Question Bank - 2

31

SRI CHAITANYA DAY-SCHOLARS EDUCATIONAL INSTITUTIONS , VJASubject : Maths Important questions

SHORT TERM-2003 (10-04-03)

1. 5 41 2 3COS X COS XCOS X dx is equal to .........

1)sin 2sin

2XX c 2)

cos3 sin3

X X c 3) sin 5 sin 4

5 4X X

c

4) sinx+sin2x+c

2.1 1( ) ( ) ( ) ( )( ). ( )

f x g x f x g xf x g x

{log(g(x))-logf(x)}dx =

1) 2

( )log( )

g xcf x

2)

21 ( )log2 ( )

g xcf x

3)2

( )log( )

g xcf x

4)

21 ( )log2 ( )

g xcf x

3) 22

( 3 3) ( 1)x

x x x

dx=................

1) 12 tan

3( 1)3x

cx

2)

1

2

1 tan3 3( 1)

xc

x

3) 12 tan

13x

cx

4)

11 tan13

xc

x

4. 32 3sin 21 cos2

x xe dxx

=

1) 3xe cotx+c 2) 3xe tan x+c 3) 3 3xe sin x+c 4) 3 3xe cos x +c

5. 2 3 / 2log

( 1)x x

x dx=

1) 1

2

logsec1

xx c

x

2)1

2

logsec1

xx c

x

3) 1

2

logcos1

xx c

x

4) 1

2

logcos1

xx c

x

6. 3 2 /3 1/3( 2 2)xe x x c then ................1) K=2 2) K=3 3) K=4 4)K=5

32

7.2

2 3 / 2

1( 1)

xx x ex

. dx=

1) 2 1

xe

cx

2) 2 1xe x c 3) 2 1x

ec

x

4) 3 / 22 1

xec

x

8. 1{ ( ) ( )} ( )xe f x f x dx g x then ( )xe f x dx =..............

1) g(x) +exf(x) 2) g(x) - ex f(x) 3) 1 { ( ) ( )}2

xg x e f x 4) 1 { ( ) ( )}2xg x e f x

10. cos .log tan 2x

x dx

1) sin log tan 2x

x x c

2) sin log tan 2x

x c

3) sin log tan 2x

x x c

4) sin log tan 2x

x x c

11.3

sin2

cos sincos

x x x xe dx

x

1) esinx(secx - x) +c 2) esinx(x - secx) +c 3) esinx(tan x - x) +c 4) esinx(secx - x) x +c

12. log 2 2x x dx =

1) 1log 2 2 sin 2 2x x

x x x c

2) 1log 2 2 sin 2 2x x

x x x c

3) 1log 2 2 sin 2 2x x

x x x c

4) 1log 2 2 sin 2 2x x

x x x c

15. (ax+b), (bx+a) = (a+b)2 has roots ..............

1) 1,(a+b)2 2) 2 2( )1, a bab

3) 2 2( )1, a b ab

ab

4) 2 2( )1, a b ab

ab

19. I f the roots of 5x2+(p+q) x +(2p+1) =0 are reciprocals of roots of qx2+2px +5 = 0 then(p,q)=.....

33

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

22. I f , are the roots of 2 0ax bx c then the roots of 2 2( 1) ( 1) 0ax bx x c x are

1) ,1 1a

2) ,

1 1a

3) 1 1,

a

4)

1 1,a

26. I f , are the roots of 2 2 8 0x x then the value of1/ 3 1/32 2

is equal to

1) -1/3 2) 1/2 3) -1 4) 1

28. If P1,P2 = 2(q1+q2) then the equation 2

1 1 0x p x q and 2

2 2 0x p x q has.......roots

1) Exactly 2 reel 2) at least 2 reel 3) 4 imaginary 4) All real

23. If the equations 2 2 2( 5 6) ( 3 2) (2 4) 0x x is an identity in x then mustbe....

1) 3 2) 1/2 3) 2 4) 1/3

25. If one root of x2 +qx+p=0 is the square of the other than q(3p-q2) =.............

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

33. The centre of the circle passing through (0,0) ,(2,0) and touching the circle2 2 64x y is..........

1) (1, 3) 2) (1, 4) 3) (1, 5) 4) (1, 63)

37. If the circles 2 2 2 0x y y c and 2 2 2 0x y y c are such that one lies completelyinside the other inside the other than ............

