ashwani goyal s tutorial topics covered binomial theorem

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1. The number of terms which are free from

redical signs in the expansion of ( )551/ 5 1/10y x+is(1) 5 (2) 6

(3) 7 (4) none of these

2. The coefficient of x5 in the expansion of

( )622 3x x= + is

(1) –4692 (2)4692

(3) 2346 (4) – 5052

3. If the ninth term in the expansion of

( )1 1132

101/ 8 log 5log 25 73 3

xx − +− −+ + is equal to 180

and x > 1, then x equals

(1) 10log 15 (2) 5log 15

(3) log 15e (4) none of these

4. If

( ) 20102 20 1 2 201 2 3 .... ,xx x a a x a x a+ + = + + + +

then a1 equals

(1) 10 (2) 20


5. The number of integral terms in the expansion of

( )10241/ 2 1/ 85 7+ is

(1) 128 (2) 129

(3) 130 (4) 131

6. If the sum of the coefficients in the expansion of

( )na b+ is 4096, then the greatest coefficient in

the expansion is(1) 924 (2) 792


7. The third term in the expansion of

10

5log1

, 1xx xx

+ > is 1000, then x equals

(1) 100 (2) 10

(3) 1 (4) 1/ 10

8. If the sum of the coefficients in the expansion

of ( )1 2n

x+ is 6561, the greatest term in the

expansion for 1/ 2x= is

(1) 4th (2) 5th

(3) 6th (4) none of these

9. If the sum of the coefficients in the expansion

of ( )1 2n

x+ is 6561, then the greatest

coefficient in the expansion is(1) 896 (2) 3594


10. If the coefficients of 4th and ( )1 1th+ terms in

the expansion of ( )293 7x+ are equal, them r

equals(1) 15 (2) 21


11. If the second, third and fourth terms in the

expansion of ( )nx y+ are 135, 30 and 10/3

respectively, then(1) n = 7 (2) n = 5

(3) n = 6 (4) none of these

12. The coefficient of the term indepdnednt of x in

the expansion of ( )9

3 23 11 2

2 3x x x

x + + −

is

(1) 1/3 (2) 19/54

(3) 17/54 (4) 1/4

13. The positive integer just greater than

( )100001 0.0001+ is

(1) 3 (2) 4


14. If [x] denotes the greatest integer less than or

equal to x, then ( )100001 0.0001 + equals

(1) 3 (2) 2


15. The greatest integer less than or equal to

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BINOMIAL THEOREM

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: 2 :

( )6

2 1+ is

(1) 197 (2) 198

(3) 196 (4) 199

16. If xm occurs in the expansion of

2

2

1n

xx

+ the

coefficient of xm is

(1)( )( )

2 !

! 2 !

n

m n m− (2)( )( )2 ! 3!3!

2 !

n

n m−

(3)

( )2 !

2 4! !

3 3

n

n m n m− +

(4) none of these

17. If ( )2 1

6 6 14n

R+

= + and

[ ],f R R= − where [.] denotes the greatest

integer function, the R f equals(1) 20n (2) 202n

(3) 202n+1 (4) none of these

18. If ( )2 1

2 1n

R+

= + and [ ]f R R= − , where [ ]

denotes the greatest integer function, the [R]equal

(1)1

ff

+ (2)1

ff

−

(3)1

ff

− (4) none of these

19. If 1,n> then ( )1 1n

x nx+ − − is divisible by

(1) x (2) 2x

(3) 3x (4) 4x20. There are two bags each of which contains n

balls. A man has to select an equal number ofballs from both the bags. The number of waysin which a man can choose at least one ballfrom each bag is

(1) 2nnC (2) ( )2n

nC

(3) 21

nC (4) 2 1nnC −

21. If nP d`enotes the product of the binomical

coefficients in the expansion of

( ) 11 ,n n

n

Px then

P++ equals

(1)1

!

n

n

+(2)

!

nn

n

(3)( )( )!

1

1

nn

n

+

+ (4)( )( )

1

!

1

1

nn

n

++

+

22. If ( )5 2 6 ,n

I f+ = + where

, 0 1,I N n N and f∈ ∈ ≤ < then I equals

(1)1

ff

− (2)1

1f

f−

+

(3)1

1f

f−

− (4)1

1f

f+

−

23. ( )2

7 4 3 1 ,n

If R f= + = + where

1 0 1,N and f∈ < < then ( )1R f− equals

(1) ( )2

7 4 3n

− (2) ( )2

1

7 4 3n

+


24. The number 100101 1− isdivisible by

(1) 100 (2) 1000

(3) 10000 (4) 100000

25. P is a set containing n elements. A subset A ofP is chosen and the set P is reconstructed byreplacing the elements of A. Asubset B of P ischosen again. The number of ways of choosingA and B such that A and B have no commonelements is

