07 Binomial Theorem Q 1.
If C0, C1, C2, ….., Cn are Binomial coefficients, then 1 1 1 1 ..... is equal to: n! 0! ( n 1)! (n 2)! 2! 0! n!
(a) 2n
(b)
2n
2 n1 n!
(c) n !
(d) none of
these Q 2.
If C0, C1, C2, …., Cn are Binomial coefficients, then C0 + C2 + C4 + …. + Cn, (n is even integer), is equal to: (a) 2n-1 (b) 2n (c) 2n+1 (d) 0
Q 3.
If C0, C1, C2, …, Cn (n is a multiple of 3) are Binomial coefficients, then C0 + C3 + C6 + C9 + …… is equal to: (a) 0
(b) 3
(c)
2n 2 3
(d)
1 n (2 + (-1)n2) 3
Q 4.
If C0, C1, C2, …., Cn are Binomial coefficients, then C1 + 2C2 + 3C3 + ….. + nCn is equal to: (a) 2n-1 (b) n 2n (c) n 2n-1 (d) none of these.
Q 5.
If C0, C1, C2,….., Cn are Binomial coefficients, then C0 + (a)
2 n 1 1 n 1
(b)
2 n 1 1 n 1
(c)
2 n 1 n 1
C1 C2 C .... n is equal to: 2 3 n 1 (d) none of these.
Q 6.
If C0, C1, C2, ….., Cn are Binomial coefficients, then C0 – C1 + C2 – C3 + ….. + (-1)nCn is equal to: (a) 2n-1 (b) 2n (c) 0 (d) none of these.
Q 7.
The value of 4C3 + 5C3 + 6C3 + 7C3 + 8C3 + 9C3 + 10C3 + 11C3 + 12C3 is equal to: (a) 13C4 – 1 (b) 13C4 (c) 13C4 + 1 (d) 12C4 – 1
Q 8.
The value of 11C2 + 11C4 + 11C6 + 11C8 is equal to: (a) 210 – 1 (b) 210 – 11 (c) 210 – 12 12
Q 9.
The value of
r 1
(a) 224 – 24C12 – 1 n
Q 10. If
r 1
2 n 1
24
(d) none of these
Cr is equal to: (b) 223 +
1 2
24
C12 – 1 (c) 224
Cr = 63, then value of n is equal to:
(d) none of these.
(a) r
(b) 5
(c) 6
(d) 3
Q 11. For 1 ≤ r ≤ n, the value of nCr + n-1Cr + n-2Cr + ….. + rCr is: (a) nCr+1 (b) n+1Cr+1 (c) n+1Cr 50
Q 12. The coefficient of x12 in the expansion
50
r 0
(a) 50C12
Q 13. If
(b) 1
Cr
stands
Cr ( x 1) 50 r 2 r is: (c) 0
n
for
(d) none of these.
Cr,
then
the
sum
(d) none of these.
of
the
series
n n 2 ! ! 2 2 n!
(C02 2C12 3C22 ...... ( 1) n ( n 1)Cn2 ) ,
Where n is an even positive integer, is equal to: (a) 0 (b) (-1)n/2 (n + 1) (c) (-1)n/2 (n + 2)
(d) (-1)n/2 (n + 3)
Q 14. If 15 Cr 15 Cr 1 15 C15 r 15 C16 r 16 C13 , then the value of r is: (a) r = 3 (b) r = 2 (c) r = 4 (d) none of these. 2
Q 15. If Cr stands for nCr then (C0 + C1) + (C1 + C2) + …. + (Cn-1 + Cn) is equal to: (a) 2n-1 (b) 2n+1 + 1 (c) 2n+1 – 1 (d) 2n+1 – 2 Q 16. The value of C1 + 3C3 + 5C5 + 7C7 + ….., where C0, C3, C5, C7, …. Are binomial coefficients is: (a) n . 2n-1 (b) n . 2n+1 (c) n . 2n (d) n . 2n-2 Q 17. If T0, T1, T2, …., Tn are terms in the expansion of (x + a) n, then the value of n n Tr (1) r Tr is: r 0 r 0 2n (a) (x + a) (b) (x2 + a2)n (c) (x2 – a2)n (d) none of these. n
Q 18. If an =
r 0
1 , then n Cr
(a) (n – 1)an
n
r 0
n
r equals: Cr
(b) nan
(c)
n an 2
Q 19. The value of 3nC0 – 8nC1 + 13nC2 – 18nC3 + ….. + n terms is: (a) 0 (b) 3n (c) 5n Q 20. IF Cr stands for nCr, then the value of 2C0 to:
(d) none of these.
