x - 23 = 0 is 1) y = -1 2) x = -3 3) x = 3 4) y = 1 85. 16x2 - 3y2 - 32x - 12y -44 = 0 represents 1) an ellipse 2) a circle 3) a parabola 4) a hyperbola 86. The line 4x + 2y = c is a tangent to the parabola y2 = 16 x then c is 1) -1 2) -2 3) 4 4) -4 87. The point of intersection of the tangents at t1 = t and t2 = 3t to the parabola y2 = 8x is

1) (6t2, 8t) 2) ( 8t, 6t2 ) 3) ( t2, 4t ) 4) (4t, t 2 ) 88. The length of the latus rectum of the parabola y2 - 4x + 4y +8 = 0 is 1) 8 2) 6 3) 4 4) 2 89. The directrix of the parabola y2 = x + 4 is

1) 4

15x 2)

4

15x 3)

4

17x 4)

4

17x

90. The length of the latus rectum of the parabola whose vertex is (2, -3 ) and the directrix x = 4 is 1) 2 2) 4 3) 6 4) 8 91. The focus of the parabola x2 = 16y is 1) ( 4, 0 ) 2) ( 0, 4 ) 3) ( -4, 0 ) 4) ( 0, -4 ) 92. The vertex of the parabola x2 = 8y -1

1)

0,

8

1 2)

0,

8

1 3)

8

1,0 4)

8

1,0

93. The line 2x + 3y + 9 = 0 touches the parabola y2 = 8x at the point

1) (0, -3) 2) ( 2, 4) 3)

2

9,6 4)

6,

2

9

The tangents at the end of any focal chord to the parabola y2 = 12x intersect on the line 1) x-3 = 0 2) x + 3 = 0 3) y + 3 = 0 4) y - 3 = 0

The angle between the two tangents drawn from the point (- 4, 4) to y2 = 16x is

1) 45 2) 30 3) 60 4) 90 96. The eccentricity of the conic 9x2 + 5y2 - 54x - 40y + 116 = 0 is

1) 3

1 2)

3

2 3)

9

4 4)

5

2

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97. The length of the semi-major and the length of semi-minor axis of the ellipse 1169144

22

yx

is

1) 26, 12 2) 13, 24 3) 12, 26 4) 13, 12 98. The distance between the foci of the ellipse 9x2 + 5y2 = 180 is 1) 4 2) 6 3) 8 4) 2 99. If the length of major and semi-minor axes of an ellipse are 8,2 and their corresponding equations are y - 6 = 0 and x + 4 = 0 then the equations of the ellipse is

1) 116

)6(

4

)4(22

yx

2) 14

)6(

16

)4(22

yx

3) 14

)6(

16

)4(22

yx

4) 116

)6(

4

)4(22

yx

100. The straight line 2x - y + c = 0 is a tangent to the ellipse 4x2 + 8y2 = 32 then c is

1) 32 2) 6 3) 36 4) 4

101. The sum of the distance of any point on the ellipse 4x2+9y2 = 36 from 0,5 and 0,5 is

1) 4 2)8 3)6 4)18 102. The radius of the director circle of the conic 9x2 + 16y2 = 144 is

1) 7 2)4 3)3 4)5

103. The locus of foot of the perpendicular from the focus to a tangent of the curve 16x2 + 25y2 = 400 is 1) x2 + y2 = 4 2) x2 + y2 = 25 3) x2 + y2 = 16 4) x2 + y2 = 9 104. The eccentricity of the hyperbola 12y2-4x2-24x+48y-127 = 0 is 1) 4 2) 3 3) 2 4) 6 105. The eccentricity of the hyperbola whose latus rectum is equal to half of its conjugate axis is

1) 2

3 2)

3

5 3)

2

3 4)

2

5

106. The difference between the focal distances of any point on the hyperbola 12

2

2

2

b

y

a

x is 24 and the

eccentricity is 2 . Then the equation of the hyperbola is

1) 1432144

22

yx

2) 1144432

22

yx

3) 131212

22

yx

4) 112312

22

yx

107. The directrices of the hyperbola x2 – 4(y - 3)2 = 16

1) y = 5

8 2) x =

5

8 3) y =

8

5 4) x=

8

5

108. The line 5x - 2y + 4k = 0 is a tangent to 4x2 - y2 = 36 then k is

1) 9

4 2)

3

2 3)

4

9 4)

16

81

109. The equation of the chord of contact of tangents from (2,1) to the hyperbola 1916

22

yx

is

1) 9x -8y -72 = 0 2) 9x + 8y + 72 = 0 3) 8x -9y -72 = 0 4) 8x + 9y + 72 = 0

110. The angle between the asymptotes to the hyperbola 1916

22

yx

is

1)

4

3tan2

1 2)

3

4tan2

1 3)

4

3tan2

1 4)

3

4tan2

1

111. The asymptotes of the hyperbola 36y2 - 25x2 + 900 = 0 are

1) y = x5

6 2) y = x

6

5 3) y = x

25

36 4) y = x

36

25

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112. The product of the perpendiculars drawn from the point (8,0) on the hyperbola to its asymptotes

13664

22

yx

is

1) 576

25 2)

25

576 3)

25

6 4)

6

25

113. The locus of the point of intersection of perpendicular tangents to the hyperbola 1916

22

yx

is

1) x2 +y2 = 25 2) x2 +y2 = 4 3) x2 +y2 = 3 4) x2 +y2 = 7 114. The eccentricity of the hyperbola with asymptotes x + 2y - 5 = 0 , 2x - y + 5 = 0 is

1) 3 2) 2 3) 3 4)2

115. Length of the semi-transverse axis of the rectangular hyperbola xy = 8 is 1) 2 2)4 3)16 4)8 116. The asymptotes of the rectangular hyperbola xy = c2 are 1) x = c , y = c 2) x = 0 , y = c 3) x = c , y = 0 4) x = 0 , y = 0 117. The co-ordinate of the vertices of the rectangular hyperbola xy = 16 are 1)(4 , 4 ) , (-4 , -4) 2)(2 , 8) , (-2 , -8) 3)(4 , 0) , (-4 , 0) 4)(8 , 0) , (-8 , 0) 118. One of the foci of the rectangular hyperbola xy =18 is 1) (6 , 6) 2)(3 , 3) 3)(4 , 4) 4)(5 , 5) 119. The length of the latus rectum of the rectangular hyperbola xy = 32 is

1) 28 2)32 3)8 4)16

120. The area of the triangle formed by the tangent at any point on the rectangular hyperbola xy = 72 and its asymptotes is 1) 36 2)18 3)72 4)144

121. The normal to the rectangular hyperbola xy = 9 at

2

3,6 meets the curve again at

1)

24,

8

3 2)

8

3,24 3)

24,

8

3 4)

8

3,24

------------------------------------------------------------------------------------------------------------------------------------------------------ 5. DIFFERENTIAL CALCULUS- APPLICAIONS-I

(THREE QUESTIONS FOR FULL TEST) ------------------------------------------------------------------------------------------------------------------------------------------------------ 1. The gradient of the curve y = -2x3 +3x + 5 at x = 2 is 1) -20 2) 27 3) -16 4) -21 2. The rate of change of area A of a circle of radius r is

1) r2 2) dt

drr2 3)

dt

drr

2 4)

dt

dr

3. The velocity v of a particle moving along a straight line when at a distance x from the origin is given by a + bv2 = x2 where a and b are constants. Then the acceleration is

1) x

b 2)

x

a 3)

b

x 4)

a

x

4. A spherical snowball is melting in such a way that its volume is decreasing at a rate of 1 cm3/min. The rate at which the diameter is decreasing when the diameter is 10 cms is

1) - min/50

1cm

2) min/

50

1cm

3) - min/

75

11cm

4)- min/

75

2cm

5. The slope of the tangent to the curve y = 3x2 + 3sinx at x = 0 is 1) 3 2)2 3) 1 4) -1 6. The slope of the normal to the curve y = 3x2 at the point whose x coordinate is 2 is

1) 13

1 2)

14

1 3)

12

1 4)

12

1

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7. The point on the curve y = 2x2 -6x – 4 at which the tangent is parallel to the x–axis is

1)

2

17,

2

5 2)

2

17,

2

5 3)

2

17,

2

5 4)

2

17,

2

3

8. The equation of the tangent to the curve y = 5

3x

at the point

5

1,1 is

1) 5y + 3x = 2 2) 5y - 3x = 2 3) 3x - 5y = 2 4) 3x + 3y = 2

9. The equation of the normal to the curve t

1 at the point

3

1,3 is

1) 3 = 27t - 80 2) 5 = 27 t -80 3) 3 = 27 t + 80 4) t

1

10. The angle between the curves 1925

22

yx

and 188

22

yx

is

1) 4

2)

3

3)

6

4)

2

11. The angle between the curve y = emx and y = e-mx for m> 1 is

1)

1

2tan

2

1

m

m 2)

2

1

1

2tan

m

m

3)

2

1

1

2tan

m

m 4)

1

2tan

2

1

m

m

12. The parametric equation of the curve x2/3 + y2/3 = a2/3 are

1) x = a sin3 , y = a cos3 2) x = a cos3 , y = a sin3

3) x = a3 sin , y = a3 cos 4) x = a3 cos , y = a3 sin

13. If the normal to the curve x2/3 + y2/3 = a2/3 makes an angle with the x-axis then the slope of the normal

is

1) -cot 2) tan 3) -tan 4) cot

14. If the length of the diagonal of a square is increasing at the rate of 0.1 cm/sec. What is the rate of

increase of its area when the side is 2

15 cm ?

1) 1.5 cm2/sec 2) 3 cm2/sec 3) 23 cm2/sec 4) 0.15 cm2/sec

15. What is the surface area of a sphere when the volume is increasing at the same rate as its radius?

1) 1 2) 2

1 3) 4 4)

3

4

16. For what values of x is the rate of increasing of x3 - 2x2 + 3x + 8 is twice the rate of increase of x

1)

3,

3

1 2)

3,

3

1 3)

3,

3

1 4)

1,

3

1

17. The radius of a cylinder is increasing at the rate of 2cm/sec and its altitude is decreasing at the rate of 3 cm/sec. The rate of change of volume when the radius is 3cm and the altitude is 5 cm is

1) 23 2) 33 3) 43 4) 53

18. If y = 6x - x3 and x increases at the rate of 5 units per second, the rate of change of slope when x = 3 is 1) -90 units/sec 2)90 units/sec 3)180 units/sec 4)-180 units/sec 19. If the volume of an expanding cube is increasing at the rate of 4cm3/sec then the rate of change of

surface area when the volume of the cube is 8 cubic cm is 1) 8 cm2/sec 2) 16 cm2/sec 3) 2 cm2/sec 4) 4 cm2/sec 20. The gradient of the tangent to the curve y = 8+4x-2x2 at the point where the curve cuts the y-axis is 1. 8 2. 4 3. 0 4. -4 21. The angle between the parabolas y2 = x and x2 = y at the origin is

1)

4

3tan2

1 2)

3

4tan

1 3)

2

4)

4

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22. For the curve x =etcost; y = etsint , the tangent line is parallel to the x-axis when t is equal to

1. -4

2.

4

3. 0 4.

