Baixe 65878214 - Binomial - Theorem - Qns e outras Exercícios em PDF para Matemática, somente na Docsity!
Binomial Theorem^1
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
1. The fourth term in the expansion of (1 - 2x) 3/2^ will be :
(A) -
x 4 (B) x 3 2
(C) - x 3 2 (D)^
x 4
2.^10 C 1 + 10 C 3 + 10 C 5 + 10 C 7 + 10 C 9 = (A) 2 9 (B) 2 10 (C) 2 10 - 1^ (D) None of these 3. C 0 C (^) r + C 1 C (^) r + 1 + C 2 C (^) r + 2 + ..... + C (^) n − r C (^) n is equal to :
(A) (^ )! ( )! ( )!
2 n n − r n +r
(C) n r n r
(C) n n r
(D) None of these
4. If the coefficient of r th^ term & (r + 4) th term are equal in the expansion of (1 + x) 20 , then the value of r will be (A) 7 (B) 8 (C) 9 (D) 10
5. C
C
C
C
C
C
1 0
2 1
3 2
n n n
C
C − 1 =
(A) n^ (^ n−^1 ) 2
(B) n^ (^ n+^2 ) 2
(C) n^ (n^ +^1 ) 2
(D) (^ n^ −^1 ) (^ n−^2 ) 2
6. C 1 + 2 C 2 + 3 C 3 + 4 C 4 + ...... + nC (^) n = (A) 2 n^ (B) n. 2 n (C) n. 2 n - 1^ (D) n. 2 n + 1 7. If p & q be positive, then the coefficients of x p^ & x q^ in the expansion of (1 + x) p + q^ will be : (A) Equal (B) Equal in magnitude but opposite in sign (C) Reciprocal to each other (D) None of these 8. The term independent of x in the
expansion of x 3 x
10
will be :
(A) 3
(B) 5
(C) 5
(D) None of these
9. If the coefficients of 5 th^ , 6 th^ and 7 th terms in the expansion of (1 + x) n^ be in A.P., then n = (A) 7 only (B) 14 only (C) 7 or 14 (D) None of these 10. The sum of the coefficients in the expansion of (1 + x − 3x 2 ) 2163 will be (A) 0 (B) 1 (C) - 1 (D) 2 2163
11.
C
C
C
C
C
C
1 0
2 1
3 2
+ 2 + 3 + ...... + 15 C
C
15 14
(A) 100 (B) 120
(C) - 120 (D) None of these
12. In the expansion of x x
^
(^1 6) , the
constant term is : (A) - 20 (B) 20 (C) 30 (D) - 30
20. In the expansion of (x 2 - 2x) 10 , the
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Binomial Theorem^2
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
coefficient of x 16 is : (A) - 1680 (B) 1680 (C) 3360 (D) 6720
13. If (1 + ax) n^ = 1 + 8x + 24x 2 + ...... , then the value of a and n is : (A) 2, 4 (B) 2, 3 (C) 3, 6 (D) 1, 2 14. If for positive integers r > 1, n > 2, the coefficient of the (3 r) th^ & (r + 2) th powers of x in the expansion of (1 + x) 2n^ are equal, then : (A) n = 2r (B) n = 3r (C) n = 2r + 1 (D) None of these 15. The sum of the series, r
n
ˇ 0
(-1) r^ n^ C (^) r
r 2 3 24
r r
r r
r
.... is :
(A)
mn mn n
(B)
m n n
(C)
m n n
(D) None of these
16. Let R = ( 5 5 11 )
2 1
n+ & f = R − [R], where [. ] denotes the greatest integer function. The value of R. f is : (A) 4 2n + 1^ (B) 4 2n (C) 4 2n - 1^ (D) 4 -2n
17. The greatest coefficient in the expansion of (1 + x) 2n + 2^ is :
(A) (^ )! ( !)
2
n n
(B)
{ }
n n
(C)
n n n
(D)
n n n +
18. If (1 + x) n^ = r
n
ˇ 0
C (^) r x r^ , then
0
2 1 1
−
C
C
C
C
C
C
n n
(A)
n n
n − −
1 ( 1 )!