1) 0, 0c 2) 0, 0c 3) 0, 0c 4) 0, 0c

39. If O is the origin A= (a,b) , B=(c,d) and the circles on OA,OB as diameters are describedthen the length of common chord is

1) ac bd

AB

2) ac bd

AB

3) ad bc

AB

4) ad bc

AB

DAILY TEST-14 (19-01-05)

5. If z =3 , then the point 1-z lies on

1) circle of radius 3, centre (1,0) 2) straight line through the origin

3) circle of radius 2, and centre (0,-1) 4) circle of radius 3 and centre (-1,0)

8. If 0

34

14. tan loga ibia ib

1) ab 2) 2 2

2a b

ab

3) 2 22ab

a b 4) 2 22ab

a b

32. If z= ire then ize is equal to

1) sinre 2) cosre 3) sr inre 4) cosrre

SR.INTERSPECIAL BATCH UNIT-TEST=1(02-01-04)

6. If a sin x= b cos x= 22 tan

1 tanc x

x then

2 2

2 2

( )a ba b

=

1) c2 2) 2c2 3) 4c2 4)8c2

14. In a le ABC, A>B if A and B satisfy the equation 3 sin x - 4 sin3 x- k= 0 ,0

35

1) -2 2)1/2 3) 2 4) 4

73. If the origin is shifted to (1,1) with out rotation of axis then the equation x =1 transformsto

1) x(x-2)=0 2) x(x+2)=0 3) x(x-1)=0 4) x(x+1)=0

74. A line has intercepts a and b on x and y axis respectively . When the axis are rotatedthrough an angle keeping the origin fixed the line makes equal intercepts on the axisthen tan =

1) a ba b

2) a ba b

3) a bb a

4) b

a

77. If u= r5 where r2 =x2+y2+z2 then 2 2 2

2 2 2

u u u

x y z

=

1) 30r3 2) 30 r2 3) 20 r3 4) 20 r2

79. If sin cos 1y x y

zx y x

then xz

x

=

1) zyx

2) zyy

3) 2zyy

4)

2 zyy

=

LIMITS

3. The value of

3

2 1

23 1lim4 1

x

x

x

x

x

is

1) 0 2) 1 3) 2 4) 3

4. The value of the limit

212sin 2 2

sin1 1sin sinlim 1 2 ........x

x

x x

xn

1) 1 2) 0 3) n 4) 1

2n

23. 20tan 2 tanlim(1 cos2 )x

x x x x

x

is?

1) 2 2) -2 3) 12 4)

12

DAILY TEST-13SPECIAL BATCH (18-01-05)

7. If the hypotenuse of a right angled triangle is four times the length of the perpendiculardrawn from the opposite vertex to it, then the difference of the two acute angles will be1) 300 2) 450 3) 150 4) 600

36

23. If r1 - r = KR then k belongs to

1) ( 0 , 4 ) 2) [ 0 , 4 ] 3) 0, 4 4) 0, 4


1. If , , are angles of a triangle then the value of

tan , tan , tani j k i j k i j k in the simplest form is1) 1 2) 2 3) 3 4) 4

2. If , ,a b c represent three concerrent edges of a rectangular parallelopiped whose lengths

are 4, 3, 2 respectively then the value of .a b c a b b c c a is1) 0 2) 48 3) 72 4) 36

9. V1 is the volume of a parallelopiped and V2 is the volume of the parallelepiped with threeconcurrent diagonals of three faces of the original parallelopiped. Then V1 : V2 =

1) 8 : 1 2) 1 : 1 3) 2 : 1 4) 1 : 2

23. A B C is a triangle whose vertices have P.Vs , ,i j k respectively. A line parallel to thevector i = j and passing through the point 2 3i j k cuts the plane of the triangle at P..then P.V of the point P is

1) 1 6

3j

2) 1 6 j 3) 5 62i j k

4) 5 6

2i j k

25. The shortest distance between a line passing through A ( 1, -1, 2 ) and parallel to the vector

2i j , and a line passing through B ( 2, 0 , 3 ) and parallel to the vector 2i j1) 1 2) 2 3) 3 4) 4


3. f (x) = tan1 3 2

12 2

3 1, sin1 3 1x x xg x

x x

and 0.4

0.40.4x

f x fLt f x g

= _______

1) 12 2)

34 3)

32

4) 1

26. 2 28 3 4 3xLt x x x x

= ____________

1) 0 2) 3) 2 4) 1 / 2

37


8. If 2 is a root of the equation ax2 + bx + c = 0 then the set of lines ax + by + c = 0 passthrough the point

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

27. If the lines a1x + b1y = 1, a2x + b2 y = 1 and a3 x + b3 y = 1 are concurrent , then the points(a1 , b1 ) , ( a2 , b2 ) and ( a3 , b3 ) form

1) an equilateral triangle 2) a right angled triangle

3) a straight line 4) a scalence triangle

28. The vertices of a triangle are A ( -1 , -7 ) B ( 5 , 1 ) and C ( 1, 4 ) . The equation of thebisector of ABC is

1) x - 7y + 2 = 0 2) x + y - 6 = 0 3) x + 2y - 7 = 0 4) x - y - 4 = 0

30. If a vertex of an equilateral triangle is the origin and the side opposite to it has theequation x + y = 1, then the orthocentre of the triangle is