2

23. If a normal makes an angle with positive x-axis then the slope of the curve at the point where

the normal is drawn is

1) -cot 2) tan 3) -tan 4) cot

24. The value of „a‟ so that the curves y = 3ex and y = x

ea

3intersect orthogonally is

1) -1 2) 1 3) 3

1 4) 3

25. If s = t3 - 4t2 + 7, the velocity when the acceleration is zero is

1) sec/3

32m 2) sec/

3

16m

3) sec/

3

16m 4) sec/

3

32m

26. If the velocity of a particle moving along a straight line is directly proportional to the square of its distance from a fixed point on the line. Then its acceleration is proportional to

1) s 2) s2 3) s3 4) s4 27. The Rolle‟s constant for the function y = x2 on [ -2, 2] is

1) 3

32 2) 0 3) 2 4) -2

28. The „c‟ of Lagranges Mean Value Theorem for the function f (x) = x2 + 2x -1; a= 0 , b= 1 is

1) -1 2) 1 3) 0 4) 2

1

29. The value of c in Rolle‟s theorem for the function f (x) = cos 2

x on [ , 3 ] is

1) 0 2) 2 3) 2

4)

2

3

30. The value of „c‟ of Lagranges Mean Value Theorem for f (x) = x when a = 1 and b = 4 is

1) 4

9 2)

2

3 3)

2

1 4)

4

1

31. x

limx

e

x2

is =

1) 2 2) 0 3) 4) 1

32. 0

limx xx

xx

dc

ba

=

1) 2) 0 3) log cd

ab 4)

)/(log

)/(log

dc

ba

33. If f (a) = 2; f ' (a) = 1 ; g (a) = -1 ; g ' (a) = 2, then the value of ax

limax

xfagafxg

)()()()( is

1) 5 2) -5 3) 3 4) -3 34. Which of the following function is increasing in ( 0, ) ?

1) x

e 2) x

1 3) - x 2 4) x -2

35. The function f (x) = x 2 - 5x + 4 is increasing in 1) ( - , 1) 2) ( 1 , 4 ) 3) ( 4 , ) 4) everywhere 36. The function f (x) = x2 is decreasing in 1) ( - , ) 2) ( - , 0 ) 3) ( 0, ) 4) ( -2, )

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37. The function y = tan x - x is

1) an increasing function in

2,0

2) a decreasing function in

2,0

3) increasing in

4,0

and decreasing in

2,

4

4) decreasing in

4,0

and increasing in

2,

4

;

38. In a given semi circle of diameter 4 cm a rectangle is to be inscribed.The maximum area of the rectangle is 1) 2 2) 4 3) 8 4) 16 39. The least possible perimeter of a rectangle of area 100 m2 is 1. 10 2. 20 3. 40 4. 60 40. If f (x) = x2 - 4x +5 on [0, 3 ] then the absolute maximum value is 1) 2 2) 3 3) 4 4) 5 41. The curve y = - e -x is 1) concave upward for x > 0 2) concave downward for x > 0 3) everywhere concave upward 4) everywhere concave downward 42. Which of the following curves is concave down? 1) y = - x 2 2) y = x 2 3) y = e x 4) y = x2 + 2x - 3 43. The point of inflexion of the curve y = x 4 is at 1) x = 0 2) x = 3 3) x = 12 4) nowhere 44. The curve y = ax3 + bx2 + cx + d has a point of inflexion at x = 1 then 1) a + b = 0 2) a + 3b = 0 3) 3a + b = 0 4) 3a + b = 1 ------------------------------------------------------------------------------------------------------------------------------------------------------

6. DIFFERENTIAL CALCULUS-APPLICATIONS-II (TWO* QUESTIONS FOR FULL TEST)

------------------------------------------------------------------------------------------------------------------------------------------------------

45. If u = x y then x

u

is equal to

1) yxy-1 2) u log x 3) u log y 4) xyx-1

46. If u = sin - 1

22

44

yx

yx and f = sinu, then f is a homogeneous function of degree

1) 0 2) 1 3) 2 4) 4

47. If u = 22

1

yx

, then

y

uy

x

ux

1) u2

1 2) u 3) u

2

3 4) - u

48. The curve y2(x-2) = x2(1+x) has 1) an asymptote parallel to x-axis 2) an asymptote parallel to y-axis 3) asymptotes parallel to both axes 4) no asymptotes

49. If x = r cos , y = r sin , then

x

r

1) sec 2) sin 3) cos 4) cosec

50. Identify the true statements in the following: (i) If a curve is symmetrical about the origin, then it is symmetrical about both axes. (ii) If a curve is symmetrical about both the axes, then it is symmetrical about the origin. (iii) A curve f (x, y) = 0 is symmetrical about the line y = x if f (x, y) = f (y, x). (iv) For the curve f (x, y) = 0, if f (x, y) = f (-y, - x), then it is symmetrical about the origin. 1. (ii), (iii) 2. (i), (iv) 3. (i), (iii) 4. (ii), (iv)

51. If u = log

xy

yx22

, then y

uy

x

ux

is

1) 0 2) u 3) 2u 4) u-1

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52. The percentage error in the 11th root of the number 28 is approximately _________ times the percentage error in 28.

1) 28

1 2)

11

1 3) 11 4) 28

53. The curve a2y2 = x2 (a2-x2) has 1. only one loop between x =0 and x = a. 2. two loops between x =0 and x = a. 3. two loops between x = -a and x = a. 4. no loop 54. An asymptote to the curve y2 (a + 2x) = x2 (3a - x) is

1) x = 3a 2) x = -2

a 3) x =

2

a 4) x = 0

55. In which region the curve y2 (a + x ) = x2 (3a - x) does not lie? 1) x > 0 2) 0 < x < 3a 3) x -a and x > 3a 4) -a < x < 3a

56. If u = y sinx, then yx

u

2

=

1) cos x 2) cos y 3) sin x 4) 0

57. If u = f

x

ythen

y

uy

x

ux

is equal to

1) 0 2) 1 3) 2u 4) u 58. The curve 9y2 = x2(4-x2) is symmetrical about 1. y-axis 2. x-axis 3. y = x 4. both the axes 59. The curve ay2 = x2 (3a-x) cuts the y-axis at 1) x = -3a, x = 0 2) x = 0, x = 3a 3) x = 0, x = a 4) x = 0 --------------------------------------------------------------------------------------------------------------------------------------------------------

7. INTEGRAL CALCULUS (THREE QUESTIONS FOR FULL TEST)

--------------------------------------------------------------------------------------------------------------------------------------------------------

60. The value of dxxx

x

2/

0

3/53/5

3/5

sincos

cos

is

1)2

2)

4

3) 0 4)


xx

2/

0cossin1

cossin

is

1) 2

2) 0 c)

4

d)


1

0

4)1( is

1) 12

1 2)

30

1 3)

24

1 4)

20

1

63. The value of dxx

x

2/

2/cos2

sin

is

1) 0 2) 2 3) log 2 4) log 4


0

4sin is

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1.16

3 2.

16

3 3. 0 4.

8

3


4/

0

32cos

is

1. 3

2 2.

3

1 3. 0 4.

3

2

66. The value of xdxx3

0

2cossin

is

1) 2) 2

3)

4

4) 0

67. The area bounded by the line y = x, the x-axis, the ordinates x = 1, x = 2 is

1) 2

3 2)

2

5 3)

2

1 4)

2

7

68. The area of the region bounded by the graph of y = sinx and y = cosx between x = 0 and

x = 4

is

1) 12 2) 12 3) 2 22 4) 2 22

69. The area between the ellipse 12

2

2

2

b

y

a

x and its auxillary circle is

1) b(a - b) 2) 2 a (a - b) 3) a (a - b) 4) 2 b ( a - b)

70. The area bounded by the parabola y2 = x and its latus rectum is

1) 3

4 2)

6

1 3)

3

2 4)

3

8

71. The volume of the solid obtained by revolving 1169

22

yx

about the minor axis is

1) 48 2) 64 3) 32 4) 128

72. The volume, when the curve y = 2

3 x from x = 0 to x = 4 is rotated about x- axis is

1) 100 2) 9

100 3)

3

100 4)

3

100

73. The volume generated when the region bounded by y = x, y = 1, x = 0 is rotated about y-axis is

1) 4

2)

2

3)

3

4)

3

2

74. Volume of solid obtained by revolving the area of the ellipse 12

2

2

2

b

y

a

x about major and minor axes are

in the ratio 1) b2 : a2 2) a2 : b2 3) a : b 4) b : a 75. The volume generated by rotating the triangle with vertices at (0 , 0) , (3 , 0) and (3 , 3) about x-axis is 1)18 2) 2 3) 36 4) 9

76. The length of the arc of the curve x2/3 + y2/3 = 4 is 1) 48 2) 24 3) 12 4) 96 77. The surface area of the solid of revolution of the region bounded by y = 2x , x = 0 and x = 2 about x-axis is

1) 8 5 2) 2 5 3) 5 4) 4 5

78. The curved surface area of a sphere of radius 5, intercepted between two parallel planes of distance 2 and 4 from the centre is

1) 20 2) 40 3)10 4) 30

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------------------------------------------------------------------------------------------------------------------------------------------------------ 8. DIFFERENTIAL EQUATIONS

(THREE QUESTIONS FOR FULL TEST) -----------------------------------------------------------------------------------------------------------------------------------------------------

79. The integrating factor of xe

x

y

dx

dy 42 is

1) log x 2) x2 3) ex 4) x

80. If cos x is an integrating factor of the differential equation dx

dy+ Py = Q , then P =

1) -cot x 2) cot x 3) tan x 4) -tan x 81. The integrating factor of dx + xdy = e-y sec2 y dy is 1) ex 2) e-x 3) ey 4) e-y

82. The integrating factor of 2

2.

log

1

xy

xxdx

dy is

1) ex 2) log x 3) x

1 4) e-x

83. The solution of dy

dx+ mx = 0 , where m < 0 is

1) x = cemy 2) x = ce-my 3) x = my + c 4) x = c 84. y = cx - c2 is the general solution of the differential equation

1) (y)2 - xy + y = 0 2) y = 0 3) y = c 4) (y)2 + xy + y = 0

85. The differential equation xydy

dx

3/1

2

5 is

1. of order 2 and degree 1 2. of order 1 and degree 2 3. of order 1 and degree 6 4. of order 1 and degree 3 86. The differential equation of all non-vertical lines in a plane is

1) dx

dy= 0 2) 0

2

2

dx

yd 3)

dx

dy= m 4) m

dx

yd

2

2

87. The differential equation of all circles with centre at the origin is 1) x dy + y dx = 0 2) x dy - y dx = 0 3) x dx + y dy = 0 4) x dx - y dy = 0

88. The integrating factor of the differential equation dx

dy+ py = Q is

1) dxp 2) dxQ 3) Qdx

e 4)

pdx

e

89. The complementary function of (D2 + 1 ) y = e2x is 1) (Ax + B)ex 2) A cos x +B sinx 3) (Ax + B)e2 x 4) (Ax + B)e-x

90. A particular integral of (D2 - 4D +4)y = e2x is

1) x

ex 2

2

2 2) x

xe

2 3) x

xe

2 4)

xe

x 2

2

91. The differential equation of the family of lines y = mx is

1) dx

dy= m 2) y dx - xdy = 0 3) 0

2

2

dx

yd 4) ydx + xdy = 0

92. The degree of the differential equation 2

23/1

1dx

yd

dx

dy

is

1) 1 2) 2 3) 3 4) 6

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93. The degree of the differential equation c=

