92Inter 1st Year MATHS 1B Guess Paper E

7/13/2019 92Inter 1st Year MATHS 1B Guess Paper E

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Maths IB

Section A

I) very short answer type question

i)Answer all questions

II)Each question carries two marks

1)If the area of the triangle formed by the straight lines x = 0 and Y = 0 and 3x+4y = a (a > 0 ) is 6. Find thevalue of a.

2)Find the equation of the straight line passing through (-4, 5) and cutting off equal non zero intercepts on the

coordinate axes.

3)Find the fourth vertex of the parallelogram whose consecutive vertices are (2, 4, -1) (3, 6, -1) and (4, 5, 1)

4)If (3, 2, -1), (4, 1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid of a tetrahedron, find the

fourth vertex.

5)Compute lim +1- 6)Compute lim

7)If Y = 7+ 3x (x > 0) then find 8)Differentiate f(x) w.r.to g(x) if f(x) = and g(x) = 9)Find the approximate value of 8210)Find the value of C in Rolls theorem for the function f(x) = + 4 on (-3, 3)

Section B

11)A (2, 3); B (-3, 4) are two given points. Find the equation of locus of P so that the area of P so that the area

of

PAB is 8.5 Sq. units.

12)When the axes are rotated through an angle of 45, the transformed equation of a curve is

17-16xy+17= 225. Find the original equation of the curve.13)Find the equation of the line perpendicular to the line 3x+4y+6 = 0 and making an intercept 4 on axis.

14)If f(x) given by f(x) = 12 < 1is continuous function on R then find the values of K.

15)Find the derivative of cosby using first principle w.r.to x.16)The distance time formula for the motion of a particle along a straight line S = - 9+ 24t 18 thenfind when and where the velocity is zero.

17)Show that the curves 6

-5x+2y = 0 and 4

+8

= 3 touch each other at

,

Section C

III)Long answer type question

i)Answer any five questions

ii)Each question carries seven marks.

18)Find the ortho centre of the triangle with the vertices (-2, -1), (6, 1) and (2, 5).

19)Show that the two pairs of lines 6-5xy-6= 0 and 6-5xy-6+x+5y-1 = 0

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20)Find the angle between the lines joining the origin to the points of intersection of the curve +2xy+ +2 +2 5= 0 and the line 3x y +1 = 021)If a ray makes the angles ,,,with four diagonals of a cube then findcos +cos +cos +cos 22)If Y = x ++ log(+ +) then prove that = 2 +23)If the tangent at any point P on the curve . = (mn 0) meets the coordinate axes in A, B thenshow that AP:BP is a constant.24)A window is in the shape of a rectangle surmounted by a semicircle. If the perimeter of the wondow is

20ft, find the maximum area.

Solutions

I

1)3x+4y-a = 0

area of triangle

||= 6||= 6= 6 64a = 12

2)Point (-4, 5)

Each line+

= 1 + = 1

(-4, 5)

-4 + 5 = 9 x + y = 9

a = 1

line x + y = 1

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

Fourth vertex (- + , + , + )= (2-3+64, 4-6+5, -1+1+1)

= (3, 3, 1)

4)(3, 2,-1) (4, 1, 1) (6, 2, 5) three vertices centroid = (4, 2, 2)G = , , (4, 2, 2) = , , 4 =

2 =

2 =

= 3 = 3 = 3

(3, 3, 3)

5)lim +1-





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Rationalize

lim

lim

lim

X =

lim

=

6)lim lim

1+ +1

lim lim1+ +1

11+0+1 =2

7)y = 7 = 7

. (2x + 3)

8)f(x) = g(x) = f()= g()= = 2. =

()() =

= 2.

9)82X = 81 = 1 f(x) = f()= = 1 = = 0.05582= f(x + ) = f(x) + 81+ 0.05582 = 9.05510)f(x) = + 4f()= ()()() = (9+4) (9-





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II)

11) A (2, 3) B (-3, 4) P (x, y)

PAB = |(3 4) +2( ) 3( 3)| | 5 +17|= 17X + 5y -17 = 17(x + 5y 34) (x + 5y) = 0

12) = 4517- 16y + 17 = 225 (1)X = cos45+ sin45= Y = -x sin45+ cos45= - (1) 17

- 16

+ 17

= 225

13)Equation of the line 3x+4y+6 = 0 (1)