1) 1 1,2 2

2) 2 2,

3 3

3) 1 1,3 3

4) 2 2,3 3


1. If 8Cos x + 15 Sin x = 15 and Cos x 0 then 8 Sin x - 15 Cos x =

1) 8 2) -8 3) 8 4) 0

7. Arrange the following in increasing order based on their values

A = Cos 1c B = - Cos 2 c C = - Cos 3c

1) A, B, C 2) B, A , C 3) B, C, A 4) C, A, B

40. When the origin is shifted to a suitable point, the equation 2x2 + y2 - 4x + 4y = 0 transformedas 2x2 + y2 - 8x + 8y + 18 = 0 . The point to which origin was shifted in1) ( 1, 2) 2) ( 1, - 2) 3) ( -1, 2 ) 4) ( -1, -2 )

WEEKEND TESTSPECIAL BATCH (09-01-05)

1. The integer n for which

0

cos 1 cos x

nx

x x eLt

x

is a finite nonzero number is

1) 1 2) 2 3) 3 4) 4

38

4. 1 11 22x

x xLt Tan Cotx x

1) -1 2) -1/2 3) 1/2 4) 1

5.2 2 2 2

20

tan tansinx

e x e xLt

x

is

1) 0 2) 8 3) 15 4) 10

6.0

2 .3 .5 7 .1111 .5 2 .3 .7

x x x x x

x x x x xxLt

1) 30 / 7755log42

2) 55/ 4230log77

3) 6 / 75log7

4) 6 / 76log7

7. 1 1log log 1 ....... log kk

n n nnLt n n n k N

1) 1 2) k 3) 2k 4) 0

9. Let f (x) = 2 2log 1 log 1

, 0sec cos

x x x xx

x x

. Then the value of f(0) so that f is

continuous at x = 0 is

1) 1s 2) 0 3) 2 4) -1

DAILY TEST - 3 : 27-03-05

1. If x = log3 243, y = log2 64 then 2x y =

1) 3 -1 2) 2 - 3 3) 3 - 2 4) 3 - 4

3. If 3log sin 2 2 cos 2, 2 2x x x then number of solutions of the given in-equality are1) 0 2) 3 3) infinite 4) 2

12. If x1 < x2 and x1 , x2 are the roots of x2 - 26x + 120 = 0 then 1 2x x

1) 5 +1 2) 5 +2 3) 3 2 4) 5 3

GRAND TEST : 10-04-058. If the projections of the line segment AB on the YZ plane, ZX plane and XY plane are

160, 153 , 5 respectively, then the projections of AB on Z axis is

1) 13 2) 12 3) 3 4) 5

16. Three normals are drawn from the point ( c , 0 ) to the parabola y2 = x. One normal is

39

always x axis. If the other two normals are perpendicular to each other, then the value ofc is

1) 34 2)

12 3)

32 4)

52

20. The equation 7 1 cos sinr

represents

1) a Parabola 2) an Ellipse

3) circle 4) a Rectangular hyperbola

75. If / 2 2

0

sinsinn

nxa

x

, then a2 - a1 , a3 - a2 , a4 - a3 are in

1) A.P 2) G.P 3) H.P 4) AGP

77. The area bounded by the x - axis , the curve y = f (x) and the lines x = 1 and x = b is equal2 1 2b for all b > 1 , the f (x) is equal to

1) 1x 2) 1x 3) 2 1x 4) 21x

x

80. The value of 0

cos x dx

, where [ ] is the greatest integer function is

1) 2

2) 0 3) 4) 2

DAILY TEST : 28-3-052. f (x) = log ( log x ) is increasing in

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

3. f (x) = 0x a aa x is decreasing in

1) a x a 2) 0 < x < a 3) -a < x < a 4) ,0 0,a a

4. If y = x3 - ax2 + 12x + 5 is increasing for all real values of x then a lies between1) -12 , 12 2) -11, 11 3) -6, 6 4) -10 , 10

5. f (x) = 225 4x is decreasing in

1) ( -3, 0 ) 2) 50,2

3) 5 ,0

2

4) 5 5,

2 2

7. The interval of Monotonicity of f (x) = log x is

40

1) R 2) 0, 3) ,0 4) R - {0}

8. When x ( -6, 8 ) the function f (x) = ( x + 6 )4 ( 8 - x )3 is

1) Wholly increasing 2) Wholly decreasing

3) First increasing and then decreasing 4) First decreasing then increasing

10. If f (x) = 1xx is decreasing in ,e then

1) e e 2) ee 3) ee 4) None

12. If f(x) = x2.ex then the interval in which f (x) increases is1) ( 0 , 2 ) 2) ( -2, 0 ) 3) ( -2, 2 ) 4) R

13. If a < 0 , then f (x) = ax axe e is monotonically decreasing for

1) x > 0 2) x < 0 3) R 4) ( 0 , 1 )