3/2

3

3

3

1

dx

yd

dx

dy

, where c is a constant is

1) 1 2) 3 3) -2 4) 2 94. The amount present in a radioactive element disintegrates at a rate proportional to its amount. The

differential equation corresponding to the above statement is ( k is negative)

1) p

k

dt

dp 2) kt

dt

dp 3) kp

dt

dp 4) kt

dt

dp

95. The differential equation satisfied by all the straight lines in xy-plane is

1.dx

dy= a constant 2. 0

2

2

dx

yd 3. y+

dx

dy= 0 4. 0

2

2

ydx

yd

96. If y = kex then its differential equation is

1) dx

dy=y 2 )

dx

dy= ky 3)

dx

dy+ ky = 0 4)

dx

dy= ex

97. The differential equation obtained by eliminating a and b from y = ae3x + be-3x is

1) 02

2

aydx

yd 2) 09

2

2

ydx

yd 3) 09

2

2

dx

dy

dx

yd 4) 09

2

2

xdx

yd

98. The differential equation formed by eliminating A and B from the relation y = ex (A cos x +Bsinx) is 1) y2 + y1 = 0 2) y2 - y1 = 0 3) y2 - 2y1 +2y = 0 4) y2 - 2y1 -2y = 0

99. If dx

dy=

yx

yx

then

1) 2xy + y2 + x2 = c 2) x2 + y2 - x + y = c 3) x2 + y2 - 2xy = c 4) x2 - y2 - 2xy = c

100. If f (x) = x and f (1) = 2, then f (x) is

1) )2(3

2 xx 2) )2(

2

3xx 3) )2(

3

2xx 4) )2(

3

2xx

101. On putting y = vx, the homogeneous differential equation x2dy +y(x + y)dx=0 becomes 1) xdv + (2v+v2)dx = 0 2) vdx + (2x+x2)dv = 0 3) v2dx - (x+x2)dv = 0 4) vdv + (2x+x2)dx = 0

102. The integrating factor of the differential equation dx

dy- y tan x = cos x is

1) sec x 2) cos x 3) etanx 4) cotx 103. The particular integral of (3D2 + D-14)y = 13e2x is

1) 26xe2x 2)13xe2x 3) xe2x 4) x

ex 2

2

2

104. The particular integral of the differential equation f (D) y = eax , where f (D) = (D-a) g(D),

g(a) 0 is

1) meax 2) )(

eax

ag 3) g(a)eax 4)

)(

xeax

ag

------------------------------------------------------------------------------------------------------------------------------------------------------ 9. DISCRETE MATHEMATICS

(THREE QUESTIONS FOR FULL TEST) ------------------------------------------------------------------------------------------------------------------------------------------------------ 105. Which of the following are statements?

(i) May God bless you (ii) Rose is a flower (iii) Milk is white (iv) 1 is a prime number 1) (i) , (ii) , (iii) 2) (i) , (ii) , (iv) 3) (i) , (iii) , (iv) 4) (ii) , (iii) , (iv)

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106. If a compound statement is made up of three simple statements, then the number of rows in the truth table is 1) 8 2) 6 3) 4 4) 2 107. If p is T and q is F, then which of the following have the truth value T ?

(i) p q (ii) ~ p q (iii) p ~q (iv)) p ~q

1) (i) , (ii) , (iii) 2) (i) , (ii) , (iv) 3) (i) , (iii) , (iv) 4) (ii) , (iii) , (iv)

108. The number of rows in the truth table of ~ [p(~q)] is 1) 2 2) 4 3) 6 4) 8

109. The conditional statement p q is equivalent to

1) p q 2) p ~q 3) ~ p q 4) p q

110. Which of the following is a tautology?

1) p q 2) pq 3) p ~p 4) p ~p

111. Which of the following is a contradiction?

1) p q 2) pq 3) p ~p 4) p ~p

112. p q is equivalent to

1) p q 2) q p 3) (p q ) ( q p) 4) (p q ) ( q p)

113. Which of the following is not a binary operation on R?

1) a * b = ab 2) a * b = a- b 3) a * b = ab 4) a * b = 22

ba

114. A monoid becomes a group if it also satisfies the 1. closure axiom 2. associative axiom 3. identity axiom 4.inverse axiom 115. Which of the following is not a group ? 1) (Zn , +n) 2) (Z , +) 3) (Z , .) 4) (R , +) 116. In the set of integers with the operation * is defined by a * b = a + b- ab, then

the value of 3 * (4 * 5) is 1) 25 2) 15 3)10 4) 5 117. The order of [7] in (Z9 , +9) is 1) 9 2) 6 3) 3 4) 1

118. The multiplicative group of cube root of unity, the order of 2

is

1)4 2)3 c)2 d)1 119. The value of [3] +11 ( [5] +11 [6] ) is 1. [0] 2)[1] 3)[2] 4)[3]

120. In the set of real numbers R, an operation * is defined by a * b = 22

ba Then the value of (3 * 4)* 5 is

1)5 2)5 2 3)25 4)50

121. Which of the following is correct ? 1. An element of a group can have more than one inverse. 2. If every element of a group is its own inverse, then the group is abelian. 3. The set of all 2x2 real matrices forms a group under matrix multiplication. 4. (a * b )-1= a -1* b -1, for all a, b G. 122. The order of -i in the multiplicative group of 4th roots of unity is 1) 4 2) 3 3) 2 4) 1 123. In the multiplicative group of nth roots of unity, the inverse of

k is (k < n)

1) 1/k 2)

-1 3) n-k 4)

n/k

124. In the set of integers under the operation * defined by a * b = a + b -1, the identity element is 1)0 2)1 3)a 4)b

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------------------------------------------------------------------------------------------------------------------------------------------------------ 10. PROBABILITY DISTRIBUTIONS

(THREE QUESTIONS FOR FULL TEST) ------------------------------------------------------------------------------------------------------------------------------------------------------

125. If

elsewhere

xkxxf

,0

30,)(

2

is a probability density function then the value of k is

1)3

1 2)

6

1 3)

9

1 4)

12

1

126. If f (x) =

xx

A,

16

12

is a p.d.f of a continuous random variable X, then the value of A is

1) 16 2) 8 3) 4 4) 1 127. A random variable X has the following probability distribution

X 0 1 2 3 4 5

P(X=x) 4

1 2a 3a 4a 5a

4

1

Then P(1 x 4) is

1) 21

10 2)

7

2 3)

14

1 4)

2

1

128. A random variable X has the following probability mass function as follows:

Then the value of is

1) 1 2) 2 3) 3 4) 4

129. X is a discrete random variable which takes the values 0, 1, 2 and P(X=0) = 169

144, P(X=1) =

169

1, then

the value of P(X =2 ) is

1) 169

145 2)

169

24 3)

169

2 4)

169

143

130. A random variable X has the following p.d.f.

X 0 1 2 3 4 5 6 7

P(X =x) 0 k 2k 2k 3k k2 2k2 7k2 + k

The value of k is

1) 8

1 2)

10

1 3) 0 4) -1 or

10

1

131. Given E(X+ c) = 8 and E(X-c) = 12 then the value of c is 1) -2 2) 4 3) -4 4) 2

132. X is a random variable taking the values 3, 4 and 12 with probabilities 4

1,

3

1and

12

5Then E (X) is

1)5 2)7 3) 6 4) 3 133. Variance of the random variable X is 4. Its mean is 2. Then E(X2) is 1)2 2)4 3)6 4)8

134. 2 = 20,

2 = 276 for a discrete random variable X. Then the mean of the random variable X is

1)16 2)5 3)2 4)1 135. Var (4X + 3) is 1)7 2)16 Var(X) 3)19 4)

X 2 3 1

P(X=x)

6

4

12

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136. In 5 throws of a die, getting 1 or 2 is a success. The mean number of successes is

1. 3

5 2.

5

3 3.

9

5 4.

5

9

137. The mean of a binomial distribution is 5 and its its standard deviation is 2. Then the value of n and p are

1.

25,

5

4 2.

5

4,25 3.

25,

5

1 4.

5

1,25

138. If the mean and standard deviation of a binomial distribution are 12 and 2. Then the value of its parameter p is

1. 2

1 2.

3

1 3.

3

2 4.

4

1

139. In 16 throws of a die getting an even number is considered a success. Then the variance of the successes is 1. 4 2. 6 3. 2 4. 256 140. A box contains 6 red balls and 4 white balls. If 3 balls are drawn at random, the probability of getting 2

white balls without replacement is

1) 20

1 2)

125

18 3)

25

4 4)

10

3

141. If 2 cards are drawn from a well shuffled pack of 52 cards, the probability that they are of the same colours without replacement, is

1) 2

1 2)

51

26 3)

51

25 4)

102

25

142. If in a Poisson distribution P(X = 0) = k, then the variance is

1) log k

1 2) log k 3) e 4)

k

1

143. If a random variable X follows Poisson distribution such that E(X2 ) = 30, then the variance of the distribution is

1. 6 2. 5 3. 30 4. 25 144. The distribution function F(X) of a random variable X is 1. a decreasing function 2. a non-decreasing function 3. a constant function 4. increasing first and then decreasing

145. For a Poisson distribution with parameter =0.25 the value of the 2nd moment about the origin is

1)0.25 2)0.3125 3)0.0625 4)0.025

146. In a Poisson distribution if P(X = 2) =P(X=3) then the value of its parameter is

1) 6 2) 2 3) 3 4) 0

147. If )( xf is a p.d. f. of a normal distribution with mean , then

dxxf )( is

1. 1 2. 0.5 3. 0 4. 0.25

148. The random variable X follows normal distribution f(x) = c25

)100(212

x

e . Then the value of „c‟ is

1) 2 2) 2

1 3) 5 2 4)

25

1

149. If )( xf is a p.d. f. of a normal variate X and X ~ N( , 2

), then

dxxf )( is

1. undefined 2. 1 3. 0.5 4. -0.5 150. The marks secured by 400 students in a Mathematics test were normally distributed with mean 65. If 120

students got more marks above 85, the number of students securing marks between 45 and 65 is 1)120 2)20 3)80 4)160

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+2 MATHEMATICS

Chapter wise Collection of Public out Of Book Questions From 26 Public Question Papers [2006-14]

2006: March/June/October 2010: March/June/October 2007: March/June/October 2011: March/June/October 2008: March/June/October 2012: March/June/October 2009: March/June/October 2013: March/June/October

2014: March/June.

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--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: I MATRICES AND DETERMINANTS --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-B [SIX MARKS]

1. Verify that (A-1 )T = (AT )-1 for the matrix

65

32A

2. Find the rank of the matrix

7431

2110

4312


4

0844

4840

412124

4. Solve the system of linear equations by determinant method. 1464;732 yxyx

5. Solve the system of linear equations by determinant method. 102;7622;43 zyxzyxzyx 6. Solve the system of linear equations by determinant method. 62;5;02 zxzyxzyx

7. Solve by determinant method. 442;22 yxyx

PART-C [TEN MARKS] 1. Solve the following non- homogeneous system of linear equations by using determinant method.

x + 2 y + z = 2 ; 2 x + 4 y + 2 z = 4 ; x - 2 y - z = 0 --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 2 VECTOR ALGEBRA --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-B [SIX MARKS]

1. For any three vectors cba ,, prove that 0,, accbba

2. For any three vectors cba ,, prove that cbaccbcba ,,

3. If ikckjbjia ,, then find accbba ,,

4. Show that the points A (1,2,3) , B (3,-1,2) , C (-2,3,1) and D (6,-4,2) are lying on the same plane. PART-C [TEN MARKS]

1. Find the vector and Cartesian equation of the plane through the point (-1,2,1) and perpendicular to the plane .0742 zyx and .0332 zyx

2. Find the vector and Cartesian equation of the plane through the point (1, 2, -2) and parallel

to the line 4

4

2

1

3

2

zyx and perpendicular to the plane 2 x + 3 y + 3 z = 8.