(1) 2n (2) 3n

(3) 4n (4) none of these

26. In example 25, the number of ways of choosingA and B such that A = B, is

(1) 2n (2) 3n

(3) 2nnc (4) none of these

27. In, example 25, the number of ways of choosingA and B such that A and B have equal numberof elements, is

(1) 2n (2) 3n



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(3) ( )22n (4) 2n

nc

28. In example 25, the number of ways of choosingA and B such that B contains just one elementmore than A, is

(1) 21

nnc − (2) 3n

(3) ( )22n (4) 2n

nc

29. In example 25, the number of ways of choosingA and B such that B is a subject of A, is

(1) 2n (2) 3n

(3) 2nnc (4) none of these

30.( )

( )0

1 log 101

1 log 10

nrn e

r rnr

e

rc

=

+−+

∑ equals

(1) 1 (2) -1

(3) n (4) none of these

31. In the expansion of ( ) ( ) ( )1 1 1 ,n n n

x y z+ + +the sum of the coefficients of the terms ofdegree r is

(1) ( )3nrC (2) 3.n

rC

(3) 3nrC (4) 3

nrC

32. The number of non -negative integral solutions

of theequation 3 33x y z+ + = is

(1) 120 (2) 135

(3) 210 (4) 520

33. 3,If n > then

( )( )0 11 1xyC x y C− − − + ( )( ) 22 2x y C− −

( )( ) ( ) ( )33 3 ... 1n

x y C x n− − − + + − −

( ) ny n C− equals

(1) 2nxy× (2) n xy

(3) xy (4) none of these

34. If 3,n > then ( )( )0 1 1xyz C x y− − − ( 1)z −

( )( )( )1 22 2 2C x y z C+ − − −

( )( )( ) 33 3 3x y z C− − − −

( ) ( )( )( )... 1n

nx n y n z n C+ + − − − − equals

(1) xyz (2) nxyz

(3) xyz− (4) none of these

35. The total number of dissimilar terms in the

expansion of ( )3

1 2 ... nx x x+ + + is

(1) 3n (2)3 23

4

n n+

(3)( )( )1 2

6

n n n+ +(4)

( )22 1

4

n n+

36. The value of

595 100

4 31

j

j

C C−

=+ ∑ is

(1) 995C (2) 100

4C

(3) 994C (4) 100

5C

37. The coefficient of 5x in the expansion of

( ) ( ) ( )21 22 301 1 ... 1x x x+ + + + + + is

(1) 515C (2) 9

5C

(3) 31 216 6C C− (4) 30 20

5 5C C+

38. The cofficient of 6x in the expansion of

( )82 31 x x+ − is

(1) 80 (2) 84

(3) 88 (4) 92

39. The digit at unit’s place in the number1995 1995 199517 11 7+ − is

(1) 0 (2) 1

(3) 2 (4) 3

40. If

1 1 4 1 1 4 1

2 24 1

n n

x x

x

+ + − + − + 5

0 1 5... ,a a x a x= + + + then n =

(1) 11 (2) 9


41. ( ) ,nIf f x x= then the value of

( ) ( ) ( ) ( )1 21 1 11 ... ,

1 2! !

nf f ff

n+ + + + where



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( )rf x denotes the rth order derivative of

( )f x with respect to x , is

(1) n (2) 2n

(3) 12n− (4) none of these

42. The expression

7 7

1 1 4 1 1 4 1

2 24 1

x x

x

+ + − + − + is a polynomial in x of degree(1) 7 (2) 5

(3) 4 (4) 3

43. rIf C be the coefficient of rx in ( )1 ,n

x+ then

the value of ( )2

0

1n

r

r

r C=

+∑ is

(1) ( )( ) 21 4 2nn n −+ +

(2) ( )( ) 11 4 2nn n −+ +

(3) ( )2 21 2nn −+ (4) ( )2 24 2nn −+

44. If the second term in the expansion

13

1

na

aa−

+

is 5 / 214 ,a then the value of 3 2/n nC C is

(1) 4 (2) 3

(3) 12 (4) 6

45. If n is an odd natural number , then ( )

0

1rn

nr rC=

−∑equals

(1) 0 (2)1

n

(3)2n

n(4) none of these

46. If n is an even natural number , then ( )

0

1rn

nr rC=

−∑equals

(1) 0 (2)1

n

(3)( ) / 2

/ 2

1n

nnC

−(4) none of these

47.0 0

1,

n n

n n nr rr r

rIf a then

C C= == ∑ ∑ equals

(1) ( )1 nn a− (2) nna

(3)2 n

na (4) none of these

48. The coefficient of the term independent of x in

the expansion of

10

2 / 3 1/ 3 1/ 2

1 1

1

x x

x x x x

+ − − − + − is(1) 210 (2) 105

(3) 70 (4) 112

49. The coefficient of rx in the expansion of

( ) 1/ 21 4x

−− is

(1)( )( )2 !