14. Stationary values of f(x) = x ( logx )2 are

1) 241,e

2) 0, 4e2 3) 240,e

4) 211,e

15. If f (x) = sin cossin cos

a x b xc x d x

is decreasing for all x then

1) ad - bc > 0 2) ad + bc < 0 3) ad - bc < 0 4) None

16. 2

3Y x x increasing for all values of x lies in the interval

1) 30,2

2) 0, 3) ,0 4) 1,3

19. f (x) = x3 + px2 + qx + r is an increasing function for all real x , then condition is1) P2 < 3q 2) P2 > 3q 3) P < 3q 4) P2 < 3r

20. The values of a and b such that x3 + 3ax2 + 3a2x + b is increasing on R - {a} are1) 1, 2 2) a,b are any real numbers

3) -1, 2 4) 1

DAILY TEST : 25-3-05

4.

21/

20

2x

x x

x

e eLtx

is

1) e1/2 2) e1/4 3) e1/3 4) e1/2

41

6.

3 3

x a

x xLta a

( a < 0 ) where [ x ] denotes the greatest integer less than or equal to x

is1) a2 - 3 2) a2 - 1 3) a2 4) a2 + 1

19. Let f(x) =[x2+1],[x] is greatest integer less than or equal to x. Then f is continues1) on [1 3] 2) x in [1 3] except four points

3) x[1 3] except seven points 4) x [1 3] except eight point

29. The general solution of Sin x 3 sin 2 x + sin 3 x = Cos x 3 Cos 2 x + Cos 3x

1) 8n

2) 2 8

n 3) 1 2 8

n n 4)

1 322

n cos

AIEEE CUMULATIVE (DATE : 13-12-04)18. The number of terms in the expansion of (x + y + z)10 is

1) 11 2) 33 3) 66 4) 132

21. I f 3 8 n I F where I and n are positive integers and 0 < F < 1 then I is1) Any integer 2) an even integer3) An odd integer 4) Can not be determined

25. A, B, C are three speakers amongst seven persons who speak at a function. The numberof ways in which it can be done if A speaks before B ad B speaks before C is1) 720 2) 840 3) 5040 5) 1680

32. tan 3 . tan 2 . tanx x x dx =

1) 1 1log sec log sec 23 2

x x C

2) 1 1log sec 3 log sec 2 log sec3 2

x x x C

3) 1 1log sec3 log sec 2 log sec3 2

x x x C

4) None

33. 24 6 log 9 49 4

x xx

x x

e e dx Ax B e Ce e

then (A, B) =

1) 3 35,2 36

2) 3 35,2 36

3) 3 35,

2 36

4) 3 35,

2 36

42

36. 3 / 22 9dx

x

=

1) 29 9x C

x

2) 29 9

x Cx

3) 2 9

x Cx

4) 2 9x C

x

SDS / SPL / UNIT TEST -4 (DATE : 30-3-05)3. If ,a b are non zero vectors and have opposite direction such that 3xa b c and a b xc

then x lies in ............

1) 11, 1,3

2) 1, 3) 0, 4) 1,3

4. I f a b i j k , 3a b i j k are the diagonals of parallelogram then its per imeter is

1) 2 2 2) 2 5 2 3) 5 2 4) 2 2 25. I f a i j k , 2b i j k are two vectors and if xi yj zk is in the angular bisector

direction of aand b then ...........

1) x = y 2) y = z

3) z - x 4) y is the A.M of x and y

6. I f , ,a b c are unit vectors and projections of , ,a b c on , ,b c a respectively are 1/2, 1, 1/2then the value of a b c =

1) 14 2) 2 3 3) 1 4) 2 2

9. I f 2 2a i j k , 2 4b i j k and p is length of projection of aandb and q is the lengthof projection of band a and if r is the sine of angle between them then ascending order ofp, q, r is ..............1) p, q, r 2) r, p, q 3) r, q, p 4) p, r, q

12. I f 2 3 4a i j k , 3 2b j k and c form r ight handed system then c = .........1) i + j + k 2) i j + k 3) i j + k 4) i j k

14. I f the vectors 2 3 2xi j k and 7xi x j k make obtuse angle then the greatest positiveintegral value of x is1) 1 2) 2 3) 3 4) 4

18. I f y = x2 is a curve in x y plane and if the normal at P whose abscissa is 2 meets x-axis atQ and O is the or igin then .OP OQ = ...........1) 12 2) 15 3) 24 4) 36

43

20. I f 2a i j k , 3b i k then the unit vector c such that abc is maximum is .......

1) 2 2

14i j k

2) 4

3 2i j k

3) 3 7

59i j k

4) 3 7

59i j k

27. The equation of the diagonal of the quadr ilateral formed by x = 0, y = 0, x + y = 1,6x + y = 3 which passes through or igin is ............1) 3x 2y = 0 2) 2x + 9y = 0 3) 9x 2y = 0 4) 2x 9y = 0