3. Find the vector and Cartesian equation of the plane which contains the line 1

1

32

1

zyx and

Perpendicular to the plane x -2 y + 3 z -2=0. --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 3 COMPLEX NUMBERS --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-B [SIX MARKS]

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1. P represents the variable complex number z. Find the locus of P, if 212 zz

2. P represents the variable complex number z. Find the locus of P, if iziz 33

3. P represents the variable complex number z. Find the locus of P, if 1353 zz

4. If P represents the variable complex number z. Find the locus of P, if 01

Re

iz

z

5. Solve the equation 044

x if one of the root is i1

6. Solve the equation 046423

xxx if i1 is a root PART-C [TEN MARKS]

1. If P represents the variable complex number z. Find the locus of P, if 11

2Im

iz

iz

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT:4 ANALYTICAL GEOMETRY --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-C [TEN MARKS] 1. Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the

following parabolas and hence sketch their graphs. 071282

yxy

2. Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the


xyy

3. Find the axis , vertex, focus, equation of directrix, latus rectum, length of the latus rectum for the


yxx

4. Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram:

092363291622

yxyx

5. Find the eccentricity, centre, foci, vertices of the following ellipses and draw the diagram:

01161001825922

yxyx

6. Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams.

0127322441222

yxyx

7. Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams.

09214367922

yxyx 8. Find the eccentricity centre, foci and vertices of the following hyperbolas and draw their diagrams.

0164363291622

yxyx --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT:6 DIFFERENTIAL CALCULUS – APPLICATIONS - II --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-C [TEN MARKS]

1. Verify xy

u

yx

u

22

for the functions:

y

xu sin

2. Prove

yx

yx

yx

yx

y

uy

x

ux cos

2

1 if

yx

yxu sin

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--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 7 INTEGRAL CALCULUS --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-C [TEN MARKS]

1. Find the area between the curves ,322

xxy x-axis and the lines 3x and 5x

2. Find the common area enclosed by the parabolas xy 2

and yx 2

3. Find the volume of the solid obtained by revolving the area of the triangle whose sides are having the equation y=0 ; x=4 and 3x-4y = 0 , about x-axis

[OR] Find the volume of the cone obtained by revolving the area of the triangle whose vertices are (0,0) ; (4,0) and (4,3) about x-axis --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 8 INTEGRAL CALCULUS --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-B [SIX MARKS] 1. Solve: xx

eeyDD 101314322

2. Solve: xyDD 5sin452

PART-C [TEN MARKS]

1. Solve : dxxxxydxxdyxdy ]tan[secsec323

2. Solve : xecyxdx

dyx

223cos621

3. Solve : xyxdx

dyx

223sec31

4. Solve: xexyDD

322sin65

5. Solve: 52sin222

xyDD

6. Solve: xexyDD 51213

2

--------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 9 DISCRETE MATHEMATICS --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-B [SIX MARKS]

1. Use the truth table to determine whether the statement ppqP ~~ is a tautology.

2. With usual symbol, prove that 55

,0 Z forms a group.

3. Find the order of all elements of the group (Z6 ,+6) 4. Find the order of all elements of the group 77 ,0 Z --------------------------------------------------------------------------------------------------------------------------------------------------------

UNIT: 10 PROBABILITY DISTRIBUTIONS --------------------------------------------------------------------------------------------------------------------------------------------------------

PART-B [SIX MARKS] 1. Find k , and of the normal distribution whose probability function is given by

.242

2

Xkexfxx

********

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SOME IMPORTANT TRIGONOMETRICAL FORMULA’S TRIGONOMETRICAL IDENTITIES:

sin = opp

hyp

cosec = hyp

opp

sin =

cosecθ

cosec =

sinθ

cos = adj

hyp

sec = hyp

adj

cos =

secθ

sec =

cos θ

tan = opp

adj

cot = adj

opp

tan =

cot θ (OR) tan =

sinθ

cosθ

cot =

tan θ (OR) cot =

cosθ

sinθ

sin2 + cos2 = 1 sin2 = 1- cos2 cos2 = 1- sin2

1+tan2 = sec2 sec2 - tan2 = 1 tan2 = sec2 -1

1+cot2 = cosec2 cosec2 - cot2 = 1 cot2 = cosec2 - 1

TRIGONOMETRICAL TABLES:

O π

6

π

4

π

3

π

2 π

3π

2 2π

sin 0 1

2

1

√2 √3

2 1 0 -1 0

cos 1 √3

2

1

√2

1

2 0 -1 0 1

tan 0 1

√3 1 √3 ∞ 0 −∞ 0

ASTC RULE: I Quadrant (90˚ − θ)

A→ All the trigonometric functions are positive

II Quadrant (90° + θ) (OR) (180˚ − θ)

S→ sin & cosec only positive, remains are negative

III Quadrant (180° + θ) (OR) (270˚ − θ)

T→ tan & cot only positive, remains are negative

IV Quadrant (270° + θ) (OR) (360˚ − θ)

C→ cos & sec only positive, remains are negative

T-Ratios of (90˚ ± θ) (OR) (270˚ ± θ)

sin↔cos ; tan↔cot ; cosec↔sec (T-functions Changes)

T-Ratios of (180˚ ± θ) (OR) (360˚ ± θ)

No change in trigonometric functions

SOME SPECIAL PROPERTIES OF TRIGONOMETRICAL FUNCTIONS: Even Function: Replace x =- x in f(x), then we get f(-x) = f(x) Odd Function: Replace x =- x in f(x), then we get f(-x) = -f(x)

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S NO Compound Angles for: sin(A±B) & cos(A±B) 1

sin(A+B) = sinA cosB + cosA sinB

2

sin(A-B) = sinA cosB – cosA sinB

3

cos(A+B) = cosA cosB – sinA sinB

4

cos(A-B) = cosA cosB + sinA sinB

5

2 sinA cosB = sin(A+B) + Sin(A-B)

6

2 cosA sinB = sin(A+B) - Sin(A-B)

7

2 cosA cosB = cos(A+B)+ Cos(A-B)

8

-2 sinA sinB = cos(A+B)- Cos(A-B) (OR) 2 sinA sinB = cos(A-B)- cos(A+B)

S NO Multiple Angles for: sin2A & cos2A

1

sin2A = 2SinA cosA (OR) sinA = 2sinA

2 cos

A

2

2

cos2A = cos2A- sin2A (OR) cosA = cos2A

2 - sin2

A

2

3

cos2A = 2 cos2A – 1 (OR) cos2A = +cos2A

2 (OR) cosA = 2 cos2

A

2 - 1

4

cos2A = 1 - 2 sin2A (OR) sin2A = −cos2A

2 (OR) cosA = 1 - 2 sin2

A

2

S NO Multiple Angles for: sin3A & cos3A

1

sin3A = 3sinA – 4sin3A (OR) sin3A = 3sinA−sin3A

4

2

cos3A = 4cos3A – 3cosA (OR) cos3A = cos3A+3cosA

4

S NO Properties of Inverse trigonometric functions:

1

sin− x + sin− y = sin− [x√1 − y2 + y√1 − x2 ]

2

sin− x - sin− y = sin− [x√1 − y2 - y√1 − x2 ]

3

tan− x + tan− y = tan− ⌈x+y

−xy⌉

4

tan− x - tan− y = tan− ⌈x−y

+xy⌉

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IMPORTANT DIFFERENTIATION & INTEGRATION FORMULA

S:NO

DIFFERENTIATION

INTEGRATION

1.

( )= n − : n

∫ dx =

+ +c : where −1

∫( + ) dx = ( + )

( + ) +c : where −1

2.

(x) = 1

(

) =

−

∫ 1 = x+ c

3.

(log x) =

∫

dx = log x + c

4.

(k) = 0 : where k is an constant ∫ = +

5.

( ) = ∫ = +

6.

( ) = ∫ =

+

7.

( ) = ∫ = − +

8.

( ) = ∫ =

−

+

9.

( ) = − ∫ = +

10.

( ) = − ∫ =

+

11.

( ) = 2x ∫ = ( ) +

12.

( ) = − ∫ = ( − ) +

13.

( ) = ∫ = ( + ) +

14.

( ) = − 2 ∫ = ( ) +

15. - ∫ 2 = +

16. - ∫ 2 = − +

17. - ∫ = +

18. - ∫ = − +

19.

Product rule:

( ) =

+

(OR)

( ) = +v

Integration by parts:

∫ = − ∫ + Bernoulli’s formula:

∫ = − + 2 − 3 + Where , , ,… →successive derivative of u

, 2 , 3 , …→Repeated integrals of v

20. Quotient rule:

(

) =

−

-

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21.

( − ) =

√ − ∫

√ − = − +

22.

( − ) =

−

√ − ∫

−

√ − = − +

23.

( − ) =

+ ∫

+ = − +

24.

( − ) =

−

√ − ∫

−

√ − = − +

25.

( − ) =

√ − ∫

√ − = − +

26.

( − ) =

−

+ ∫

−

+ = − +

Some important standard results of integrals

27. ∫ ( )

( ) = [ ( )] +

28. ∫ ( )

√ ( ) = 2 [√ ( )] +

29. ∫

+ =

− (

) + c (OR) ∫

+ = − +

30. ∫

− =

2 (

−

+ ) + c

31. ∫

− =

2 (

+

− ) + c

32. ∫

√ + = [x+√ 2 + 2 ] + c

33. ∫

√ − = [x+√ 2 − 2 ] + c

34. ∫

√ − = − (

) + c (OR) ∫

√ − = − +

35. ∫√ 2 + 2 =

2√ 2 + 2 +

2 [x+√ 2 + 2 ] + c

36. ∫√ 2 − 2 =

2√ 2 − 2 −

2 [x+√ 2 − 2 ] + c

37. ∫√ 2 − 2 =

2√ 2 − 2 +

2 − (

) + c

38. ∫ sinbx =

+ [ − ]+ c

39. ∫ cosbx =

+ [ + ]+ c

40. Gamma Integral: ∫ −

=

Reduction formulae

41. = ∫ , then = −

− +

−

−2

42. = ∫ , then =

− +

−

−2

43. ∫

= ∫

= −

−3

−2 −5

−4 …

2

3.1 (when n is odd)

44. ∫

= ∫

= −

−3

−2 −5

−4 …

2

2 (when n is even)

Relationship Between Exponential & Logarithmic Functions

45. e o x = x e = 1 e = ∞ e− = 0

46. loge = 1 log1 = 0 log∞ = ∞ log0 = -∞

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COME BOOK PUBLIC ONE WORD (MAR/JUN/SEP: 2006-14) 1. MATRICES AND DETERMINANTS

1. (AT)-1 is equal to 1) A-1 2) AT 3) A 4) (A-1)T 2. Which of the following is not elementary transformation? 1) Ri Rj 2) Ri 2Ri+Rj 3) Ci Cj+Ci 4) Ri Ri+Cj 3. In echelon form, which of the following is incorrect?

1. Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. 2. The first non-zero entry in each non-zero row is 1.