! 2

r

r (2) 2rrC

(3)( )1.35.... 2 1

2 !r

r

r


50. The value of 1 3 5 71. 3. 5. 7. ....,C C C C+ + + +

where 0 ' 1' 2 '.... nC C C C are the binomial

coefficients in the expansion of ( )1 ,n

x+ is

(1) 1.2nn − (2) 2.2nn −

(3) ( ) 11 2nn n −− (4) none of these

51. The value of 2 2 21 3 51 . 3 . 5 . ...C C C+ + + , is

(1) ( ) 2 11 2 .2n nn n n− −− + (2) ( ) 21 2nn n −−

(3) ( ) 31 .2nn n −− (4)none of these

52. If in the expansion of( ) ( )1 1 ,m n

x x+ − the

coefficients of x and 2x are 3 and -6

respectively, then m is(1) 6 (2) 9

(3) 12 (4) 24

53. If A and B are coefficients of r n rx and x −



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respectively the expansion of ( )1 ,n

x+ then

(1) A = B (2) A B≠

(3) A Bλ= for someλ (4)none of these

54. coefficient of

104

2

3 3

2x in

x− −

is

(1)406

226(2)

504

289

(3)450

263(4)none of these

55. In the expansion of ( )nx a+ the sum of evem

terms is E and that of odd terms is O, then2 2O E− is equal to

(1) ( )2 2 nx a+ (2) ( )2 2 n

x a−

(3) ( )2nx a− (4) none of these

56. The number of terms in the expansion of

( )2021 2 ,x x+ + when expanded in descending

powers of x, is(1) 20 (2) 21

(3) 40 (4) 41

57. The largest coefficient in the expansion of

( )241 x+ is

(1) 2424C (2) 24

13C

(3) 2412C (4) 24

11C


( )2 3 4n

x y z+ − is

(1) 1n+ (2) 3n+

(3)( )( )1 2

2

n n+ +(4) none of these

59. In the expansion of

153

2

1,x

x −

the constant

term is

(1) 159C (2) 0

(3) 159C− (4) 1


10

2

3

2

x

x −

is

(1)405

256(2)

504

259

(3)450

263(4) none of these

61. Given positive integers 1, 2r n> > and the

coefficients of ( )3r th and ( )2r th+ terms in

the binomial expansion of ( )21

nx+ are equal.

then(1) n = 2 r (2) n =3 r

(3) n =2 r +1 (4) none of these


( ) ( )9 9

1 5 2 1 5 2x x+ + − is

(1) 5 (2) 7

(3) 9 (4) 10

63. The sum of the series 10

20

0r

r

C=∑ is

(1) 202 (2) 192

(3)19 20

10

12

2C+ (4)

19 2010

12

2C−

64. The value of the expansion

547 52

4 31

j

j

C C−

=+ ∑ is

equal to

(1) 475C (2) 52

5C

(3) 524C (4) none of these

65. If the coefficient of rth,

( ) ( )1 2r thand r th+ + terms in the expansion

of ( )141 x+ are in A.P, then the value of r is

(1) 5 (2) 6

(3) 7 (4) 9

66. The value of ( )0

1n

k nk

k

C=

−∑ is

(1) 1− (2) 2k

(3) 2n (4) 0.



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67. If | | 1,x < then the coefficient of nx in the

expansion of ( )22 31 ...x x x+ + + is

(1) n (2) 1n−

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

68. If rC stands for 'n

rC then the sum of first

( )1n + terms of the series

( ) ( ) ( )0 1 2 32 3 ...aC a d C a d C a d C− + + + − + +is

(1)2n

a(2) na


69. If ( )2 20 1 21 ....

nx x C C x C x+ + = + + + , then

the value of 0 1 1 2 2 3 ...C C C C C C− + − is

(1) 3n (2) ( )1n−


70 If ( )6

2 1 I F+ = + where 0 1F≤ < and

,I N∈ then the value of I is.

(1) 196 (2) 197

(3) 198 (4) 199

71. If the coefficients of the second, third and fourth

terms in the expansion of ( )1n

x+ are in AP,

then the value of n is(1) 2 (2) 7

(3) 6 (4) 8

72. If A and B arwe coefficients of nx in the

expansions of ( )21

nx+ and ( )2 1

1n

x−+

respectively, then

(1) A B= (2) 2A B=

(3) 2A B= (4) none of these

73. If the binomial coefficients of 2nd, 3rd and 4thterms in the expansion of

( ) ( ) 310 10loglog 10 3 5 22 2

xm

x− − +

, are in AP

and the 6th term is 21, then the value (s) of x is(are)

(1) 1 ,3 (2) 0, 2

(3) 4 (4) -1.