(d) none of these.
22 23 2 n 1 C1 C2 .... ( 1) n 1 is equal 2 3 ( n 1)
(a) 0, if n is even
(b) 0, if n is odd
(c)
1 1 , if n is even (d) s , if n is n 1 n 1
odd. Q 21. The value of C02 + C12 + C22 + …. + Cn2, where C0, C1, C2, ….. Cn are Binomial coefficients is: (a) 2n (b) 2n+1 (c) 2nCn (d) none of these. Q 22. The value of C0 + (C0 + C1) + (C0 + C1 + C2) +…. + (C0 + C1 + …. + Cn-1), where C0, C1, C2, …, Cn are Binomial coefficients, is: (a) n2 . 2n-1 (b) n . 2n-1 (c) n . 2n (d) none of these. Q 23. The value of 4nC0 + 4nC4 + 4nC8 + …. + 4nC4n is: (a) 24n (b) 24n-2 (c) 22n-1 + 24n-2 C Q 24. k k k 1 Ck 1 4
3
(a) 50
(d) 22n-1 (-1)n + 24n-2
2
(where, Ck stands for 4Ck), is equal to: (b) 100
(c) 90
(d) 150
Q 25. The coefficient of xn-1 in the expansion of (x – 1) (x – 2) …. (x – n) is equal to: n( n 1) n(n 1)(2n 1) (a) (b) n (c) (d) none of these. 2 6 Q 26. The coefficient of xn-2 in the expansion of (x – 1) (x – 2) …. (x – n) is: (a) (n) 2
(c) (n 2 ) n 2
(b) n 2
Q 27. aC0 + (a + b) C1 + (a + 2b) C2 + …. + (a + nb) Cn is equal to: (a) (2a + nb)2n (b) (2a + nb)2n-1 (c) (na + 2b)2n
(d)
1 (n 2 ) n 2 2
(d) (na + 2b)2n-1
Q 28. If (1 – x + x2)n = a0 + a1x + a2x2 + …. + a2nx2n, hten a0 + a2 + …. + a2nis equal to: (a)
3n 1 2
(b)
3n 1 2
n (c) 3
1 2
( 1) r equals: n Cr r 0 n (c) n 2
n (d) 3
1 2
n
Q 29. If n is an odd natural number, then (a) 0
(b) 1/n
Q 30. If (1 + x)n = C0 + C1x + …… + Cnxn, then: (a)
2n n!
(b)
(n 1) n n!
(d) none of these.
C1 C C C 2 2 3 3 n n is: C0 C1 C2 Cn 1 n( n 1) (c) 2
(d)
n( n 1) 2
Q 31. The value of C0 + 5C1 + 9C2 + 13C3 + …. to (n + 1) terms is equal to: (a) 22n (b) 2n (n + 1) (c) 2n (1 + 2n) (d) none of these Q 32. The term independent of x in the expansion of (a) 9C6 (23)
(b) 9C5 (2)4
Q 33. The coefficient of x in the expansion of 10
C5
2 3
9
is equal to:
(c) 9C7 (2)5
5
(a)
x2 2 x 2
x 2
2 3x 2
(d) none of these.
10
is:
5
(b) 10C5 25
(c) 10C0 2-5
(d) none of these
Q 34. If the ratio of the 3rd term from the beginning to the 3rd term from the end in the expansion of (1 + 2)n is 1/8, then n is equal to: (a) 8 (b) 12 (c) 14 (d) 10 Q 35. The middle term in the expansion of (1 + x)20 is : (a) 20C9x9 (b) 20C10x5 (c) 20C10x10
(d) none of these. 11
1 Q 36. The product of middle terms in the expansion of x x
1 x
(b) 11C5 11C6
(a) 11C5 11C6
is equal to
(c) 11C5 11C6(x)
(d) none of these.
Q 37. (u + v)6 + (u - v)6. (where u and v are rational numbers), is: (a) a rational number (b) an irrational number (c) an integer (d) none of the above. Q 38. If Pn = nC0 nC1 nC2 ….. nCn, then (a)
(n 1) n 1 ( n 1)!