(B)
n n
1
(C) (^ )
n n
n n
19. The coefficient of x 4 in the expansion of (1 + x + x 2 + x 3 ) n^ is : (A) n^ C 4 (B) n^ C 4 + n^ C (^2) (C) n^ C 4 + n^ C 2 + n^ C 4. n^ C (^2) (D) n^ C 4 + n^ C 2 + n^ C 1. n^ C (^2) 20. If x < 1, then the value of,
1 + n
x +x
n n x x
^
2
will be :
(A) 1 1
^
x x
n (B) 2 1
x x
n
(C) 1
x x
n (D) 1 1
x x
n
21. If the value of x is so small that x 2 and greater powers can be neglected,
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Binomial Theorem^4
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(A)
n n
(B)
n n
(C)
n n
(D) None of these
31. If (1 + x − 2x 2 ) 6 = 1 + a 1 x + a 2 x 2 + ...... + a 12 x 12 , then ethe expression, a 2 + a 4 + a 6 + ...... + a 12 has the value (A) 32 (B) 63 (C) 64 (D) None of these 32. If the 6th term in the expansion of the binomial ,
ˇ 2 log (^) ( 10 − (^3) ) (^) + (^52) ( − (^2) ) log 3
x (^) x m
is equal to 21
and it is known that the binimial co- efficient of the 2 nd, 3 rd^ & 4 th^ terms in the expansion represent respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base 10), then x = (A) 0 (B) 1 (C) 2 (D) 3
33. If n is an integer greater then 1, then a - n^ C 1 (a - 1) + n^ C 2 (a - 2) + ...... + (- 1) n^ (a - n) = (A) a (B) 0 (C) a 2 (D) 2 n 34. The sum of the coefficients of even power of x in the expansion of, (1 + x + x 2 + x 3 ) 5 is : (A) 256 (B) 128 (C) 512 (D) 64 35. Given that 4 th^ term in the expansion
of, 2
10 ˇ + ^
x (^) has the maximum
numerical value, the rang of value of x for which this will be true is given by :
(A) - 64 21 < x < - 2
(B) - 64
< x < 2
(C) 64
< x < 4 (D) None of these
36. If n is a positive integer and C (^) k = n^ C (^) k ,
then the value of k
n
ˇ 1
k 3 C C
k k −
1
2
(A) n^ (n^ +^1 ) (^ n^ +^2 ) 12
(B) n^ (n^ +^1 ) 12
2
(C) n^ (n^ +^2 )^ (^ n+^1 ) 12
2
(D) None of these
37. 1 - 1 8
(A) 2
(B) 2
(C)
(D) None of these
69. If a 1 , a 2 , a 3 , a 4 are the coefficients of
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Binomial Theorem^5
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
any four consecutive terms in the expansion of (1 + x) n^ , then a a a
1 1 +^2
3 3 +^4
(A)
a a a
2 2 +^3
(B) 1
2 ( )
a a a
2 2 + 3
(C) 2 2 2 3
a a + a
(D) 2 3
2 3
a a +a
39. If the sum of the coefficients in the expansion of (α^2 x 2 - 2 αx + 1) 51 vanishes, then the value of α is : (A) 2 (B) - 1 (C) 1 (D) - 2 40. The value of x, for which the 6 th^ term in the expansion of,
( ) ( )
2 1 2 1
9 7 1 5 3 1
7 log ( / ) log
x x
− −
is
84, is equal to : (A) 4 (B) 3 (C) 2 (D) 1
41. Let n and k be positive integers such
that n ≥ k^ (^ k+^1 ) 2
. The number of solutions (x 1 , x 2 , ...... x (^) k ) , x 1 ≥ 1, x 2 ≥ 2 , ...... x (^) k ≥ k , all integers, satisfying x 1 + x 2 + ...... + x (^) k = n, is : (A) m^ C (^) k - 1 (B) m^ C (^) k + 1 (C) m^ C (^) k (D) None of these
{where m = 1 2
(2n - k 2 + k - 2)}
42. The number of integral terms in the expansion of, (5 1/2^ + 7 1/6^ ) 642 is : (A) 106 (B) 108 (C) 103 (D) 109 43. The coefficient of x in the expansion
of 1 2
1 ˇ + − − x x (^) in ascending pwers of x, when x < 1, is :
(A) 0 (B) 1 2
(C) - 1 2
(D) 1
44. The value of, 15 0 2 15 1 2 15 2
C − C + C^2 - ...... - 15
15 C (^2) is : (A) 15 (B) - 15 (C) 0 (D) 51
ANSWERS
1. B 2. A 3. A 4. C 5. C 6. C
- A 8. B 9. C 10. C 11. B 12.A
- A 14. C 15. A 16. A 17. B 18.C
- D 20. A 21. B 22. C 23. D 24.C
- C 26. A 27. C 28. B 29. A 30.B
- D 32. AC33. B 34. C 35. A 36.D
- C 38. C 39. C 40. CD41. A 42.B
- D 44. C
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