Slope =

Slope of the line r to (1) =

Y = mx + c

m = c = -4

y = - 4

4x 3y -4 = 0

14)f(x) = 12 < 1

limf(x) = 2limf(x) = lim k k = 2

k 2 = 0

(k 2) (k + 1) = 0 k = 2

K = -1

15) f(x) = cosf()= lim ()() lim

()





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lim

-2 lim

. sin

-2 . a . sin

= -a sin16)S = - 9+ 24t 18V =

= 3+ 18t +24 = 0- 6t +8 = 0

(t 2) (t 4) = 0

T = 2 t = 4

T = 2 S = 8-36+48-18= 56-54 = 2

Section CIII

18)A (-2, -1), B 96, -1), C (2, 5) are the vertices of

Slope of BC ==

=

AD is to BC

Slope of AD =

Eq. of AD is

Y + 1 =(x + 2)

2x-3y+1 = 0 (1)

Slope of BE isY + 1 = - (x -6)2x + 3y 9 = 0 (2)

By solving (1) and (2)

3 -9 2 3

-3 1 2 -3

A

B CD

0

E

(-2, -1)

(6, -1) (2, -5)





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=

=

=

=

X = = 2 y =

=

Coordinates of ortho centre 0 are (2,)

19)H6-5xy-6= (3x+2y)(2x-3y) andS 6-5xy-6+x+5y-1= (3x+2y - 1)(2x-3y + 1)

Clearly, H = 0 represents a pair of St. lines and S = 0 also represents a pair of lines. Further the lines represents

by H = 0 are parallel to the lines represented by S = o. therefore, the four lines form a rectangle.

But the distance of each lines 3x + 2y 1 = 0. 2x 3y + 1` = 0 from origin

. Hence the rectangle is square.

20)

Eqn. of the curve is +2xy+ +2 +2 5= 0 (1)Eqn of AB is 3x y +1 = 0

Y 3x = 1 (2)

Homogenizing (1) with the help of (2) combined equation of OA, OB is

+2xy+ +2.1+2.1 5(1)= 0+2xy+ +2( 3) +2( 3) 5( 3)= 0+2xy+ +2 6 + 2 6 5( + 9 6)= 0-5-2xy + 3-5- 45+ 30xy = 025- 14xy + = 0Suppose Q is the angle b/w OA and AB

cos= ||() = ||() =

=

=

= cos

B

A





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21) let each side of the cube be of length a. let one of the vertices of the cube be the origin 0 and the co-

ordinate axes be along the 3 edges , and passing through the origin. The co-ordinates of thevertices of the cube w.r.t the frame of reference OABC are as shown in figure. The diagonals of the cube are

, ,, and (a, a, a), (a, a, -a), (-a, a, a) and (a, -a, a) are direction ratios of these diagonals.

20x - 2 + f(x) = 20x - 2 f()= 0 20 4x - (( +4)x = 20X =

f()= -4 - < 0f(x) is max when x =

=

=

Max area = 2xy + =

+

()

=

() =

()() =

Sq feet.

Let the direction cosinesof the given ray be (l, m, n).. if this ray makes the angles ,,,with the fourdiagnols of the cube, then

cos

=

||

||

=

ly cos= || cos= || cos= || cos + cos + cos + cos = {| + + | + | + | + | + +| +| + |}

=[4( + + )]=

Since + + = 1

B (0, 4, 0) D (a, a, 0)

P (a, a, a)

A (a, a, 0)

F (a, 0, a)

0 (0, 0, 0)

C (0,0, a)

E (a, a, a)

Z





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X

Y

B

A0

P(x1, y1)

22)Given Y = x + + log + + = 2 + + 1+ 1+ 2

=

+ ++

=

+ ++

=

+ + = + + +=2 +

23)Eq. of the curve = Differentiating w.r.t x + = 0n = - . =

.. = -

Slope of the tgt at P(x1, y1) =

Eqn of the tgt at P is

y y1=

(x x1)

nx1y nx1y1= -my1x + mx1y1ny1x + nx1y = mx1y1+nx1y1= (m+n) x1y1() + () = 1

+

= 1

OA =

, OB = Co ordinates of A are ,0and B are 0, Let P divide AB in the ratio K:l

Coordinates of P are

() ,() = (,)

=

= (1) = 1=

(2)





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x

y

x

Dividing (1) by (2) =

= P divides AB in the ratio n:m

i.e. AP:PB = n:m = constant.

24)Perimeter = 2x + 2y +.x = 202y = 20 2x = 2x 10 +



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92Inter 1st Year MATHS 1B Guess Paper E