(b)
Pn 1 is equal to: Pn
(n 1) n1 n!
(c)
n 1
nn n!
(d) n ! 9
1 Q 39. The term in independent of x in the expansion of 6 x 3 is equal to: x
(a) – 9C3
(b) 9C4
(c) 9C2
1 x
(d) none of these.
5
Q 40. The coefficient of x-2in the expansion of x 2 is equal to: (a) 5C2
(b) 5C3 5
Q 41. The value of 95C4 +
j 1
100 j
C3 is equal to:
(c) 5C4
(d) 5C5
(a) 99C5
(b) 100C4
(c) 99C4
(d) 100C5
10
2 Q 42. The coefficient of x in x 2 x 7
(a) 20
is equal to:
(b) – 20
(c) 0
(d) none of these
Q 43. The coefficient of x100 in the expansion of (1 + x3)100 is equal to: (a) 100C32 (b) 0 (c) 100C33 10
Q 44. The 7th term from the end in the expansion of x 1 (a) 10C4 24 2 x
(b)
10
(d) none of these.
2 x2
is equal to:
(c) 10 C3 23 x
C4 2 4
(d)
none
of
these. Q 45. The number of terms in the expansion of (x + y + z)10 is: (a) 66 (b) 55 (c) 50
(d) none of these
Q 46. If the coefficient of rth term in the expansion of (1 + x)20 is equal to its (r + 2)th term then r is equal to: (a) 10 (b) 9 (c) 11 (d) none of these. Q 47. The value of (a) 10
183 7 3 3.18.7.25 is equal to: 36 6.243.2 15.81.4 20.27.8 15.9.16 6.3.32 64 (b) 1 (c) 20 (d) 2
Q 48. The positive integer just greater than (1 + 0.0001)10000 is: (a) 2 (b) 3 (c) 4
(d) 5
Q 49. The greatest term in the expansion of (1 + 2)12 is : (a) 7th (b) 8th (c) 6th
(d) 5th 11
11
1 Q 50. If the coefficient of x in ax 2 bx 7
equal, then ab is equal to: (a) 1 (b) 2
and the coefficient of x (c) 3
Q 51. The value of (7.995)1/3 is correct to four decimal places is (a) 1.9995 (b) 1.9996 (c) 1.9990
-7
1 in ax 2 bx
(d) 4 (d) 1.9991
Q 52. If the third term in the expansion of x x log x is 106 then x must be: (a) 1 (b) 10 (c) 10 (d) 10-3/5 10
5
Q 53. The expansion [x2 + (x6 – 1)1/2]5 + [x2 – (x6 – 1)1/2]5 is a polynomial of degree:
are
(a) 8
(b) 10
Q 54. (1 + x + x2 + …)-n is equal to: (a) (1 + x)n (b) (1 + x)2n
(c) 13
(d) none of these.
(c) (1 – x)n
(d) none of these.
Q 55. The expansion of (1 + 2x)n for non-positive integral index is valid only when: (a) – 1 < x < 1
(b)
1 1 x 2 2
(c) x R
(d) none of these.
Q 56. The coefficient of xn in the expansion of (1 + 2x + 3x2 + ….)-n/2 is equal to: (a) 1 (b) (-1)n (c) 0 (d) none of these. Q 57. Sum of coefficient in the expansion of (x + y – z)10 is: (a) 1 (b) 0 (c) 310
(d) none of these.
Q 58. Sum of the coefficients in the expansion of (2x + y + z)5 is: (a) 1 (b) 45 (c) 1
(d) none of these.
Q 59. The coefficient of xn in the expansion of (1 + 2x + 3x2+ ….)1/2 is equal to: (a) 1 (b) n (c) – n (d) none of these. Q 60. If p and q are the coefficients of xn in (1 + x)2n and (1 – 4x)-1/2, | x | < (a) p = q
(b) p = 2q
(c) q = 2p
1 , then: 4
(d) p + q = 2nCn
Q 61. If | x | < 1, then expansion of (1 + x)-1 + (1 – x)-1 is equal to: (a) expansion of (1 – x2)-1 (b) expansion of 2(1 – x2)-1 (c) expansion of
1 (1 – x2)-1 2
(d) none of these.