28. Two sides of a rhombus ABCD are parallel to the lines y = x + 2 and y = 7x = 3. I f thediagonals of the rhombus intersect at 91, 2) and the ver tex A is on Y-axis then A could be1) (0, 3) or (0, 5) 2) (0, 2) or 90, 10 3) (0, 2) or (0, 50 4) (0, 0) or (0, 5/2)

30. The equation of the straight line parallel to 2x y + 10 = 0 and bisecting the area of squareformed by the axes and the lines x = 4 and y = 4 is ..........1) 2x y + 3 = 0 2) 2x y 2 = 0 3) 2x y + 2 = 0 4) 2x y + 4 = 0

35. The line midway between the pair of parallel lines x2 + 6xy + 9y2 x 3y 2 = 0 is .......1) 2x + 6y 1 = 0 2) 2x + 6y 3 = 0 3) x + 3y 1 = 0 4) x + 3y 3 = 0

36. I f the slopes of two lines of 6x3 17x2y + 11xy2 2y3 = 0 are reciprocal to each other thenthat of third line is ........1) 2 2) 3 3) 1/2 4) 4

39. The figure formed by ( y mx)2 = a2(1 + m2) and (y nx)2 = a2(1 + n2) forms .........1) Parallelogram 2) Rhombus 3) Square 4) Rectangle

MODEL AIEEE - 20074. I f | Z | = 1, Then the point representing the complex number 1 + 3z lies on

1) a circle 2) a straight line 3) a parabola 4) a hyperbola

13. I f the focus of parabola ( 2 8y x always lies between lines x + y = 2 andx + y = 8, then the number of integral values of is

1) 0 2) 7 3) 6 4) 5

14. The maximum area of the tr iangle formed by a normal to the ellipse 2x2 + y2 = 2 and thecoordinate axes is

1) 12 2)

12 2 3)

14 2 4)

98

15. I f f(x) = maximum of {sinx, cos x} x R, then the minimum value of f(x) is

1) 1 2) 112

3) 12 4)

12

16. I f the roots of ax2 + bx c = 0 are diminished by same quantity, then the value of theexpression in a, b, c which does not change is

1) 2

2

4b aca

2)

4b ca

3)

22

4b aca

4)

2 22

a bc

44

17. A man swims at a speed of 5 km / hr . He wants to cross a canal 120 metres wide, in adirection perpendicular to the direction of flow. I f the canal flows at 4 km . hr , the direc-tion and the time taken by the man to cross the canal are

1) Tan1(3/4), 2.4 m 2) 1 3 / 4 ,1.44secTan

3) 1 1/ 2 ,100secTan 4) 1 3 / 4 ,2.4Tan m

19. Equation of the sphere with centre in the positive octant which passes through the circlex2 + y2 + z2 = 4, z = 0 and is cut by the plane x + 2y + 2z = 0 in a circle of radius 3 is

1) 2 2 2 6 4 0x y z x 2) 2 2 6 4 0x y x

3) 2 2 2 6 4 0x y z z 4) 2 2 6 6 4 0x y x y

20. One die has 3 faces marked as 1, 2 faces marked as 2, and 1 face marked as 3. Another diehas 1 face marked as 1, 2 faces marked as 2, and 3 faces marked as 3, the probability of themost probable sum of numbers that turn up when both dice are thrown together is

1) 3

36 2) 8

36 3) 2236 4)

1436

24. Radius of the circle that can be drawn passing through the point 90, 1) (0, 6) and touchingthe x-axis is1) 5/3 2) 3/2 3) 7/2 4) 9/2

26. I f x, x + 1, 2x + 2 are in G.P, then the next term of the sequence is1) 8/3 2) 8 3) 4 4) 4/3

27. I f s = 10 200

.m

i m ii

c c , then the value of m for which s is maximum is

1) 10 2 )12 3 )15 4) 20

BINOMIAL THEOREM

2. The 7th term in 10

21 yy

then expanded in descending powers of y is

1) 210y2 2) 210y 3) 10C5 y2 40 210y

16. The no of non-zero terms in the expansion of (x = a)2n+1 (x a)2n+1 is1) n +1 2) 2n +2 3) n 4) n 1

22. The value of (x 1)4 + 4(x 1)3 + 6(x 1)2 + 4 ( x 1) + 1 is1) x2 2) x3 3) x4 4) None

23. In the expansion of (1 + x)n if the 2nd and 3rd terms are respectively a, b then x =

1) 2 2a b

a

2)

2

2a

a b3) 2

a

a b 4) None

45

30. The value of x for which the 6th term in the expansion of (p + q)7 is 84 where p = 12log 9 72 x

and q = 12

11 log 3 152

x

is

1) 1 or 2 2) 1 or 2 3) 1 or 3 4) None

31. At what x the 6th term in the expansion of (p + q)7 is 21 where p = 10log 10 32x

and

q = 102 log 35 2 x

1) 0 or 1 2) 0 or 1 3) 0 or 2 4) None

32. I f y = ax5 and x increases by 4 % then using first three terms of binomial expansion theincrease in y is1) 20% 2) 20.6% 3) 21% 4) 21.6%

33. The expansion

7 71 1 3 1 1 3 1

2 23 1x x

x

is a polynomial in x of degree

1) 7 2) 5 3) 4 4) 3

34. The expansion 1023

2x

x

is a polynomial of nth degree in x. Then n =

1) 10 2) 20 3) 40 4) 60

38. In the expansion of 2475 3 2 the rational term is1) T14 2) T16 3) T15 4) T7

47. The coefficient of xk in the expansion of 2

2 1N

xx

IS

1)

2 !4 ! 2 !