3. The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.

4. Two rows can have same number of zeros before the first non-zero entry

4. In the system of 3 linear equations with three unknowns, if 0 and one of zyx

or ,, is non-zero then

the system is 1. consistent 2. inconsistent 3. consistent and the system reduces to two equations 4. consistent and the system reduces to a single equation. 5. Every homogeneous system(linear) 1. is always consistent 2. has only trivial solution 3. has infinitely many solution 4. need not be consistent

6. If BAA ,)( then the system is


7. If BAA ,)( = the number of unknowns then the system is


8. BAA ,)( then the system is


9. In the system of 3 linear equations with three unknowns, BAA ,)( = 1,

then the system 1.has unique solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent 10. In the homogeneous system with three unknowns, )( A = number of unknowns then the system has

1. only trivial solution 2. reduces to 2 equations and has infinitely many solutions 3. reduces to a single equations and has infinitely many solutions 4. is inconsistent

11. In the system of three linear equations with three unknowns, in the non-homogeneous system BAA ,)(

= 2, then the system 1. has unique solution 2. reduces to two equations and has infinitely many solutions

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3. reduces to a single equations and has infinitely many solutions 4. is inconsistent 12. In the homogeneous system )( A < the number of unknowns then the system has

1. only trivial solution 2. trivial solution and infinitely many non-trivial solutions 3. only non-trivial solutions 4. no solution 13. If (A) = r , then which of the following is correct?

1. all the minors of order r which do not vanish. 2. A has atleast one minor of r which does not vanish and all higher order minors vanish. 3. A has atleast one (r+1) order minor which vanishes. 4. all (r+1) and higher order minors should not vanish. 14. Cramer’s rule is applicable only (with three unknowns) when

1. 0 2. 0 3. 0,0 x

4. 0zyx

2. VECTOR ALGEBRA

1. The angle between the vectors ji

and kj

is

1. 3

2.

3

2 3. -

3

4.

3

2

2. If cba

,, are mutually perpendicular unit vectors, then cba

1. 3 2. 9 3. 33 4. 3

3. Let u

, v

and w

be vector such that 0

wvu . If 3u

, 4v

and 5w

, then uwwvvu

is

1. 25 2. -25 3. 5 4. 5

4. The projection of ji

on z-axis is

1. 0 2. 1 3. -1 4. 2


22 on kji

52 is

1. 30

10 2.

30

10 3.

3

1 4.

30

10


3 on kji

24 is

1. 21

9 2. -

21

9 3.

21

81 4. -

21

81

7. The work done in a moving particle from the point A with position vector kji

762 to the point B, with

position vector kji

53 by a force F

kji

3 is

1. 25 2. 26 3. 27 4. 28

8. The angle between the two vectors a

and if ba

= ba

is

1.4

2.

3

3.

6

4.

2

9. The unit normal vectors to the plane 522 zyx are

1. kji

22 2. kji

223

1 3. kji

22

3

1 4. kji

22

3

1

10. The length of the perpendicular from the origin to the plane 261243. kjir

is

1. 26 2. 169

26 3. 2 4.

2

1


is

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1. 30

7 2.

7

30 3.

7

30 4.

30

7

12. Chord AB is a diameter of the sphere 1862 kjir

with coordinate of A as (3, 2, -2). Then the

coordinates of B is 1. (1, 0, 10) 2. (-1, 0, -10) 3. (-1, 0, 10) 4. (1, 0, -10)

13. The centre and radius of the sphere )42( kjir

5 are

1. (2, -1, 4) and 5 2. (2, 1, 4) and 5 3. (-2, 1, 4) and 6 4. (2, 1, -4) and 5

14. The centre and radius of the sphere )43(2 kjir

4 are

1.

2,

2

1,

2

3, 4 2.

2,

2

1,

2

3and 2 3.

2,

2

1,

2

3, 6 4.

2,

2

1,

2

3and 5

15. The vector equation of a plane passing through a point whose position vector is a

and perpendicular to a vector n

is

1. nanr

2. nanr

3. nanr

4. nanr

16. The non-parametric vector equation of a plane passing through a point whose position vector is a

and parallel

to u

and v

is

1. 0,, vuar

2. 0vur

3. 0 vuar

4. 0vua

17. The non-parametric vector equation of a plane passing through the points whose position vectors

are a

. b

and parallel to v

is

1. 0 vabar

2. 0 vabr

3. 0vba

4. 0bar

18. The non-parametric vector equation of a plane passing through three non-collinear points whose

position vectors are cba

,, is

1. 0 acabar

2. 0bar

3. 0cbr

4. 0cba

19. The vector equation of a plane passing through the line of intersection of the planes

11

. qnr

and 22

. qnr

is

1. 0..2211

qnrqnr

2. 2121

.. qqnrnr

3. 2121

qqnrnr

4. 2121

qqnrnr

20. The vector equation of a plane whose distance from the origin is p and perpendicular to a unit vector n̂ is

1. pnr

. 2. qnr ˆ.

3. pnr

4. pnr ˆ.

21. The angle between the line btar

and the plane qnr

is connected by the relation

1. q

na

cos 2.

nb

nb

cos 3. n

ba

sin 4. nb

nb

sin

3. COMPLEX NUMBERS 1. If )72()68( iiiba then the values of ‘a’ and ‘b’ are

1. 15,8 2. 15,8 3. 9,15 4. 8,15

2. The cube roots of unity are

1. in G.P. with common ratio , 2. in G.P. with common difference 2

3. In A.P. with common difference 4. in A.P. with common difference 2

3. The arguments of nth roots of a complex number differ by

1. n

2 2.

n

3.

n

3 4.

n

4

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4. Which of the following statements is correct? 1. negative complex numbers exist 2. order relation does not exist in real numbers 3. order relation exist in complex numbers 4. )1( i > )23( i is meaningless


1. || z 2. 2|| z 3. ||2 z 4. 2

||2 z

6. If |||| 21zzzz then the locus of z is

1. a circle with centre at the origin

2. a circle with centre at 1

z

3. a straight line passing through the origin

4. is a perpendicular bisector of the line joining 1

z and 2

z

7. The number of values of q

p

i sincos where p and q are non-zero integers prime to each other, is

1. p 2. q 3. p+q 4. p-q


is

1. sin 2. sin2 3. sini 4. sin2 i

9. Polynomial equation P(x)=0 admits conjugate pairs of imaginary roots only if the coefficients are 1. imaginary 2. complex 3. real 4. either real or complex


1. ||)(Re zz 2. ||)(Im zz 3. 2|| zzz 4. ||)(Re zz

11. Which of the following is incorrect regarding nth roots of unity? 1. The number of distinct roots is n

2. the roots are in G.P. with common ratio n

cis2

3. the arguments are in A.P. with common difference n

2.

4. product of the roots is 0 and the sum of the roots is 1 . 12. Which of the following are true?

1. If n is a positive integer then ninin

sincossincos


sincossincos

3. If n is a fraction then nin sincos is one of the values of nin )sincos( .


sincossincos

1. (i), (ii), (iii), (iv) 2. (i), (iii), (iv) 3. (i), (iv) 4. (i) only 13. The principal value of zarg lies in the interval

1.

2,0

2. ],( 3. ,0 4. ]0,(


1. ||||||2121

zzzz 2. ||||||2121

zzzz

3. ||||||2121

zzzz 4. ||||||2121

zzzz


1. z is the mirror image of z on the real axis 2. The polar form of z is ,r

3. –z is the point symmetrical to z about the origin 4. The polar form of -z is ,r


a. zz )(Re b. zzmI )( c. zz d. nnzz

1. (a), (b) 2. (b), (c) 3. (b), (c) and (d) 4. (a), (c) and (d)

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17. If Z = 0, then the arg(Z) is

1. 0 2. 3. 2

4. Indeterminate

4. ANALYTICAL GEOMETRY

1. The equation of the major and minor axes of 149

22

yx

are

1. 2,3 yx 2. 2,3 yx 3. 0,0 yx 4. 0,0 xy

2. The equation of the major and minor axes of 123422 yx are

1. 2,3 yx 2. 0,0 yx 3. 2,3 yx 4. 0,0 xy

3. The length of the major and minor axes of 123422 yx are

1. 32,4 2. 3,2 3. 4,32 4. 2,3

4. The equation of the latus rectum of 1916

22

yx

are

1. 7y 2. 7x 3. 7x 4. 7y


22

yx

are

1. 0,5 2. 5,0 3. 5,0 4. 0,5

6. The equations of transverse and conjugate axes of the hyperbola 36002514422 yx are

1. 0;0 xy 2. 5;12 yx 3. 0;0 yx 4. 12;5 yx


1. 0124 yx 2. 0124 yx 3. 0124 xy 4. 0124 xy

8. The equation of chord of contact of tangents from the point (5, 3) to the hyperbola 246422 yx is

1. 012109 yx 2. 012910 yx 3. 012109 yx 4. 012910 yx

9. The point of contact of the tangent cxmy and the parabola axy 42 is

1.

m

a

m

a 2,

2 2.

m

a

m

a,

2

2 3.

2

2,

m

a

m

a 4.

m

a

m

a 2,

2

10. The point of contact of the tangent cxmy and the ellipse 12

2

2

2

b

y

a

x is

1.

c

ma

c

b22

, 2.

c

b

c

ma22

, 3.

c

b

c

ma22

, 4.

c

b

c

ma22

,

11. If ''1

t , ''2

t are the extremities of any focal chord of a parabola axy 42 then ;

21tt is

1. 1 2. 0 3. 1 4. 2

1

12. The normal at ''1

t on the parabola axy 42 meets the parabola at ''

2t then

1

1

2

tt is

1. 2

t 2. 2

t 3. 21

tt 4. 2

1

t

13. The chord of contact of tangents from any point on the directrix of the hyperbola 12

2

2

2

b

y

a

x passes through


14. The point of intersection of tangents at ''1

t and ''2

t to the parabola axy 42 is

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1. )),((2121

tattta 2. ))(,(2121

ttatta 3. )2,(2

atat 4. ))(,(2121

ttatta

15. If the normal to the R.H. 2cxy at ''

1t meets the curve aagain at ''

2t then

2

3

1tt

1. 1 2. 0 3. -1 4. -2

16. The locus of the foot of perpendicular from the focus on any tangent to the parabola axy 42 is

1. 2222bayx 2. 222

ayx

3. 2222bayx 4. 0x

17. The condition that the line 0 nmylx may be a normal to the ellipse 12

2

2

2

b

y

a

x is

1. 02223

nmalmal 2. 2

222

2

2

2

2)(

n

ba

m

b

l

a

3. 2

222

2

2

2

2)(

n

ba

m

b

l

a 4.

2

222

2

2

2

2)(

n

ba

m

b

l

a

18. The condition that the line 0 nmylx may be a normal to the parabola axy 42 is

1. 02223

nmalmal 2. 2

222

2

2

2

2)(

n

ba

m

b

l

a

3. 2

222

2

2

2

2)(

n

ba

m

b

l

a 4.