74. If the 6th term in the expansion of

82

108 / 3

1logx x

x +

is 5600, then x equals

(1) 1 (2) log 10e

(3) 10 (4) xdoes not exist

75. If the 4th term in the expansion of 1

n

axx

+ is

5

2, then the value of a and n are

(1)1

62, (2) 1, 3

(3)1

32, (4) cannot be found

76. The coefficients of mx and ( ),nx m n N∈ in the

expansion of ( )1m n

x++ are

(1) equal

(2) equal but opposite in sign

(3) reciprocal to each other

(4) none of these

77. If the (r +1)th term in the expansion of

213

3

a b

b a

+

contains a and b to one and the

same power, then the value of r is(1) 9 (2) 10

(3) 8 (4) 6

78. If the coefficients of second, third and fourth

terms in the expansion of ( )21

nx+ are in AP,

then

(1) 22 9 7 0n n+ + = (2) 22 9 7 0n n− + =

(3) 22 9 7 0n n− − = (4) none of these


( )112 31 x x x+ + + is

(1) 900 (2) 909

(3) 990 (4) 999

80. If



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( )62 2 31 2 31 2 1 ...x x C C x C x+ − = + + + +

, 1212 ,C x+ then the value of

2 4 6 12...C C C C+ + + + is

(1) 30 (2) 32


81. If the coefficient of the middle terms in the

expansion of ( )2 11

nx

++ is p and the coefficient of

middle terms in the expansion of ( )2 11

nx

++ are q

and r, then

(1) p q r+ = (2) p r q+ =

(3) p q r= + (4) 0p q r+ + =

82. If 1 2 3 4, , ,a a a a are the coefficients of any four

consecutive terms in the expansion of ( )1 ,n

x+

then 1 3

1 2 3 4

a a

a a a a+

+ + is equal to

(1)2

2 3

a

a a+ (2)2

2 3

1

2

a

a a+

(3)2

2 3

2a

a a+ (4)3

2 3

2a

a a+

83. The coefficient of ( )( )2 0 1x r n≤ ≤ − in the

expansion of ( ) ( )21( 3) 3 2nnx x x

−−+ + + + +

( ) ( ) ( )3 2 13 2 ... 2

n nx x x

− −+ + + + + is

(1) ( )3 2n r nrC − (2) ( )3 2n n r n r

rC − −−

(3) ( )3 2n r n rrC −+ (4) none of these

84. If

( )2 2 20 1 2 21 ... ,

n nnx x a a x a x a x− + = + + + +

then( )0 2 4 2... na a a a+ + + + is equal to

(1)3 1

2

n +(2)

3 1

2

n −

(3)13 1

2

n− +(4)

13 1

2

n− −

85. The coefficient of mx in

( ) ( ) ( )1 ,1 1 ... 1

p p nx x x

++ + + + + +

p m n≤ ≤ is

(1) 11

nmC+

+ (2) 11

nmC−

−

(3) nmC (4) 1

nmC +

86. In the third term in the expansion of

10

5log1 xx

x +

is 1000, then the value ofx is

(1) 10 (2) 100


87. If the coefficient of 7x in the expansion of

( )112 1 1a x b x− −+ is equal to the coefficient

of 7x− in ( )111 2 ,ax b x− −− then ab =

(1) 1 (2) 2

(3) 3 (4) 4


( ) ( )5 421 1x x+ + is

(1) 30 (2) 60


89. The greatest term in the expansion of

201

3 13

+ is

(1)25840

9(2)

24840

9

(3)26840

9(4) none of these

90. If 0' 1' 2 '..., nT T T T represent the terms in the

expansion of ( ) ,n

x a+ then the value of

( )2

0 2 4 6...T T T T− + − + ( )2

1 3 5 ...T T T− + + is

(1) ( )2 2 nx a− (2) ( )2 2 n

x a+

(3) ( )2 2 na x− (4) none of these

91. The total number of terns in the expansion of

( )100x y+ + ( )100

x y− after simplification is

(1) 50 (2) 51



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92. If the coefficients of three consecutive terms in

the expansion of ( )1n

x+ are in the ratio 1 : 7 :