3! 3!

n

n k n k 2)

2 !

4 ! 2 !3! 3!

n

n k n k

3) 2 !

4 ! 2 !3! 3!

n

n k n k 4)

2 !

4 ! 2 !3! 3!

n

n k n k

52. The coefficient of x2y3z4 in the expansion of (ax by + cz)9 is1) 1260 a2b3c4 2) 1220 a2b3c4 3) 1260 a2b3c4 4) 1220 a2b3c4

46

54. The coeff icient of p6q3 in 9

23qp

is

1) 17929

2)

17929 3)

17909

4) 1790

9

61. I f a, b, c, d are the coefficients of 2nd , 3rd, 4th and 5th terms respectively in ( 1 + x)n then

, ,a b ca b b c c d are in .........

1) G.P 2) H.P 3) A.P 4) A.G.P

64. Coefficient of x5 in (1 + x)3 (1 + x2)4 is1) 0 2) 4 3) 10 4) 22

65. Coefficient of x4 in (1 + x 2x2)6 is1) 45 2) 45 4) 195 4) 195

66. Coefficient of x5 in (2 x + 3x2)6 is1) 4692 2) 4692 3) 5052 4) 5052

67. The coefficient of 1/x in (1 + x)n (1 + 1/x)n is1) 2ncn1 2)

2ncn 3 )1 4) ncn1

68. The coefficient of xn in (1 + x)n (1 + 1/x)n is1) 2ncn 2) 1 3) 0 4)

2ncn1

69. The term independent of x in (1 + x + 2x3) 9

23 12 3

xx

is

1)827 2) 0 3)

1754 4)

127

72. If the coefficient of (2r + 4)th term is equal to coefficient of (r 2) the term in (1 + x)18 then r =1) 6 2) 4 3) 2 4) 5

74. Coefficient of x20 in (x +1) (x + 2) ..........(x +21) is1) 210 2) 0 3) 1 4) 231

76. I f the coefficients of xr1, xr, xr+1 in (1 + x)14 are in A.P. then r =1) 5 2) 4 3) 3 4) 2

77. I f the coefficient of 2nd, 3rd, and 4th terms in (1 + x)2n are in A.P then n =1) 1 2) 7 3) 14 4) 7/2

78. The coefficient of x6 in (1 + x + x2 + x3 + x4 + x5)6 is1) 2770 2) 2772 3) 456 4) 2770

79. The coefficient of x9 in (x +2) (x + 4) (x + 8) ........ (x + 1024) is1) 2046 2) 1024 3) 55 4) 0

47

83. I f (x 3)50 = a0 + a1x + a2x2 + ...... + a50 x50 then a49 =

1) 150 2) 150 3) 4950 49 3c 4) None

84. I f the term containing x3 in 1n

x

n

is 78 when x = 2 and n is a positive integer then n =

1) 7 2) 8 3) 9 4) 10

85. I f n N and n > 3 and the coefficient of x4k in the expansion of 3

2 1n

xx

is not zero then

n 2k is1) a multiple of 3 2) a multiple of 2 3) a multiple of 5 4) a positive integer

88. I f a term independent o x were to exist in the expansion of 21 n

xx

then x, must be

1) a multiple of 3 2) a multiple of 2 3) a multiple of 5 4) a multiple of 4

89. There is a term containing x2r in the expansion of 3

2

1 nx

x

. Then n 2r is

1) a multiple of 3 2) a positive integer

3) a multiple of 5 4) an odd positive integer

90. p, q are positive integers and p > q. I f (1 + x)p + q ( 1 x)p q is expanded as a polynomial inx, then coefficient of x2 is1) 2p2 q 2) 2q2 p 3) 2p2 + q 4) 2q2 + p

91. The coefficient of x4 in the expansion of (1 +x + x2 + x3)n is1) nc4 2) 4 2

n nc c 3) 4 2 2 4.n n n nc c c c 4) 4 2 1 2.

n n n nc c c c

92. I f (1 + 2x + x2)n = 2

0.

