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Question Bank No: 1

1. If the third term in the binomial expansion if ${(1 + x)}^{m}$ is - $\frac{1}{8}$ $χ^{2}$ , then the rational vlaue of m is

a)		2
b)		1/2
c)		3
d)		4

2. If $| x |$ < 1,then the coefficient of $x^{n}$ in the expansion of ${(1 + x + x^{2} + x^{3} ......)}^{2}$ 2 is

a)		n
b)		n - 1
c)		n + 2
d)		n + 1

3. The value of ${}^{13}{C_{2}}$ + ${}^{13}{C_{3}}$ + ${}^{13}{C_{4}}$ + ..... + ${}^{13}{C_{13}}$ is

a)		$2^{13}$ - 13
b)		$2^{13}$ - 14
c)		an odd number $\neq$ $2^{13}$ - 12
d)		an even number $\neq$ $2^{13}$ - 14

4. If the coefficient of rth term in the expansion of ${(1 + x)}^{20}$ = coefficient of (r + 2)th term, then r equals

a)		12
b)		8
c)		10
d)		9

5. If the binomial expansion of ${(a + bx)}^{- 2}$ is $\frac{1}{4}$ -3x + ........, where a > 0, then (a, b) is

a)		(2, 12)
b)		(2, 8)
c)		( - 2, 12)
d)		None of these

6. The total number of terms in the expansion of ${(x + y)}^{100}$ + ${(x - y)}^{100}$ after simplification is

a)		50
b)		51
c)		202
d)		None of these

7. If y = $\frac{1}{3}$ + $\frac{1.3}{3.6}$ + $\frac{1.3.5}{3.6.9}$ + ....., then the value of $y^{2}$ + 2y is

a)		2
b)		-2
c)		0
d)		None of these

8. The 4th term in the expansion of ${(- x - \frac{1}{x})}^{5}$ , x > 0, is

a)		Independent of x
b)		positive
c)		negative
d)		the only middle term

9. The coefficient of $x^{2}$ in the expansion of ${(1 + 2 x + 3 x^{2} + .....)}^{1 / 2}$ is

a)		1
b)		n + 1
c)		-1
d)		n

10. Constant term in the expansion of ${(x - \frac{1}{x})}^{10}$ is

a)		-252
b)		152
c)		252
d)		-152

11. The coefficient of $x^{4}$ in ${(\frac{x}{2} - \frac{3}{x^{2}})}^{10}$ is equal to

a)		$\frac{405}{256}$
b)		$\frac{504}{259}$
c)		$\frac{450}{263}$
d)		None of these

12. If the sum of the coefficients in the expansion of (a + b)n is 4096, then the greatest coefficient in the expansion is

a)		924
b)		792
c)		1594
d)		None of these

13. The number of terms which are free from radical signs in the expansion of ${(y^{1 / 5} + x^{1 / 10})}^{55}$ is

a)		5
b)		6
c)		7
d)		None of these

14. In Pascal’s triangle, each row is bounded by

a)		2
b)		1
c)		-1
d)		None of these

15. The expansion of $\frac{1}{\sqrt{6 - 3 x'}}$ , in powers of x, is valid if

a)		x < 2
b)		$\| x \|$ < 2
c)		x > 2
d)		$\| x \|$ > 2

16. The term independent of x in the expansion of ${(2 x + \frac{1}{3 x})}^{6}$ is

a)		$\frac{160}{9}$
b)		$\frac{80}{9}$
c)		$\frac{160}{27}$
d)		$\frac{80}{3}$

17. If the expansion of ${(x^{2} + \frac{2}{x})}^{n}$ for positive integer n has a term independent of x, then n is

a)		23
b)		18
c)		16
d)		0

18. If the sum of the coefficients in the expansion of ${(1 + 2 x)}^{n}$ is 6561, to greatest term in the expansion for x =1/2 is

a)		4th
b)		5th
c)		6th
d)		None of these

19. The middle term in the expansion of ${(\frac{x^{2}}{2} - \frac{2}{x})}^{9}$ is

a)		- 56
b)		14
c)		- 14
d)		None of these

20. When the number of terms in the expansion of binomial ${(x + a)}^{n}$ , where n is a +ve integer, is even, then there is/are only