2

222

2

2

2

2)(

n

ba

m

b

l

a

19. The locus of the point of intersection of perpendicular tangents to the parabola axy 42 is

1. latus rectum 2. directrix 3. tangent at the vertex 4. axis of the parabola

20. The locus of the point of intersection of perpendicular tangents to the ellipse 12

2

2

2

b

y

a

xis

1. 2222bayx 2. 222

ayx 3. 2222bayx 4. 0x

5. DIFFERENTIAL CALCULUS- APPLICATIONS-I 1. The law of the mean can also be put in the form 1. 10),(')()( hahfafhaf

2. 10),(')()( hahfafhaf

3. 10),(')()( hahfafhaf

4. 10),(')()( hahfafhaf

2. spitaloHl 'ˆ' rule cannot be applied to 3

1

x

x as 0x because f(x) = x+1 and g(x) = x+3are

1. not continuous 2. not differentiable 3. not in the indeterminate form as 0x 4. in the indeterminate form as 0x

3. x

x

x tanlim

0

is

1. 1 2. -1 3. 0 4. 4. f is differentiable function defined on an interval I with positive derivative. Then f is 1. increasing on I 2. decreasing on I 3. strictly increasing on I 4. strictly decreasing on I 5. If f has a local extremum at a and if )(' af exists then

1. )(' af <0 2. )(' af >0 3. )(' af =0 4. )(" af =0

6. 0

xx is a root of even order for the equation )(' xf = 0, then 0

xx is a

1. maximum point 2. minimum point 3. inflection point 4. critical point 7. The statement: “ If f has a local extremum (minimum or maximum) at c and if )(' cf exists then 0)(' cf ” is

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1. the extreme value theorem 2. Fermat’s theorem 3. Law of mean 4. Rolle’s theorem 8. Identify the false statement: 1. all the stationary numbers are critical numbers 2. at the stationary point the first derivative is zero 3. at critical numbers the first derivative need not exist 4. all the critical numbers are stationary numbers 9. Identify the correct statements: a. Every constant function is an increasing function b. Every constant function is a decreasing function c. Every identity function is an increasing function

d. Every identity function is a decreasing function 1. a, b and c 2. a and c 3. c and d 4. a, c and d

10. Which of the following statement is incorrect? 1. Initial velocity means velocity at t = 0 2. Initial acceleration means acceleration at t = 0 3. If the motion is upward, at the maximum height, the velocity is not zero 4. If the motion is horizontal, v = 0 when the particle comes to rest.

11. Which of the following statements are correct (1

m and 2

m are slopes of two lines)

a. If the two lines are perpendicular then 1

m2

m = -1

b. If 1

m2

m = -1, then two lines are perpendicular

c. If 1

m = 2

m , then the two lines are parallel

d. If 2

1

1

mm then the two lines are perpendicular

1. b, c and d 2. a, b and d 3. c and b 4. a and b 12. The statement “If f is continuous on a closed interval [a, b], then f attains an absolute maximum value )(cf

and an absolute minimum value )(df at some number c and d in [a, b] ” is

1. The extreme value theorem 2. Fermat’s theorem 3. Law of mean 4. Rolle’s theorem 13. The point that separates the convex part of a continuous curve from the concave part is 1. the maximum point 2. the minimum point 3. the inflection point 4. critical point 14. Identify the correct statement: a. a continuous function has local maximum then it has absolute maximum b. a continuous function has local minimum then it has absolute minimum c. a continuous function has absolute maximum then it has local maximum d. a continuous function has absolute minimum then it has local minimum 1. a and b 2. a and c 3. c and d 4. a, c and d 15. One of the conditions of Rolle’s theorem is 1. f is defined and continuous on (a, b)

2. f is differentiable on [a, b]

3. )()( bfaf

4. f is differentiable on ],( ba 6. DIFFERENTIAL CALCULUS- APPLICATIONS-II

1. The curve 22)2()1( xxy is not defined for

1. 1x 2. 2x 3. 2x 4. 1x

2. The curve 22)()( bxaxy , 0, ba and ba does not exist for

1. ax 2. bx 3. axb 4. ax

3. The curve )6()2(22

xxxy exists for

1. 62 x 2. 62 x 3. 62 x 4. 62 x

4. The curve )1()1(22

xxxy is defined for

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1. 11 x 2. 11 x 3. 11 x 4. 11 x 7. INTEGRAL CALCULUS

1. If xdxIn

nsin , then

nI

1. 2

1 1cossin

1

n

nI

n

nxx

n 2.

2

1 1cossin

1

n

nI

n

nxx

n

3. 2

1 1cossin

1

n

nI

n

nxx

n 4.

n

nI

n

nxx

n

1cossin

1 1

2. dxxf

a

)(

2

0

= 0 , if

1. f(2a-x) = f(x) 2. f(2a-x) = -f(x) 3. f(x) = -f(x) 4. f(-x) = f(x)

3. dxxf

b

a

)( is

1.

a

dxxf

0

)(2 2.

b

a

dxxaf )( 3.

b

a

dxxbf )( 4.

b

a

dxxbaf )(

4. The arc length of the curve y = f(x) from x=a to x=b is

1.

b

a

dxdx

dy2

1 2.

d

c

dxdy

dx2

1

3. 2

b

a

dxdx

dyy

2

1 4.2

d

c

dxdy

dxy

2

1

5. The surface area obtained by revolving the area bounded by the curve y= f(x), the two ordinates x=a, x=b and x-axis, about x-axis is

1.

b

a

dxdx

dy2

1 2.

d

c

dxdy

dx2

1

3. 2

b

a

dxdx

dyy

2

1 4.2

d

c

dxdy

dxy

2

1

6. dxexx45

0

is

1. 6

4

6 2.

54

6 3.

64

5 4.

54

5

7. dxxemx 7

0

is

1. m

m

7

2.

7

7

m

3.

17

m

m 4.

8

7

m

8. The area bounded by the curve x = f(y) to the left of y-axis and between the lines y=c and y=d is

1. xdy

d

c

2. - xdy

d

c

3. ydx

d

c

4. - ydx

d

c

9. The area bounded by the curve x = f(y), y-axis and the lines y=c and y=d is rotated about y-axis. Then the volume of the solid is

1. dyx

d

c

2

2. dxx

d

c

2

3. dxy

d

c

2

4. dyy

d

c

2


10. dxxf

a

)(

0

+ dxxaf

a

)2(

0

=

1. dxxf

a

)(

0

2. 2 dxxf

a

)(

0

3. dxxf

a

)(

2

0

4. dxxaf

a

)(

2

0

11. dxxf

a

)(

2

0

= ,)(2

0

dxxf

a

if

1. f(2a-x) = f(x) 2. f(a-x) = f(x) 3. f(x) = -f(x) 4. f(-x) = f(x)

12. dxex

x

26

0

is

1. 7

2

6 2.

62

6 3. 62

6 4. 62

7

8. DIFFERENTIAL EQUATIONS


32

2

2

4

dx

dy

dx

yd are

1. 2, 1 2. 1, 2 3. 2, 4 4. 4,2

2. The order and degree of the differential equation 22')'1( yy are

1. 2, 1 2. 1, 2 3. 2, 2 4. 1, 1


3

3

3

2

2

dx

yd

dx

dyy

dx

yd are

1. 2, 3 2. 3, 3 3. 3, 2 4. 2,2


2

3)'(" yyy are

1. 2, 3 2. 3, 3 3. 3, 2 4. 2,2

5. The order and degree of the differential equation 22)"()"(' yxxyy are

1. 2, 2 2. 2, 1 3. 1, 2 4. 1, 1 6. The order and degree of the differential equation sinx(dx+dy) = cosx(dx-dy) are 1. 1, 1 2. 0, 0 3. 1, 2 4. 2, 1 7. The differential equation corresponding to xy = c2 where c is an arbitrary constant, is 1. 0" xxy 2. 0" y 3. 0' yxy 4. 0" xxy


2

xdy

dxx

dx

dy

are

1. 2, 2 2. 2, 1 3. 1, 2 4. 1, 3

9. The order and degree of the differential equation 22)"()"(' yxyy are

1. 1, 1 2. 1, 2 3. 2, 1 4. 2,2 10. In finding the differential equation corresponding to y = emx where m is the arbitrary constant, then m is

1. 'y

y 2.

y

y ' 3. y’ 4. y

9. DISCRETE MATHEMATICS 1. Which of the following are not statements? (i) Three plus four is eight. (ii) The sun is a planet. (iii) Switch on the light. (iv) Where are you going? 1. (i) and (ii) 2. (ii) and (iii) 3. (iii) and (iv) 4. (iv) only 2. Which of the following are statements?


(i) 7+2<10. (ii) The set of rational numbers is finite. (iii) How beautiful you are! (iv) Wish you all success. 1. (iii), (iv) 2. (i), (ii) 3. (i), (iii) 4. (ii), (iv) 3. Let p be “Kamala is going to school” and q be “There are twenty students in the class”.“Kamala is not going to school or there are twenty students in the class” stands for 1. qp 2. qp 3. p 4. qp

4. If p is true and q is unknown then 1. p is true 2. )( pp is false 3. )( pp is true 4. qp is true

5. If p is true and q is false then which of the following is not true? 1. qp is false 2. qp is true 3. qp is false 4. qp is true

6. ‘+’ is not a binary operation on 1. N 2. Z 3. C 4. Q-{0} 7. ‘-’ is a binary operation on 1. N 2. Q-{0} 3. R-{0} 4. Z 8. ‘ ’ is a binary operation on 1. N 2. R 3. Z 4. C-{0} 9. In congruence modulo5, },25/{ zkkxzx represents

1. 0 2. 5 3. 7 4. 2 .

10. In the group (G, .), G = {1, -1, i, -i} the order of -1 is 1. -1 2. 1 3. 2 4. 0


Z , order of [3] is

1.4 2. 3 3. 2 4. 1

12. In ,S , Syxxyx ,, then ‘ ’ is

1. only associative 2. only commutative 3. associative and commutative 4. neither associative nor commutative 13. In (N, *), x*y = max{x, y}, x, y N, then (N, *) is 1. only closed 2. only semi group 3. only monoid 4. a group


1. 5 2. 3 3. 4 4. 2


1. 5 2. 3 3. 4 4. 2 16. In the group (G, .), G = {1, -1, i, -i} the order of i is 1. 2 2. 0 3. 4 4. 3

10. PROBABILITY DISTRIBUTIONS 1. A discrete random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of values 2. If X is a discrete random variable then P(X a) = 1. P(X<a) 2. 1 – P(X a) 3. 1 – P(X<a) 4. 0 3. If X is a continuous random variable then P(a<X<b) = 1. P(a X b) 2. P(a<X b) 3. P(a X<b) 4. all the three above 4. A discrete random variable X has probability mass function p(x), then 1. 1)(0 xp 2. 0)( xp 3. 1)( xp 4. 1)(0 xp

5. Which of the following is or are correct regarding normal distribution curve ? a. symmetrical about the line X = (mean)

b. Mean = median = mode c. unimodal d. Points of inflection are at X =

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1. a, b only 2. b,d only 3. a.b,c only 4. all 6. For a standard normal distribution the mean and variance are

1. 2, 2. , 3. 0, 1 4. 1, 1

7. If X is a discrete random variable then which of the following is correct ? 1. 1)(0 xF 2. 1)(0)( FandF

3. )(][1

nnn

xFxFxXP 4. F(x) is a constant function

8. If X is a continuous random variable then which of the following is incorrect ? 1. )()(' xfxF 2. 0)(1)( FandF

3. )()(][ aFbFbxaP 4. )()(][ aFbFbxaP


(i) E(aX+b) = aE(X)+b (ii) 2

122)'('

(iii) iancevar2 (iv) var(aX+b) = a2var(X)

1. all 2. (i), (ii), (iii) 3. (ii), (iii) 4. (i), (iv) 10. Which of the following is not true regarding the normal distribution? 1. skewness is zero. 2. mean = median = mode

3. the points of inflection are at X = 4. maximum height of the curve is 2

1

11. A continuous random variable takes 1. only a finite number of values 2. all possible values between certain given limits 3. infinite number of values 4. a finite or countable number of values 12. If X is a continuous random variable then P(X a) = 1. P(X<a) 2. 1 – P(X>a) 3. P(X>a) 4. 1-P(X a-1)

**************


SECTION- A NOTE: Choose the most suitable answer from the given four alternatives 40 x 1= 40

1. If dx

dy=

yx

yx

then

1) 2xy + y2 + x2 = c 2) x2 + y2 - x + y = c 3) x2 + y2 - 2xy = c 4) x2 - y2 - 2xy = c 2. The order and degree of the differential equation sinx(dx+dy) = cosx(dx-dy) are 1) 1, 1 2) 0, 0 3) 1, 2 4) 2, 1 3. If a compound statement is made up of three simple statements, then the number of rows in the truth table is 1) 8 2) 6 3) 4 4) 2 4. Which of the following is a tautology?