Q 62. The remainder when (27)15 is divided by 13 equal to: (a) 1 (b) 0 (c) 12
(d) 8
Q 63. The remainder when (25)15 is divided by 13 equal to: (a) 1 (b) 12 (c) 0
(d) none of these.
Q 64. The digit at unit’s place in the number (13)1225 + (11)1915 – (23)1225 is equal to: (a) 0 (b) 2 (c) 1 (d) none of these. Q 65. If x is numerically very small as compared with 1, then (1 – 7x)1/3 (1 +2x)-3/4 is equal to: (a) 1 +
23x 6
(b) 1 -
25x 6
(c) 1 -
23x 6
Q 66. The greatest integer less than or equal to ( 2 1) 6 is equal to: (a) 196 (b) 197 (c) 198
(d) 1 +
(d) 199.
25x . 6
Q 67. If x is very large and n is a negative integer or a proper fraction, then an approximate n
1 x is equal to: x
value of (a) 1+
x n
(b) 1
n x
(c) 1
1 x
1 x
(d) n 1
Q 68. The first three terms in the expansion of (1 + x)-3 are: (a) 1 + 3x + 6x2 (b) 1 – 3x + 6x2 (c) 1 – 2x + 3x2
(d) none of these
Q 69. The coefficient of x2y3z5 in the expansion of (x + y + z)10 is (a) 1260 (b) 5040 (c) 2520
(d) none of these.
Q 70. If (1 + x + x2)n = a0 + a1x + axx2 + …… + a2nx2n, then: (a) a0 - a2 + a4 – a6 + ….. 0, if n is even (b) a1 – a3 + a5 – a7 + …. = 0, if n is even (c) a0 – a2 + a4 – a6 + …. = 0, if n = 4k, k I+ (d) a1 – a3 + a5 – a7 + ….. = 0, if n = 4k + 1, k Q 71. The coefficient of x50 in the expansion of (1 + x)41 (1 – x + x2)40 is: (a) 0 (b) 50C41 (c) 81C50 (d) 81C49 n Q 72. If the second, third and fourth terms in the expansion of (a + b) are 135, 30 and 10/3 respectively, then: (a) n = 3 (b) n = 2 (c) n = 7 (d) n = 5 Q 73. If (1 + ax)n = 1 + 8x + 24x2 + ….., then the values of a and n are equal to: (a) 2, 4 (b) 2, 3 (c) 3, 6 (d) 1, 2 Q 74. The sum of the coefficients in (1 + x – 3x2)2143 is equal to: (a) 22143 (b) 0 (c) 1
(d) – 1.
Q 75. If | x | < 1, then the coefficient of xn in the expansion of (1 + 2x + 3x2 + 4x3 + …)1/2 is: (a) n (b) n + 1 (c) 1 (d) – 1. Q 76. The coefficients of xn in the expansion of (1 + x)2n and (1 + x)2n-1 are in the ratio of : (a) 1 : 2 (b) 1 : 3 (c) 3 : 1 (d) 2 : 1 Q 77. The coefficient of x7 in the expansion of (1 – x4) (1 + x9) is equal to: (a) 27 (b) – 24 (c) 48 (d) – 48 Q 78. The value of nC0 + n+1C1 + n+2C2 + …. + n+kCk is equal to: (a) n+k+1Ck (b) n+k+1Cn+1 (c) n+kCn+1
(d) none of these.
Q 79. The largest coefficient in the expansion of (4 + 3x)25 is: 11
4 3
(a) 25C11 . 325
11
3 4
(b) 25C11.425
(c) 25C14.414.311 (d) none of these.