42, then the value of n is(1) 50 (2) 70

(3) 10 (4) 5

93. If the second, third and fourth term in the

expansion of ( )nx a+ are 240, 720 and 1080

respectively, then the value of n is(1) 15 (2) 20

(3) 10 (4) 5

94. The value of

3 3

6

18 7 3.18.7.25

3 6.243.2 15.181.4 20.27.8

+ ++ + + is

15.9.16 6.3.32 64+ + +(1) 10 (2) 1

(3) 2 (4) 20

95. If the coefficients of ( ) ( )2 4 2r th and r th+ −

terms in the expansion of ( )181 x+ are equal,

then the value of r is(1) 5 (2) 6

(3) 7 (4) 9

96. The middle term in the expansion of

101

xx

+ is

(1)10

1

1C

x(2) 10

5C

(3) 106C (4) 10

7 .C x

97. The 14th term from the end in the expansion of

( )17

x y− is

(1) ( )517 6

5C x y− (2) ( )1117 3

6C x y

(3) 17 13/ 2 24C x y (4) none of these

98. If [x] denotes the greatest integer less than orequal to x and F = R – [R] where

( )2 1

5 5 11n

R+

= + , then RF is equal to

(1) 2 14 n+ (2) 24 n

(3) 2 14 n− (4) none of these

99. If [x] denotes the greatest integer less than or

equal to x, then ( )2 1

6 6 14n+ +

(1) is an even integer (2) is an odd integer

(3) depends on n (4) none of these

100. If n N∈ such that

( )7 4 3n

+ ( )7 4 3 1 ,n

F+ = + where I N∈

and 0 1.F< < Then the value of

( )( )I F I F+ − is

(1) 0 (2) 1

(3) 27 n (4) 22 n

101. If the ratio of the 7th term from the berinning tothe seventh term from the end in the expansion

of 3

12

3

x +

is 1

6, then x is

(1) 9 (2) 6


102. The sum of the coefficients in the expansion of

the polynomial ( )214321 3x x+ − is

(1) –1 (2) 1



( )512 2 2 1x xα α− + vanishes, then the value of

α is(1) 2 (2) –1

(3) 1 (4) –2

104. If ,n N∈ then the sum of the coefficients in the

expansion of the binomial ( )5 4n

x y− is

(1) 1 (2) –1

(3) 1 (4) 0


( )21 3 10n

x x− + is a and if the sum of the

coefficients in the expansion of ( )21n

x+ is b,

then(1) a = 3b (2) a = b3

(3) b = a3 (4) none of these

106. If the coefficient of ( )1r th+ term in the



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expansion of ( )21

nx+ be equal to theat of

( )3r th+ term, then

(1) 1 0n r− + = (2) 1 0n r− − =

(3) 1 0n r+ + = (4) none of these

107. The coefficient of the middle term in the

expansion of ( )21

nx+ is

(1)( )1.3.5.... 2 1

2!

nn

n

−

(2)( )

( )2

1.3.5.... 2 12

!nn

n

−

(3)( )( )

22

2 !2

!nn

n (4) none of these

108. If n is even, then the greatest coefficient in the

expansion of ( )nx a+ is

(1) 12

nnC

+ (2) 12

nnC

−

(3)2

nnC (4) none of these

109. If n is even and rth has the greatest coefficient

in the binomial expansion of ( )1n

x+ , then

(1)2

nr = (2) 1

2

nr = +

(3) 12

nr = − (4) none of these

110. If there is a term containing

32

2

1n

rx in xx

− +

,

then(1) n – 2r is a positive integral multiple of 3

(2) n – 2r is even (3)n – 2r is odd

(4) none of these

111. If n is even positive integer, then the conditionthat the greatest term in the expansion of

( )1n

x+ may have the greatest coefficient also

is

(1)2

2

n nx

n n

+< <+ (2)

1

1

n nx

n n

+ < <+

(3)4

4 4

n nx

n

+< <+ (4) none of these

112. If the fourth term in the expansion of

6

1/121

log 1x x

x

+ + is equal to 200 and

x > 1 x is equal to

(1) 210 (2) 10


113. The interval in which x must lie so that thenumerically greatest term in the expansion of

( )211 x− has the numerically greatest

coefficient is

(1)5 6

,6 5

(2)5 6

,6 5

(3)4 5

,5 4

(4)4 5

,5 4

114. The interval in which x must lie so that the

greatest term in the expansion of ( )21

nx+ has

the greatest coefficient is

(1)1

,1

n n

n n

− −

(2)1

,1

n n

n n

+ +

(3)2

,2

n n

n n

+ +

(4) none of these

115. If the rth, ( )1r th+ and ( )2r th+ coefficients

of ( )1n

x+ are in AP, then n is a root of the

equation

(1) ( )2 24 1 4 2 0x x r r− + + − =

(2) ( )2 24 1 4 2 0x x r r+ + + − =

(3) ( )2 24 1 4 2 0x x r r+ + + + =

(4) none of these

116. The remainder when 599 is divided by 13 is(1) 6 (2) 8

(3) 9 (4) 10

117. 0, 1, 2,..., nIf C C C C denote the binomial

coefficient in the expansion of ( )1 ,n

x+ then the



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value of 1

n

rr

rC=

∑ is

(1) 1.2nn − (2) ( )1 2nn+

(3) ( ) 11 2nn −+ (4)( ) 12 .2nn −+

118. If 0, 2, 2,... nC C C C denote the binomial coefficient

in the expansion of( )1 ,n

x+ then the value of

( )0

1n

rr

r C=

+∑ is

(1) 2nn (2)( )2 11

nn

−+

(3) ( ) 12 2nn −+ (4)( ) 22 2nn −+

119. If 0, 1, 2,..., nC C C C denote the binomial cofficeint

in the expansion of ( )1 ,n

x+ then the value

of ( ) ( )0 1 22aC a b C a b C+ + + + + ...+

( ) na nb C+ is

(1) ( )2na nb+ (2)( ) 12na nb −+

(3) ( ) 12 2na nb −+ (4)( )2 2 .na nb+

120. Let( )0

1n

n rr

r

x C x=

+ = ∑ and

31 2

0 1 2 1

2 3 ... n

n

C CC Cn

C C C C −

+ + + ( )11 ,n n

k= +

then the value of k is

(1)1

2(2) 2

(3)1

3(4) 3

121. The value of ( )0

1n

r nr

r

C=

−∑ is

(1) -1 (2) 2n

(3) 2 n− (4) 0

122. Let( )0

1n

n rr

r

x C x=

+ = ∑ and0

,1

nr

r

Ck

r=

=+∑ then

the value of k is

(1)12 1

1

n

n

+ ++

(2)12 1

1

n

n

+ −+

(3)2 1

1

n

n

++

(4)2 1

1

n

n

−+

123. If 0, 1, 2,... nC C C C are binomial coefficient in the

expansion of( )1 ,n

x+ then the value of

( )31 20 ... 1

2 3 4 1

n nC CC CC

n− + − + + −

+ is

(1) 0 (2) 1

1n+

(3)2

1

n

n+(4)