a)		One middle term
b)		two middle term
c)		no middle term
d)		None of these

21. If $| x | < 1 then 1 + n (\frac{2 x}{1 + x}) + \frac{n (n + 1)}{2!} (\frac{2 x}{1 + x})^{2} + ..... is$

a)		$(\frac{2 x}{1 + x})^{n}$
b)		$(\frac{1 + x}{2 x})^{n}$
c)		$(\frac{1 - x}{1 + x})^{n}$
d)		$(\frac{1 + x}{1 - x})^{n}$

22. The coefficient of $x^{n} in binomial expansion {(1 - x)}^{- 2} is$

a)		2n
b)		n+1
c)		$\frac{2 n}{2!}$
d)		n

23. Coefficient of $x^{2} in {(1 + 3 x)}^{\frac{1}{2}} (1 - 2 x)^{\frac{1}{3}} is$

a)		$\frac{- 55}{72}$
b)		$\frac{55}{72}$
c)		$\frac{72}{55}$
d)		none of these

24. If the sum of the coefficient in the expansion of ( $\propto^{2} x^{2} - 2 \propto x + 1)^{51} vanishes then \propto is equal to$

a)		2
b)		-1
c)		1
d)		-2

25. The expansion of $\frac{1}{\sqrt{6 - 3 x}} in powers of x is valid if$

a)		x $< 2$
b)		$\| x \| < 2$
c)		x $> 2$
d)		$\| x \| > 2$

26. The expansion ${(1 - 4 x)}^{\frac{- 1}{2}} is valid for$

a)		$\| x \| < 1$
b)		$\| x \| > 1$
c)		$\| x \| < \frac{1}{4}$
d)		$\| x \| > \frac{1}{4}$

27. The coefficient of $x^{7} in the expansion of {(1 - x}^{4}) {(1 + x)}^{9} is$

a)		27
b)		-24
c)		48
d)		-48

28. The smallest +ve integer n for which n! $< (\frac{n + 1}{2})^{n} is$

a)		1
b)		2
c)		3
d)		4

29. $49^{n} - 16 n - 1 is divisible by$

a)		3
b)		64
c)		19
d)		29

30. The sum of the coefficient in the expansion of ${(1 - x)}^{10} is$

a)		0
b)		1
c)		$10^{2}$
d)		1024

31. The number of terms in the expansion of $[({a + 4 b}^{3}) ({a - 4 b)}^{3}]^{2}$

a)		6
b)		7
c)		8
d)		32

32. In the expansion of ${(1 + x)}^{5} the sum of the coefficient of the term is$

a)		80
b)		16
c)		32
d)		64

33. The coefficient of $x^{n} in the expansion of {(1 + x)}^{2 n}$ and $(1 + x)^{2 n - 1} are in the ratio$

a)		1:2
b)		1:3
c)		3:1
d)		2:1

34. The value of ${nc}_{0} - {nc}_{1} + {nc}_{2} - {nc}_{3} + ...... (- 1)^{n} {nc}_{n} is$

a)		$2^{n} - 1$
b)		n
c)		$2^{n}$
d)		0

35. If n is a +ve integer then the number of terms in the expansion of ${(x + a)}^{n} is$

a)		n
b)		(n+1)
c)		n-1
d)		none of these

36. If n is +ve integer then ${2.4}^{2 n + 1} + 3^{3 n + 1} is divisible by$

a)		2
b)		9
c)		11
d)		27

37. The middle term of (x+ $\frac{1}{x})^{10} is$

a)		${10 c}_{6}$
b)		${10 c}_{5}$
c)		${10 c}_{4}$
d)		none of these

38. In the expansion of ${(1 + x)}^{n} coefficient of 2^{nd}, 3^{rd}, 4^{th} term are in Ap, then x =$

a)		5
b)		6
c)		4
d)		7

39. The sum of the series $(\begin{matrix} 20 \\ 0 \end{matrix}) - (\begin{matrix} 20 \\ 1 \end{matrix}) + (\begin{matrix} 20 \\ 2 \end{matrix}) - (\begin{matrix} 20 \\ 2 \end{matrix})$ +..............+ $(\begin{matrix} 20 \\ 10 \end{matrix}) =$

a)		$\frac{1}{2} (\begin{matrix} 20 \\ 10 \end{matrix})$
b)		0
c)		$(\begin{matrix} 20 \\ 10 \end{matrix})$
d)		$- (\begin{matrix} 20 \\ 10 \end{matrix})$