Q 80. If (1 + x + x2) = a0 + a1x + a2x2 + …… + a2nx2n then a0 + a3 + a6 + ….. equals: (a) 3n (b) 3n-1 (c) 0 (d) 1 ......1 is a: Q 81. The number 111 91 times
(a) prime number (b) divisible by 3 (c) divisible by
10 7 1 (d) none of the above 9
Q 82. The sum of the coefficients of even power of x in the expansion of (1 + x + x2 + x3)5 is equal to: (a) 510 (b) 512 (c) 1022 (d) 1024. Q 83. The number of integral terms in the expansion of (51/2 + 71/8)1028 is: (a) 128 (b) 129 (c) 130 (d) 131. Q 84. If in the expansion of (1 + x)m (1 + x)n, the coefficients of x and x2 are 3 and – 6 respectively, the m is equal to: (a) 6 (b) 9 (c) 12 (d) 24. n + r 1
n is equal to: r 2 n 1 n 2 (b) 2 (c) r 1 r
n r
Q 85. For 2 ≤ r ≤ n, + 2 n 1 r 1
(a)
n 2 r
(d) 2
Q 86. In the binomial expansion of (1 – b)n, n 5, the sum of the 5th and 6th terms is zero. Then a/b equals: (a)
n5 6
(b)
n4 5
10 20 , (where i 0 m i m
Q 87. The sum
i
(a) 5
(c)
5 n4
(b) 10
(a) 2
1 10 2 n 102 C1 n 81n 81n 81 (b) 0 n
Q 90. If x + y = 1, then
r 0
(a) 1
n
6 n5
p = 0 if p > q ) is maximum when m is: q
(c) 15
Q 88. Coefficient of t24 in (1 + t2)12 (1 + t12) (1 + t24) is: (a) 12C6 + 3 (b) 12C6 + 1 (c) 12C6 Q 89. The value of
(d)
2n
(d) 20. (d) 12C6 + 2
103 2 n 102 n C2 n C3 .... n is: 81 81 (c) 1/2 (d) 1
Cr x r y n r equals: (b) n
(c) nx
(d) nv.
Q 91. The coefficient of xn in the expansion of (1 – 2x + 3x2 – 4x3 + ….)-n is: ( 2n)! 1 ( 2n)! ( 2n)! (a) (b) (c) (d) none of these. 2 ( n!) 2 ( n!) 2 n! Q 92. If the third term in the binomial expansion of (1 + x)n is m is: (a) 2
(b) ½
1 2 x , then the rational value of 8
(c) 3
(d) 4.
Q 93. The greatest coefficients in the expansion of (1 + x)2n+2 is: ( 2n)! (2n 2)! (2n 2)! (a) (b) (c) 2 2 n!( n 1)! ( n!) [( n 1)!]
(d)
(2n)! n!(n 1)!
Q 94. The number of non-zero terms in the expansion of (2 + 2 x)7 + (2 - 2 x)7 is: (a) 3 (b) 4 (c) 0 (d) 8. Q 95. Consider the following statements: n
r
(1) If x + y = 1, then
r 0 2n
(2) If (1 + 2x + x2)n =
n
Cr x r y n r is equal to nx.
a x r 0
r
r
, then ar is equal to nCr
(3) If A and B are coefficients of xn in the expansions of (1 + x)2n and (1 + x)2n-1 respectively, the A = 2B. Which of these is/are correct? (a) only 2 (b) only 3
(c) 1 and 3
(d) none of these.
Answers 1. 8. 15.
(c) (c) (d)
2. 9. 16.
(a) (b) (d)
3. 10. 17.
(d) (d) (c)
4. 11. 18.
(c) (b) (c)
5. 12. 19.
(b) (a) (a)
6. 13. 20.
(c) (c) (b)
7. 14. 21.
(a) (a) (c)
22. 29. 36. 43. 50. 57. 64. 71. 78. 85. 92.
(b) (a) (a) (b) (a) (a) (c) (a) (a) (c) (b)
23. 30. 37. 44. 51. 58. 65. 72. 79. 86. 93.
(d) (d) (a) (a) (b) (b) (c) (d) (b) (b) (b)
24. 31. 38. 45. 52. 59. 66. 73. 80. 87. 94.
(a) (c) (a) (a) (c) (a) (b) (a) (b) (c) (b)
25. 32. 39. 46. 53. 60. 67. 74. 81. 88. 95.
(b) (a) (a) (a) (d) (a) (b) (d) (b) (d) (c)
26. 33. 40. 47. 54. 61. 68 75. 82. 89.
(d) (c) (c) (b) (c) (b) (b) (c) (b) (d)
27. 34. 41. 48. 55. 62. 69. 76. 83. 90.
(b) (d) (b) (b) (b) (a) (c) (d) (b) (a)
28. 35. 42. 49. 56. 63. 70. 77. 84. 91.
(a) (b) (b) (b) (b) (b) (b) (d) (c) (b)