1

1n−

+124. If n is an integer than unity, then the value of

( ) ( ) ( ) ( )1 2 31 2 .... 1nn n na C a C a C a n− − + − − + + − − is

(1) 0 (2) 1

(3) n (4) –1

125. The value of the sum of the series

0 1 2 33. 8 13 18 ...n n n nC C C C uj− + − +(1) 0 (2) 3n


126. If ( ) 20 1 21 .... ,

n nnx C C x C x C x+ = + + + +

then 2 2 21 2 .... nC C C+ + + is equal to

(1) 2 22 n− (2) 2n

(3)( )( )2

2 !

2 !

n

n (4)( )( )2

2 !

!

n

n

127. If ( ) 20 1 21 .... ,

n nnx C C x C x C x+ + + + + +

then for n odd,

( )2 2 2 2 20 1 2 3 ... 1

n

nC C C C C− + − + + − is equal to

(1) 2 22 n− (2) 2n

(3)( )( )2

2 !

2 !

n

n (4)( )( )2

2 !

!

n

n

128. If( ) 20 1 21 ...

nx C C x C x+ = + + + + ,n

nC x then

for n odd,

( )2 2 2 2 20 1 2 3 ... 1

n

nC C C C C− − − + + − is equal



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to

(1) 0 (2) 2 22 n−

(3)( )( )2

2 !

2 !

n

n (4) 22 n

129. If 0' 1' 2 '... nC C C C arecofficeints in the binomial

expansion of ( )1n

x+ and n is even, then the

value of ( )2 2 2 2 20 1 2 3 ... 1

n

nC C C C C− + − + + −is

(1) 0 (2)

( ) / 2

2

!1

!2

n n

n−

(3) ( ) ( )( )2

2 !1

!

n n

n− (4)

( )( )2

2 !

!

n

n

130. If 0' 1' 2'... nC C C C arecofficients in the binomial

expansion of ( )1 ,n

x+ then

0 2 1 3 2 4 2... n nC C C C C C C C−+ + + + is equal to

(1)( )

( ) ( )2 !

2 ! 2 !

n

n n− + (2) ( )

( ) )2

2 !

2 !

n

n−

(3)( )

( ) )2

2 !

2 !

n

n+ (4) none of these

131. The value of

2 3 4 11

0 1 2 3 10

2 2 2 22 ...

2 3 4 11C C C C C+ + + + + is

(1)113 1

11

−(2)

112 1

11

−

(3)311 1

11

−(4)

211 1

11

−

132. If m, n, r, are positive integers such that r < m, n,

then 1 1 2 2 ...m m n m nr r rC C C C C− −+ + + +

1 1m n n

r rC C C− + equals

(1) ( )2nrC (2) m n

rC+

(3) m n m nr r rC C C+ + + (4) none of these

133. The value of

2 32 2 2

1 2 3

1 10 10 10

81 81 81 81n n n

n n n nC C C− + − +

210...

81

n

n+ is

(1) 2 (2) 0

(3) 1/2 (4) 1

134. If x+y =1, then 0

nn r n r

rr

r C x y −

=∑ equals

(1) 1 (2) n

(3) nx (4) ny

135. If x+y =1,then2

0

nn r n r

rr

r C x y −

=∑ equals

(1) n x y (2) n x ( x + y n )

(3) n x (n x + y ) (4) none of these

136. The term independent of x in the expansion of

4 31 1

x xx x

− + is

(1) -3 (2) 0

(3) 1 (4) 3

137. The positive value of a so that the coefficients

of 5x and 15x are equal in the expansion of

102

3

ax

x +

(1)1

2 3(2)

1

3

(3) 1 (4) 2 3.