40. In the binomial expansion of ${(a - b)}^{n}, n \geq 5,$ the sum of 5th and 6th terms is zero, then $\frac{a}{b}$ =

a)		$\frac{6}{n - 5}$
b)		$\frac{n - 5}{6}$
c)		$\frac{n - 4}{5}$
d)		$\frac{5}{n - 4}$

41. If $\frac{1}{(1 - ax) (1 - bx)} = a_{0} + a_{1} x + a_{2} x^{2} + ....,$ then $a_{n} =$

a)		$\frac{a^{n + 1} - b^{n + 1}}{b - a}$
b)		$\frac{b^{n + 1} - a^{n + 1}}{b - a}$
c)		$\frac{b^{n} - a^{n}}{b - a}$
d)		$\frac{a^{n} - b^{n}}{b - a}$

42. For natural numbers m and n if, ${(1 - x)}^{m}$ ${(1 + x)}^{n} = 1 + a_{1} x + a_{2} x^{2} + ..... and a_{1} = a_{2} = 10,$ then (m,n) is

a)		(45, 35)
b)		(35, 45)
c)		(20, 45)
d)		(35, 20)

43. If the coefficients of $r^{th}, {(r + 1)}^{th}, {(r + 2)}^{th}$ terms in the expansion of ${(1 + x)}^{n}$ are in A.P., then n and r satisfy

a)		$n^{2} - n (4 r + 1) + 4 r^{2} + 2 = 0$
b)		$n^{2} - n (4 r - 1) + 4 r^{2} - 2 = 0$
c)		$n^{2} - n (4 r - 1) + 4 r^{2} + 2 = 0$
d)		$n^{2} - n (4 r + 1) + 4 r^{2} - 2 = 0$

44. $C_{r} = (\begin{matrix} n \\ r \end{matrix}),$ then $a_{n} = \sum_{r = 0}^{n} \frac{1}{C_{r}}$ , then $\sum_{r = 0}^{n} \frac{r}{C_{r}} =$

a)		( $n - 1) a_{n}$
b)		n $a_{n}$
c)		$\frac{n a_{n}}{2}$
d)		none of these

45. The coefficient of $x^{n} in the expansion of (1 + x) {(1 - x)}^{n}$ is

a)		n-1
b)		${(- 1)}^{n} (- n)$
c)		${(- 1)}^{n - 1} {(n - 1)}^{2}$
d)		${(- 1)}^{n - 1} n$

46. The coefficient of the middle term in the binomial expansion is powers of x of ${(1 + α x)}^{4} and {(1 - α x)}^{6}$ is the same if $α =$

a)		$- \frac{5}{3}$
b)		$\frac{3}{5}$
c)		$\frac{- 3}{10}$
d)		$\frac{10}{3}$

47. The number of integral terms in the expansion of ${(\sqrt{3} + 5^{\frac{1}{8}})}^{256}$ is

a)		32
b)		33
c)		34
d)		35

48. The sum of the coefficients in teh expansion of ${(a + b)}^{n}$ is 4096. The greatest coefficient in the expansion is

a)		1594
b)		792
c)		924
d)		2924

49. The coefficients of $x^{p}$ and $x^{q}$ in the expansion of ${(1 + x)}^{p + q}$ are

a)		equal
b)		equal with opposite signs
c)		reciprocals to each other
d)		none of these

50. The positive integer just greater than ${(1 + 0.0001)}^{10000}$ is

a)		4
b)		5
c)		2
d)		3