1) p q 2) pq 3) p ~p 4) p ~p 5. The value of [3] +11( [5] +11 [6] ) is 1. [0] 2)[1] 3)[2] 4)[3] 6. In (N, *), x*y = max{x, y}, x, y N, then (N, *) is 1)only closed 2) only semi group 3) only monoid 4) a group 7. Given E(X+ c) = 8 and E(X-c) = 12 then the value of c is

1) -2 2) 4 3) -4 4) 2 8. If the mean and standard deviation of a binomial distribution are 12 and 2. Then the value of its parameter p is

1)2

1 2)

3

1 3)

3

2 4)

4

1

9. For a Poisson distribution with parameter =0.25 the value of the 2nd moment about the origin is 1)0.25 2)0.3125 3)0.0625 4)0.025 10. Which of the following is not true regarding the normal distribution? 1)skewness is zero. 2)mean = median = mode

3)the points of inflection are at X = 4) maximum height of the curve is 2

1

11. The least possible perimeter of a rectangle of area 100 m2 is 1) 10 2) 20 3) 40 4) 60 12. The law of the mean can also be put in the form 1) 10),(')()( hahfafhaf 2) 10),(')()( hahfafhaf

3) 10),(')()( hahfafhaf 4) 10),(')()( hahfafhaf

13. If x = r cos , y = r sin , then

x

r

1) sec 2) sin 3) cos 4) cosec

14. If u = ysinx, thenyx

u

2

=

1) cosx 2) cosy 3) sin x 4) 0

MODEL QUESTION PAPER -I PART III- MATHEMATICS

(English Version) Time Allowed: 3 hours Maximum Marks: 200

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x

2/

2/cos2

sin

is

1) 0 2) 2 3) log 2 4) log 4 16. The area bounded by the line y = x, the x-axis, the ordinates x = 1, x = 2 is

1) 2

3 2)

2

5 3)

2

1 4)

2

7

17. The length of the arc of the curve x2/3 + y2/3 = 4 is 1) 48 2) 24 3) 12 4) 96

18. dxxf

a

)(

0

+ dxxaf

a

)2(

0

=

1) dxxf

a

)(

0

2) 2 dxxf

a

)(

0

3) dxxf

a

)(

2

0

4) dxxaf

a

)(

2

0

19. The solution of dy

dx+ mx = 0 , where m< 0 is

1) x = cemy 2) x = ce-my 3) x = my + c 4) x = c 20. The differential equation of all non-vertical lines in a plane is

1) dx

dy= 0 2) 0

2

2

dx

yd 3)

dx

dy= m 4) m

dx

yd

2

2

21. If x2 + y 2 = 1, then the value of iyx

iyx

1

1 is

1) x –iy 2) 2x 3) -2iy 4) x + iy 22. If z represents a complex number then arg (z) + arg ( z ) is

1) 4

2)

2

3) 0 4)

4

23. The conjugate of i13 + i14 + i15 + i16 is 1) 1 2) -1 3) 0 4) –i 24. The cube roots of unity are

1) in G.P. with common ratio 2) in G.P. with common difference 2

3) In A.P. with common difference 4) in A.P. with common difference 2

25. The length of the latus rectum of the parabola y2 - 4x + 4y +8 = 0 is 1) 8 2) 6 3) 4 4) 2 26. The tangents at the end of any focal chord to the parabola y2 = 12x intersect on the line 1) x-3 = 0 2) x + 3 = 0 3) y + 3 = 0 4) y - 3 = 0 27. The eccentricity of the hyperbola 12y2-4x2-24x+48y-127 = 0 is 1) 4 2) 3 3) 2 4) 6


1) 0124 yx 2) 0124 yx 3) 0124 xy 4) 0124 xy

29. The slope of the tangent to the curve y = 3x2 + 3sinx at x = 0 is 1) 3 2)2 3) 1 4) -1 30. The angle between the parabolas y2 = x and x2 = y at the origin is

1)

4

3tan2

1 2)

3

4tan

1 3) 2

4)

4

31. If A = [2 0 1], then rank of AAT is 1)1 2)2 3)3 4)0

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32. If A is a matrix of order 3, then det(kA) 1) k3det (A) 2) k2det (A) 3) k det (A) 4) det (A)

33. If A =

50

00, then A12 is

1)

600

00 2)

1250

00 3)

00

00 4)

10

01

34. Equivalent matrices are obtained by 1)taking inverses 2) taking transposes 3)takingadjoints 4) taking finite number of elementary transformations

35. If 0

cba , 5,4,3 cba


and b

is

1)6

2)

3

2 3)

3

5 4)

2

36. If kjiQSkjiPR

23,2 then the area of the quadrilateral PQRS is

1) 35 2) 310 3) 2

35 4)

2

3

37. If the magnitude of moment about the point kj

of a force kjai

acting through the point

ji

is 8 then the value of a is

1) 1 2) 2 3) 3 4) 4


kji

acting on a particle, if the particle is displaced from A (3,3,3)

to the point B(4,4,4) is 1) 2 units 2) 3 units 3) 4 units 4) 7 units

39. If a

=3, b

4 and ba

=9, then ba

is

1) 3 7 2) 63 3) 69 4) 69


is

1) 30

7 2)

7

30 3)

7

30 4)

30

7

--------------------------------------------------------------------------------------------------------------------------------------------------------

SECTION- B NOTE: Answer any ten questions. Question No.55 is compulsory and choose nine from remaining questions. 10*6=60

41. If A=

41

21 and verify that the result A (adj A) = (adj A)A = A . I2.

42. State and prove reversal law for inverses of matrices.

43. With usual notations prove that C

c

B

b

A

a

sinsinsin

44. Show that diameter of a sphere subtends a right angle at a point on the surface. 45. P represents the variable complex number z. Find the locus of P, if Im (z2) = 4 46. The tangent at any point of the rectangular hyperbola xy = c2 makes intercepts a, b and the normal at the point makes intercepts p, q on the axes. Prove that ap+bq = 0.

47. Find the equations of the tangent and normal to the curve 22

xxy at the point (1, -2).

48. Find the intervals of concavity and the points of inflection of the function 3

1

)1()( xxf

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49. If byaxzeV

and z is a homogeneous function of degree n in x and y prove that

Vnbyaxy

Vy

x

Vx )(

.

50. Evaluate

2

0

9

4sin dx

x

51. i) Solve. (D2+D+1)y = 0. ii) Form the differential equations for the given function y = ax2+bx+c by eliminating arbitrary constants {a, b}.

52. Show that (pq ) (p q ) is a tautology

53. 20% of the bolts produced in a factory are found to be defective. Find the probability that in a sample of 10 bolts chosen at random exactly 2 will be defective using (i) Binomial distribution (ii) Poisson distribution.

[e-2 = 0.1353]. 54. A discrete random variable X has the following probability distributions.

X 0 1 2 3 4 5 6 7 8

P(X=x) a 3a 5a 7a 9a 11a 13a 15a 17a

(i) Find the value of a (ii) Find P(x<3) (iii) Find P(3<x<7)

55. a) i) Find the least positive integer n such that 11

1

n

i

i.

ii) Prove that if ,13 then 1

2

31

2

3155

ii

(OR)

b) Show that (R – {0}, . ) is an infinite abelian group. Here . denotes usual multiplication --------------------------------------------------------------------------------------------------------------------------------------------------------

SECTION- C NOTE: Answer any ten questions. Question No.70 is compulsory and choose nine from remaining questions. 10*10 =100 --------------------------------------------------------------------------------------------------------------------------------------------------------

56. Discuss the solutions of the system of equations for all values of . x+y+z = 2, 2x+y-2z =2, x+y+4z =2. 57. Prove that cos(A-B) = cosAcosB +sinAsinB 58. Find the vector and Cartesian equation to the plane through the point (-1, 3, 2) and perpendicular to the planes x+2y+2z=5 and 3x+y+2z=8.

59. If and are the roots of the equation 0422

xx , then prove that 3

sin21

n

innn

and

deduct 99 .

60. Assume that water issuing from the end of a horizontal pipe, 7.5m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?

61. A kho-kho player in a practice session while running realises that the sum of the distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him if the distance between

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the poles is 6m. 62. Show that the line x-y+4=0 is a tangent to the ellipse x2+3y2=12. Find the co-ordinates of the point of

contact.

63. Evaluate x

x

xsin

0

)(lim

64. Use differentials to find an approximate value for the given number y = 4302.102.1

65. Find the area of region enclosed by y 2 = x and y = x-2.

66. Solve: xeyDDx

3cos1222

67. A cup of coffee at temperature 100C is placed in a room whose temperature is 15C and it cools to 60C in 5 minutes. Find its temperature after a further interval of 5 minutes.

68. Show that the set G of all matrices of the form

xx

xx, where 0 Rx , is a group under matrix

multiplication.

69. The air pressure in a randomly selected tyre put on a certain model new car is normally distributed with

mean value 31 psi and standard deviation 0.2 psi.

(i)What is the probability that the pressure for a randomly selected tyre

(a) between 30.5 and 31.5 psi (b) between 30 and 32 psi.

(ii) What is the probability that the pressure for a randomly selected tyre exceeds 30.5 psi

[ Area table: P(0<z<2.5) = 0.4938 ]. 70. a) At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km/hr and ship B is sailing north at 25 km/hr. How fast is the distance between the ships changing at 4.00 p.m?.

(OR) b) Derive the formula for the volume of a right circular cone with radius ‘r’ and height ‘h’

LIFE IS GOOD FOR ONLY TWO THINGS, DISCOVERING MATHEMATICS AND TEACHING

MATHEMATICS

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PART-A NOTE: i) All questions are compulsory. 40 x 1 =40

ii) Choose the most suitable answer from the given four alternatives and write the option code and the corresponding answer.