138. If n is a positive integer and

2

3'

1 1

nn k

k kk k

CC C then k

C= −

=

∑ equals

(1)( )( )1 2

12

n n n+ +(2)

( ) ( )21 2

12

n n n+ +

(3)( )( )2

1 2

12

n n n+ +(4) none of these

139. The coefficient of 50x in the expression

( ) ( ) ( )1000 999 99821 2 1 3 1 ...x x x x x+ + + + + + +



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: 12 :

10001001x is

(1)100050C (2) 1001

50C

(3) 100250C (4)1000

51C

140. The coefficient of 5x in ( ) 3/ 221 2 3 ...x x−

+ + +

is(1) 21 (2) 25


141. If | | 1,x < then the coefficient of nx in the

expansion of ( )22 3 41 ...x x x x+ + + + + is

(1) n (2) n- 1

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

142. The general term in the expansion of ( )3/ 41 2x−

is

(1) 23

2 !rx

r

−(2)

3

2 !

rr

rx

r

−

(3) ( )3

2 2 !

rr

rx

r


143. If rC stands for 'n

rC then the sum of the series

2 ! !2 2

!

n n

n

( ) ( )2 2 2 20 1 22 3 ... 1 1 ,

n

nC C c n C − + + + − + where n is an even positive integer, is equal to

(1) 0 (2) ( ) ( )/ 21 1

nn− +

(3) ( ) ( )/ 21 2

nn− + (4) ( )1 .

nn−

144. The coefficient of nx in the expansion of

( )( )1

1 3x x− − is

(1)

1

1

3 1

2.3

n

n

+

+

−(2)

1

1

3 1

3

n

n

+

+

−

(3)

1

1

3 12

3

n

n

+

+

−

(4) none of these

145. If ( ) 20 1 21 ... ...,

n r rx a a x a x a x−− = + + + +

then 0 1 2 ... ra a a a+ + + + is equal to

(1)( )( ) ( )1 2 ...

!

n n n n r

r

+ + +

(2)( )( ) ( )1 2 ...

!

n n n r

r

+ + +

(3) ( )( ) ( )1 2 ... 1

!

n n n n r

r

+ + + −

(4) none of these


( ) 121 9 20x x−

− + is

(1) 5 4n n− (2) 1 15 4n n+ +−

(3) 1 15 4n n− −− (4) none of these

147 The coefficient of nx in the expansion of

( )( )

2

3

1

1

x

x

+

− is

(1) 2 2 1n n+ + (2) 22 1n n+ +

(3) 22 2 1n n+ + (4) 2 2 2n n+ +


( )2 31 2 3 4 ...n

x x x−

− + − + is

(1)( )2 !

!

n

n(2)

( )( )2

2 !

!

n

n

(3)( )( )2

2 !1

2 !

n

n (4) none of these

149. If ( )1r th+ term is the firsr negative term in the

expansion of ( )7 / 21 ,x+ then the value of r is

(1) 5 (2) 6

(3) 4 (4) 7


( ) 322x x−

− is

(1) 67485 (2) 67548

(3) 67584 (4) 67845



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: 13 :


( )21 ...n

x x−

+ + + is

(1) 1 (2) ( )1n−

(3) n (4) n +1

152. If x be very small compared with unity such that

( )( )

231 1,

1 1

x xa bx

x x

+ + −= +

+ + +then the values of

a and b are

(1)5

1,6

a b= = (2)5

1,6

a b= = −

(3)5

1,3

a b= = (4)5

1,3

a b= = −

153. If x is very small magnitude compared with a

such that

1/ 2 1/ 2 2

22 ,

a a xk

a x a x a

+ = + + −


(1)1

4(2)

1

2

(3)3

4(4) 1

154. If the binomial expansion of ( ) 2a bx

−+ is

13 ....,

4x− + , then (a, b) =

(1) (2, 12) (2) (2, 8)

(3) (–2, –12) (4) none of these

155. If nr rC C= and

( )( ) ( ) ( )0 1 1 2 1 0

1...

!

n

n

nC C C C C C k

n−

++ + + = ,


(1) 0 1 2.... nC C C C (2) 2 2 21 2 .... nC C C

(3) 1 2 .... nC C C+ + + (4) none of these

156. If the third term in the binomial expansion of

( )1m

x+ is 21,

8x− then the rational of m is

(1) 2 (2) 1/2

(3) 3 (4) 4

157. If p is nearly equal to q and 1n > , such that

( ) ( )( ) ( )

1 1,

1 1

kn p n q p

n p n q q

+ + − = − + +

then the value of

k is

(1) n (2)1

n

(3) 1n+ (4)1

1n+

158. If 2 33 6 10 ...,y x x x= + + + then x =

(1)2 3

2 2

4 1. 4 1.4.7....

3 3 2 3 . 3y y− +

(2)2 3

2 2

4 1. 4 1.4.7....

3 3 2 3 . 3y y− + −

(3)2 3

2 2

4 1. 4 1.4.7....

3 3 2 3 . 3y y+ +

(4) none of these

159. The sum of the series

2 4 6

1 1.4 1 1.4.7 11 ...