nk

kr

a x then ak =

1) nn kc 2) 1.n nk kc c 3) 2nck 4) 2nck+193. I f in the expansion of (1 +x)m(1 x)n the coefficient of x and x2 are 3 and 6 respectively

then m is1) 6 2) 9 3) 12 4) 24

94. The term independent of x in the expansion of 11

2 15 2

x

x x

is

1) 5th term 2) 6th term 3) 11th term 4) No term

48

95. The coefficient of a4b6 in the expansion of 101 2

3b

a

is

1) 4480243 2)

2240243 3)

12109 4)

112029

96. The middle term in the expansion of (p3 + pq)28 is1) 28 14 5614c q p 2)

28 14 144c q p 3)

28 14 1515c q p 4)

28 44 1515c p q

99. The two numer ically greatest terms in the expansion of (3a 2b)11 when a = 1, b = 3 are1) 8th term, 7th term 2) 9th term, 8th term 3) 9th term, 10th term 4) 10th , 11th term

101. I f the sum of the coefficients in the expansion of (a + b)n is 2048 then the greatest coefficientin the expansion of is1) 12c5 2)

11c5 3) 12c7 4)

11c7104.The sum of the binomial coefficients of the terms occupying the even places in the bino-

mial expansion of (1 +x)n. I f the binomial coefficient of the third term is greater by 9 thanthat of the second term is1) 32 2) 64 3) 512 4) 1024

106. I f the sum of the coefficients in the expansion of (1 3x + 10x2)n is p and the sum of thecoefficients in the expansion of (1 + x2)n is q then1) a = b3 2) a = 3b 3) b = a3 4) b \ 3a

107.The sum of the binomial coefficients of 3rd, 4th terms from the beginning and from theend of (x + y)n is 440. Then n =1) 10 2) 11 3) 12 4) 13

109.Sum of the coefficients in the expansion of (1 + px)n is x1. I f p is doubled and n is halved, s2is the sum of the coefficients in te resulting binomial expansion then1) s2 > s1 2) s1 > s23) s1 = s2 4) Can not be determined

110. I f the sum of the coefficients in the expansion of (a2x2 2ax + 1)51 vanishes then the valueof a is1) 2 2) 1 3) 1 4) 2

112.The range of values of x for which the 4th term in 1032

8x

is the maximum valued term is

1) 602,23

2) 612,21

3) 642,21

4) 643,21

114. I f n is an odd positive integer (1 + x + x2 + x3)n = 3

0

nr

r

r

a x then a0 a1 + a2 a3 + ...... a3n =

1) 4n 2) 0 3 )1 4) 1

115. I f (2 x)50 = a0 + a1x + a2x2 + ....... a50x50 then 1.a1 + 2.a2 + 3.a3 + ........ + 50.a50 =1) 500 2) 100 3) 50 4) 50

49

119. I f (1 + x 2x2)8 = a0 + a1x + a2x2 + ..... + a16x16 then a1 + a3 + ......... + a15 =1) 255 2) 128 3) 128 4) 127

124. I f p = 9950 +10050 and q = (101)50 then1) p > q 2) p < q

3) p = q 4) can not be determined

128.119 + 911 is divisible by1) 9 2) 11 3) 81 4) 100

131.The greatest integer less than or equal to 62 3 is1) 2700 2) 2702 3) 2701 4) 2703

132.The greatest integer less than or equal to 43 1 is1) 55 2) 56 3) 57 4) 28

134.The middle term in the expansion of (1 + 3x + 3x2 + x3)2n is1) 3n n

nc x 2) 2n n

nc x 3) 6 33

n n

nc x 4) 6 23

n n

nc x

138.9

18

0r

r

c =

1) 17 18 92 c 2) 17 18

912 .2

c 3) 18 18 912 .2

c 4) 18 18 912 .2

c

140. 20

nn

r

r

c

1) 2 1 22 n nn

c 2) 2 1 212 .2

n n

nc 3) 2 1 2

12 .2

n n

nc 4) 2 1 2

12 .2

n n

nc

142.2.5c1 + 5.5c2 + 8.5c3 + ....... to 10 terms =1) 7 x 26 + 1 2) 8 x 26 + 1 3) 9 x 26 + 1 4) 5 x 27 + 1

143. 1

1 2 .n

n

n

n

n c

=

1) 0 2) 2 3) 2 4) (n+4)2n 1

144.10 1010 10

3 101 210 10 10 10

0 1 2 9

2. 3. ....... 10c cc cc c c c

1) 210 2) 110 3) 55 4) 136

147. 0 1 1 2 2 3 49 50

0 1 2 50

. ....... . ........

c c c c c c c c

c c c c

=

1) 5051

502)

5150! 3)

505150!

4) 5051

50!