--------------------------------------------------------------------------------------------------------------------------------------------------------

1. The curve )(22222

xaxya has

1) a loop between x = a and x = -a 2) two loops between x = -a and x = 0; x = 0 and x = a 3) two loops between x = 0 and x = a 4) no loop 2. The area bounded by the curve x = f(y), y-axis and the lines y=c and y=d is rotated about y-axis. Then

the volume of the solid is

1) dyx

d

c

2

2) dxx

d

c

2

3) dxy

d

c

2

4) dyy

d

c

2

3. The differential equation corresponding to xy = c2 where c is an arbitrary constant, is 1) 0" xxy 2) 0" y 3) 0' yxy 4) 0" xxy

4. In congruence modulo5, },25/{ zkkxzx represents

1) 0 2) 5 3) 7 4) 2

5. If X is a continuous random variable then P(X a) = 1) P(X<a) 2) 1 – P(X>a) 3) P(X>a) 4) 1-P(X a-1) 6. If the equation -2x + y + z = l ; x - 2y +z = m ; x + y -2z = n such that l+m+n = 0, then the system has 1) a non-zero unique solution 2) trivial solution 3) infinitely many solution 4) No solution 7. The line 4x + 2y = c is a tangent to the parabola y2 = 16 x then c is 1) -1 2) -2 3) 4 4) -4 8. The distance between the foci of the ellipse 9x2 + 5y2 = 180 is 1) 4 2) 6 3) 8 4) 2 9. The asymptotes of the hyperbola 36y2 - 25x2 + 900 = 0 are

1) y = x5

6 2) y = x

6

5 3) y = x

25

36 4) y = x

36

25

10. The equation of the normal to the curve t

1 at the point

3

1,3 is

1) 3 = 27t - 80 2) 5 = 27 t -80 3) 3 = 27 t + 80 4) t

1

11. If the volume of an expanding cube is increasing at the rate of 4cm3/sec then the rate of change of surface area when the volume of the cube is 8 cubic cm is

1) 8 cm2/sec 2) 16 cm2/sec 3) 2 cm2/sec 4) 4 cm2/sec 12. The order of -i in the multiplicative group of 4th roots of unity is 1) 4 2) 3 3) 2 4) 1 13. The curve y2(x-2) = x2(1+x) has 1) an asymptote parallel to x-axis 2) an asymptote parallel to y-axis

MODEL QUESTION PAPER -II PART III- MATHEMATICS

(English Version)

Time Allowed: 3 hours Maximum Marks: 200

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3) asymptotes parallel to both axes 4) no asymptotes


x

2/

0

3/53/5

3/5

sincos

cos

is

1)2

2)

4

3) 0 4)


xx

2/

0cossin1

cossin

is

1) 2

2) 0 3)

4

4)


844

422

211

is

1)1 2) 2 3)3 4)4 17. If I is unit matrix of order n, where k≠0 is a constant, then adj(kI)= 1) kn(adj I) 2) k(adj I) 3) k2(adj I)) 4) kn-1(adj I)

18. The quadratic equation whose roots are i 7 is

1) x2 + 7 = 0 2) x2 - 7 = 0 3) x2 + x +7 = 0 4) x2 - x -7 = 0

19. If 0

cba , 5,4,3 cba


and b

is

1)6

2)

3

2 3)

3

5 4)

2

20. If yxbacacbcba

)()()( then

1) 0

x 2) 0

y

3) x

and y

are parallel 4) 0

x (or) 0

y (or) x

and y

are parallel

21. The curved surface area of a sphere of radius 5, intercepted between two parallel planes of distance 2 and 4 from the centre is

1) 20 2) 40 3)10 4) 30 22. The differential equation of all non-vertical lines in a plane is

1) dx

dy= 0 2) 0

2

2

dx

yd 3)

dx

dy= m 4) m

dx

yd

2

2

23. The degree of the differential equation c =

3/2

3

3

3

1

dx

yd

dx

dy

, where c is a constant is

1) 1 2) 3 3) -2 4) 2 24. The differential equation satisfied by all the straight lines in xy-plane is

1)dx

dy= a constant 2) 0

2

2

dx

yd 3) y+

dx

dy= 0 4) 0

2

2

ydx

yd

25. Which of the following are statements? (i) May God bless you (ii) Rose is a flower

(iii) Milk is white (iv) 1 is a prime number 1) (i) , (ii) , (iii) 2) (i) , (ii) , (iv) 3) (i) , (iii) , (iv) 4) (ii) , (iii) , (iv) 26. In the homogeneous system )( A < the number of unknowns then the system has

1) only trivial solution

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2) trivial solution and infinitely many non-trivial solutions 3) only non-trivial solutions 4) no solution 27. The non-parametric vector equation of a plane passing through the points whose position vectors are

a

. b

and parallel to v

is

1) 0 vabar

2) 0 vabr

3) 0vba

4)

0bar

28. If a line makes 450, 600 with positive direction of axes x and y then the angle it makes with the z-axis is

1) 300 2) 900 3) 450 4) 600

29. If cba

,, are non-coplanar and ][][ accbbaaccbba

then ],,[ cba

is

1) 2 2) 3 3) 1 4) 0 30. If a compound statement is made up of three simple statements, then the number of rows in the truth table is 1) 8 2) 6 3) 4 4) 2 31. The point of inflexion of the curve y = x 4 is at 1) x = 0 2) x = 3 3) x = 12 4) nowhere 32. A box contains 6 red balls and 4 white balls. If 3 balls are drawn at random, the probability of getting

2 white balls without replacement is

1) 20

1 2)

125

18 3)

25

4 4)

10

3

33. For a Poisson distribution with parameter =0.25 the value of the 2nd moment about the origin is 1)0.25 2)0.3125 3)0.0625 4)0.025 34. If the mean and standard deviation of a binomial distribution are 12 and 2. Then the value of its parameter p is

1)2

1 2)

3

1 3)

3

2 4)

4

1

35. If the amplitude of a complex number is 2

, then the number is

1) purely imaginary 2) purely real 3) 0 4) neither real nor imaginary 36. The equation of the plane passing through the point (2 , 1 , -1) and the line of intersection of the

planes ;0)3.( kjir

and 0)2.( kjr

is

1) x + 4y -z = 0 2) x + 9y +11z = 0 3) 2x + y - z +5 =0 4) 2x -y +z = 0

37. If x 2 + y 2 = 1, then the value of iyx

iyx

1

1 is

1) x –iy 2) 2x 3) -2iy 4) x + iy 38. Polynomial equation P(x) = 0 admits conjugate pairs of imaginary roots only if the coefficients are

1) imaginary 2) complex 3) real 4) either real (or) complex

39. The condition that the line 0 nmylx may be a tangent to the rectangular hyperbola 2cxy is

1) 22222nmbla 2) ln

2am 3) 22222

nmbla 4) 224 nlmc

40. If a real valued differentiable function y = f (x) defined on an open interval I is increasing then

1) 0dx

dy 2) 0

dx

dy 3) 0

dx

dy 4) 0

dx

dy

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--------------------------------------------------------------------------------------------------------------------------------------------------------

PART-B NOTE: i) Answer any ten questions. 10 X 6 = 60

ii) Question No. 55 is compulsory and choose any nine questions from the remaining: --------------------------------------------------------------------------------------------------------------------------------------------------------

41. Prove that (AB)-1 = B-1A-1, where A and B are two non-singular matrices.


1012

7436

3124

43. (i) If dcba and ,dbca show that da and cb parallel.

(ii) Find the angle between 42 zyx and 42 zyx

44. Find the vector and Cartesian equation of the sphere on the join of the points A and B having position

vectors kji 762 and kji 342 respectively as a diameter. Find also the centre and

radius of the sphere.

45. (i) Graphically prove that 321321 zzzzzz

(ii) Simplify :

86

33

sincos4sin4cos

3sin3cos2sin2cos

ii

ii

46. The orbit of the earth around the sun is elliptical in shape with sun at a focus. The semi major axis is of length 92.9 million miles and eccentricity is 0.017. Find how close the earth gets to sun and the greatest possible distance between the earth and the sun.

47. Prove that

2,0,tansin

xxxx

48. Resistance to motion F, of a moving vehicle is given by .1005

xx

F Determine the minimum value of

resistance.

49. A circular template has a radius of 10 cm 02.0 . Determine the possible error in calculating the

area of the templates. Find also the percentage error.

50. Find the area of the region bounded by ,362

yx y - axis , y = 2 and y = 4.

51. (i) Construct the truth tables for the given statement qqp ~

(ii) Show that ,11

Gaaa

a group.

52. Show that the set G = {

11

10,

01

11,

10

01} is a group under of matrix multiplication.

Determine whether G forms an abelian group?

53. A continuous random variable X has p.d.f. 23 xxf , 10 x , Find a and b such that.

(i) aXPaXP and (ii) 05.0 bXP

54. In a gambling game a man wins ` 10 if he gets all heads or all tails and loses ` 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.

55. a) For what values of x and y, the numbers yix2

3 and iyx 42

are complex conjugate of

each other? [OR]

b) Solve: 2a

dx

dyxy


--------------------------------------------------------------------------------------------------------------------------------------------------------

PART-C NOTE: i) Answer any ten questions. 10 X 10 = 100

ii) Question No. 70 is compulsory and choose any nine questions from the remaining. --------------------------------------------------------------------------------------------------------------------------------------------------------

56. Find the adjoint of the matrix

110

432

433

A and verify the result

.A A )A adj ( )A adj (A

57. Find the vector and Cartesian equation of the plane containing the line 3

1

3

2

2

2

zyx

and parallel to the line .1

1

2

1

3

1

zyx

58. Show that the lines 31

1

1

1 zyx

and

1

1

2

1

1

2

zyxintersect and find their point of

intersection. 59. If sincossincos ibandia , then Prove that

(i) cos ( + ) =

2 * +

+ (ii) sin ( - ) =

2 *

−

+

60. The girder of a railway bridge is in the parabolic form with span 100 ft. and the highest point on the arch is 10 ft, above the bridge. Find the height of the bridge at 10 ft, to the left or right from the midpoint of the bridge. 61. The arch of a bridge is in the shape of a semi –ellipse having a horizontal span of 40 ft and 16 ft high at the centre. How high is the arch, 9 ft from the right or left of the centre. 62. Find the equation of the rectangular hyperbola which has for one of its asymptotes the line

052 yx and passes through the points (6,0) and (-3,0).

63. Find the equations of those tangents to the circle 5222 yx , which are parallel to the straight line

632 yx .

64. Find the intervals of concavity and the points of inflection for the given functions 432

212 xxxy

65. Find the area of the region bounded by the ellipse 12

2

2

2

b

y

a

x

66. Find the perimeter of the circle with radius a.

67. Solve: x

eydx

dy

dx

yd 3

2

2

223 when 0,2log yx and 0,0 yx

68. A drug is excreted in a patients urine. The urine is monitored continuously using a catheter. A patient

is administered 10 mg of drug at time t = 0 , which is excreted at a Rate of 213t mg/h.

(i) What is the general equation for the amount of drug in the patient at time t > 0 ? (ii) When will the patient be drug free?

69. Show that the set QbabaG ,/2 is an infinite abelian group with respect to addition.

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70. a) If v

euw2

where y

xu and ,log xyv find

x

w

and

y

w

[OR] b) The mean score of 1000 students for an examination is 34 and S.D. is 16. (i) How many candidates can be expected to obtain marks between 30 and 60 assuming the normally of the distribution and (ii) Determine the limit of the marks of the central 70% of the candidates.

[ Area table : P(0<z<0.25) = 0.0987, P(0<z<1.63) = 0.4484 , P(0<z<1.04) = 0.35 ]

TIME IS ALWAYS RUNNING, ONCE LAPSED IS LAPSE FOREVER, YOU CANNOT

RE-CAPTURE TIME.

TIME IS PRECIOUS! TIME IS GOD!

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