3 1.2 3 1.2.3 3+ + + + is

(1)3

2(2)

1/33

2

(3)1

3(4)

1/31

3

160. If 1 1.3 1.3.5

...,3 3.6 3. 6.9

y= + + + then the value of

2 2y y+ is

(1) 2 (2) –2


161. If ( )2

2

0

1 2 ,nn r

rr

x x a x=

+ + =∑ then ar =

(1) ( )2nrC (2) 1.n n

r rC C +

(3) 2nrC (4) 2

1n

rC +

162. The coefficient of 3x in

6

5

3

3x

x

+

is



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: 14 :

(1) 0 (2) 120

(3) 420 (4) 540

163. The number of non-zero terms in the expansion

of ( ) ( )991 3 2 1 3 2x x+ + − is

(1) 9 (2) 0

(3) 5 (4) 10

164. The coefficient of y in the expansion of

( )52 /y c y+ is

(1) 29 c (2) 10 c

(3) 10 c3 (4) 20 c2

165. The greatest coefficient in the expansion of

( )101 x+ is

(1)10!

5! 6! (2) ( )2

10!

5!

(3)10!

5!7! (4) none of these

166. The approximate value of ( )1/37.995 correct to

four decimal places is(1) 1.9995 (2) 1.9996

(3) 1.9990 (4) 1.9991

167. The coefficient of x4 in the expansion of

( )2 31n

x x x+ + + is

(1) 4nC (2) 4 2

n nC C+

(3) 4 1 4 2.n n n nC C C C+ +

(4) 4 2 1 2.n n n nC C C C+ +

168. The term independent of x in the expansion of

( ) ( )1 1 1/n n

x x+ + is

(1) ( )2 2 2 20 1 22 3. .... 1 nC C C n C+ + + + +

(2) ( )2

0 1 ... nC C C+ + +

(3) 2 2 20 1 .... nC C C+ + +

(4) none of these

169. The expression

( ) ( )5 51/ 2 1/ 23 31 1x x x x + − − − is a

polynomial of defree(1) 5 (2) 6

(3) 7 (4) 8

170. The coefficient of x53 in the expansion

( )100

100100

0

3 .2m m

mm

C x−

=

−∑ is

(1) 10047C (2) 100

53C

(3) 10053C− (4) 100

100C−171. The value of

( )0 1 2 33 5 7 ..... 2 1 nC C C C n C+ + + + + is equal

to

(1) 2n (2) 12 .2n nn −+

(3) ( )2 . 1n n+ (4) none of these

172. The largest term in the expansion of ( )503 2x+

where 1/ 5x= is

(1) 5th (2) 51st

(3) 7th (4) 6th

173. In the expansion of ( )501 ,x+ the sum of the

coefficients of odd powers of x is(1) 0 (2) 249

(3) 250 (4) 251

174. The term independent of x in

10

2

3

3 2

x

x

+

is

(1) None (2) 101C

(3) 5/12 (4) 1

175. If the coefficients of x7 and x8 in ( )2 / 3n

x+ are

equal , then n is equal to(1) 56 (2) 55

(3) 45 (4) 15

176. If the r th term in the expansion of

( )102/ 3 2 /x x− contains x4, then r is equal to

(1) 2 (2) 3

(3) 4 (4) 5

177. If the third term in the expansion of

105log xx x + is 106 then x may be

(1) 1 (2) 10

(3) 10–5/2 (4) 102

178. The value of x, for which the 6th term in the



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: 15 :

expansion of ( )

( ) ( )1

2

12

7log 9 7

1/5 log 3 1

12

2

x

x

−

−

+

+

+

i

84 is equal to(1) 4 (2) 3

(3) 2 (4) 1

179. In the expansion of ( ) ( )21 ,

nx n N+ ∈ the

coefficients of ( )1p th+ and ( )3p th+ terms

are equal, then

(1) 2p n= − (2) 1p n= −

(3) 1p n= + (4) 2 2p n= −

180. In the expansion of

154

3

1x

x −

the coefficient

of x39 is(1) 1365 (2) – 1365

(3) 455 (4) – 455

181. The value of 2 2 20 1 23. 5. ...C C C+ + + to ( )1n+

terms, is

(1) 2 11

nnC−

− (2) ( )2 12 1

n

nn C−+

(3) ( ) 2 112 1 . n

nn C−−+

(4) ( )2 12 112 1

nnn nC n C

−−−+ +

182. The value of

( ) ( )1 1 1

....,! 2! 2 ! 4! 4 !n n n+ + +

− − us

(1) ( )22

1 !

n

n

−

− (2)12

!

n

n

−

(3)2

!

n

n(4) ( )

2

1 !

n

n−

183. The coefficients of x7 and x8 in the expansion of

23

nx +

are equal, then n is equal to

(1) 35 (2) 45


184. If ( )621 2x x+ − 2 121 2 121 ....a x a x a x= + + + +

then 2 4 6 12....a a a a+ + + + =

(1) 30 (2) 65

(3) 31 (4) 63

185. The ratio of the coefficient of x15 to the term

independent of x in

152 2

xx

+ is

(1) 1/4 (2) 1/16

(3) 1/32 (4) 1/64


( )10x y x+ + is

(1) 11 (2) 33

(3) 66 (4) 1000

187. If ( )2 1

6 6 14n

m+

+ = and if f is the fractional

part of m, then fm is equal to

(1) 115n+ (2) 120n+




ashwani goyal s tutorial topics covered binomial theorem

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