50

148. 2 2 2 21 2 33. 5. ....... 2 1 . nc c c n c =

1) 21 1n nn C 2) 2. n nn c 3) 21 n

nn c 4) 2 11

n

nn c

151. I f 2nr r

c c then 2 2 2 2 21 2 3 4 22. 3. 4. .....2 . nc c c c n c

1)

11 2 !1 !

nn

n

2)

1 2 !1 !

nn

n

3)

11 2 !! 1 !

nn

n n

4)

11 2 !! 1 !

nn

n n

153. I f (1 + x)n = 20 1 2 ...... nnc c x c x c x then 1 3 5 7 ........c c c c

1) 2n1 2) 2n/2 sin 4n

3) / 22 cos 4n n 4) 0

154. I f (1 + x)n = 20 1 2 ...... nnc c x c x c x then 0 4 8 ......c c c

1) 12 22 2 cos

4

n

n n 2) 12 22 2 sin

4

n

n n 3) 12 2 cos 4n n n 4) 1 22 2 sin

4

n

n n

163. I f (1 2x + x2)10 = a0 + a1x + a2x2 + ....... a20x20 then a1 + a2 =1) 60 2) 170 3) 190 4) 80

164.The coefficient of x99 in (1 + 3x + 6x2 +10x3 + ...... to )2/3 is (|x| < 1)1) 24 2) 25 3) 100 4) 4950

167.The coefficient of x17 in the expansion of 1 51

x

x

us

1) 6 2) 9 3) 9 4) 6

171.The range of x for which the expansion of (2 9x2)11/2 is valid is

1) 2 2,

3 3

2) 2 2,3 3

3) 2 2,

3 3

4) 2 2, ,

3 3

174. I f a > 0, b > 0 and if the first two terms in the expansion of 21 1

1 nx a bx

are 2 and 3x

then a =1) 1 2) 1/2 3) 1/2 4) 1

176. I f x is small so that x2 and higher powers of x are neglected then

4

9 2 3 41x x

x

=

1) 439

4x

2) 439

4x

3) 439

2x

4) 439

2x

51

178.Coefficient of x when 1 1 2 1 31 1 2 1 3

x x x

x x x

is expanded as a power ser ies of x is

1) 11 2) 49 3) 41 4) 12

180. I f 1 + 2. x1 + 3. x2 + 4.x3 + ........ +0n0 = 1/3 where |x|. 1 then x =

1) 1 + 2 2) 1 2

2

3) 1 2

2

4) 1 2

182. I f the coefficient of x2r in the expansion of 22 11

x

x

is 13 then r =

1) 7 2) 6 3) 5 4) 4

184. I f nN, then 1

1

11k

kk

n

=

1) n(n 1) 2) n(n + 1) 3) n2 4) (n +1)2

185. If sn denotes the sum of first \n\ natural numbers then 2 11 2 3 ...... ........nns s x s x s x =

1) (1 x)1 2) (1 x)2 3) (1 x)3 4) (1 x)4

187. 2 4 61 1.3 1 1.3.5 11 . . ......

10 1.2 10 1.2.3 10

1) 7

22)

5 27

3) 1/ 25

7

4) 5 2

3

195.The first negative term in the expansion of (1 +x)3/4 is1) T2 2) T4 3) T3 4) T7

198.Larger of 199100 + 200100 and 201100 is1) 199100 + 200100 2) 201100

3) can not be determined 4) None

200.The positive integer which is just greater than (1 + 0.0001)10000 is1) 3 2) 4 3) 5 4) 6

202. I f a is small in compar ison with x then 1/ 2 1/ 2

x x

x a x a

1) 2

2

213a

x 2)

2

2

324a

x 3)

2

2

435a

x 4)

2

2

213a

x

205. I f x = 1 1.3 1.3.5 ......3 3.6 3.6.9

then x2 + 2x 2 =

1) 0 2) 1 3) 2 4) 1

52

206.1.4 1.4.7 1.4.7.10 .......5.10 5.10.15 5.10.15.20

1) 3 5 42 5

2) 3 5 52 4

3) 3 5 53 4

4) 3 5 43 5

207.3 3.5 3.5.7 ......

4.8 4.8.12 4.8.12.16

1) 4 2 5

4

2) 6 2 5

4

3) 2 2 5

4

4) 3 2 5

4

208. I f y = 2x + 3x2 + 4x3 + ........ then x in terms of y is

1) 2 31 1.3 1.3.5 .......2 2.4 2.4.6

y y y 2) 2 31 1.3 1.3.5 .......2 2.4 2.4.6

y y y

3) 2 31 1.3 1.3.5 .......2 2.4 2.4.6

y y y 4) 2 31 1.3 1.3.5 .......2 2.4 2.4.6

y y y

209. I f the coefficients of rth (r+1)th, (r +2)th terms in the expansion of (1 + x)n are in H.P. then1) (n 2r)2 = n + 2 2) (n +2r)2 = n + 2 3) (n 2r)2 = n 4) (n 2r)2 + n = 0

210. 0 0 1 0 1 2 0 1 2 1......... ..... nc c c c c c c c c c (if x is even)

1) n.2n1 2) n.2n 3) 2n 4) 1

2!

n

nn

n

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