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Question Bank No: 2

1. $(\begin{matrix} 30 \\ 0 \end{matrix}) (\begin{matrix} 30 \\ 10 \end{matrix}) - (\begin{matrix} 30 \\ 1 \end{matrix}) (\begin{matrix} 30 \\ 11 \end{matrix}) + (\begin{matrix} 30 \\ 2 \end{matrix})$ $(\begin{matrix} 30 \\ 12 \end{matrix}) - ........ + (\begin{matrix} 30 \\ 20 \end{matrix}) (\begin{matrix} 30 \\ 30 \end{matrix}) =$

a)		$(\begin{matrix} 30 \\ 11 \end{matrix})$
b)		$(\begin{matrix} 60 \\ 10 \end{matrix})$
c)		$(\begin{matrix} 30 \\ 10 \end{matrix})$
d)		$(\begin{matrix} 65 \\ 55 \end{matrix})$

2. If $(\begin{matrix} n - 1 \\ r \end{matrix}) = (k^{2} - 3) (\begin{matrix} n \\ r + 1 \end{matrix})$ , then k $\in$

a)		$(- \infty, - 2]$
b)		$[2, \infty)$
c)		$[\sqrt{3}, 2]$
d)		$[- 2, \sqrt{3})$ $\cup (\sqrt{3}, 2]$

3. $2^{k} (\begin{matrix} n \\ 0 \end{matrix}) (\begin{matrix} n \\ k \end{matrix}) - 2^{k - 1} (\begin{matrix} n \\ 1 \end{matrix}) (\begin{matrix} n - 1 \\ k - 1 \end{matrix}) + 2^{k - 2} (\begin{matrix} n \\ 2 \end{matrix}) (\begin{matrix} n - 2 \\ k - 2 \end{matrix}) - ..... + {(- 1)}^{k} (\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} n - k \\ 0 \end{matrix}) =$

a)		$(\begin{matrix} n \\ k - 1 \end{matrix})$
b)		$(\begin{matrix} n + 1 \\ k \end{matrix})$
c)		$(\begin{matrix} n \\ k \end{matrix})$
d)		$(\begin{matrix} n + 1 \\ k + 1 \end{matrix})$

4. The coefficeint of $x^{24} in {(1 + x^{2})}^{12} (1 + x^{12}) (1 + x^{24}) is$

a)		$(\begin{matrix} 12 \\ 6 \end{matrix})$
b)		$(\begin{matrix} 12 \\ 6 \end{matrix})$ +1
c)		$(\begin{matrix} 12 \\ 6 \end{matrix})$ +2
d)		$(\begin{matrix} 12 \\ 6 \end{matrix})$ +3

5. The sum $\sum_{i}^{m} (\begin{matrix} 10 \\ i \end{matrix})$ $(\begin{matrix} 20 \\ m - i \end{matrix})$ , where $(\begin{matrix} p \\ q \end{matrix}) = 0 if p < q,$ is maximum for m=

a)		5
b)		10
c)		15
d)		20

6. Let $T_{n}$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If $T_{n + 1} - T_{n} = 21, then n =$

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

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

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

8. For any positive integers $n \geq m,$ $(\begin{matrix} n \\ m \end{matrix}) + 2 (\begin{matrix} n - 1 \\ m \end{matrix})$ +3 $(\begin{matrix} n - 2 \\ m \end{matrix})$ +.......+(n-m+1) $(\begin{matrix} m \\ m \end{matrix})$ =

a)		$(\begin{matrix} n + 2 \\ m \end{matrix})$
b)		$(\begin{matrix} n + 1 \\ m \end{matrix})$
c)		$(\begin{matrix} n + 2 \\ m + 1 \end{matrix})$
d)		$(\begin{matrix} n + 2 \\ m + 2 \end{matrix})$

9. For any positive integers with n $\geq m$ , $(\begin{matrix} n \\ m \end{matrix})$ + $(\begin{matrix} n - 1 \\ m \end{matrix}) + (\begin{matrix} n - 2 \\ m \end{matrix}) + ..... + (\begin{matrix} m \\ m \end{matrix})$ =

a)		$(\begin{matrix} n + 1 \\ m \end{matrix})$
b)		$(\begin{matrix} n + 1 \\ m + 1 \end{matrix})$
c)		$(\begin{matrix} n \\ m + 1 \end{matrix})$
d)		$(\begin{matrix} n + m \\ m \end{matrix})$

10. For $2 \leq r \leq n, (\begin{matrix} n \\ r \end{matrix})$ +2 $(\begin{matrix} n \\ r - 1 \end{matrix})$ + $(\begin{matrix} n \\ r - 2 \end{matrix})$ =

a)		$(\begin{matrix} n + 1 \\ r - 1 \end{matrix})$
b)		2 $(\begin{matrix} n + 1 \\ r + 1 \end{matrix})$
c)		2 $(\begin{matrix} n + 2 \\ r \end{matrix})$
d)		$(\begin{matrix} n + 2 \\ r \end{matrix})$

11. If in the expansion of ${(1 + x)}^{m} {(1 - x)}^{n},$ the coefficient of x and $x^{2}$ are 3 and -6 respectively, then m=

a)		6
b)		9
c)		12
d)		24

12. $\sum_{r = 0}^{n}$ ${(- 1)}^{r} \frac{(\begin{matrix} n \\ r \end{matrix})}{(\begin{matrix} r + 3 \\ r \end{matrix})}$ =

a)		$\frac{2}{n + 3}$
b)		$\frac{6}{n + 2}$
c)		$\frac{3}{n + 2}$
d)		$\frac{3}{n + 3}$

13. The sum of the rational terms in the expansion of ${(\sqrt{2} + 3^{\frac{1}{5}})}^{10}$ is

a)		28
b)		36
c)		41
d)		54

14. Let n $\in N$ and ${(1 + x + x^{2})}^{n}$ = $\sum_{r = 0}^{2 n} a_{r} x^{r}$ . Then ${a_{0}}^{2} -$ ${a_{1}}^{2} +$ ${a_{2}}^{2} - ..... {... + a}_{2 n}^{2} =$

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

15. Let n $\in N .$ If the coefficient of 2nd, 3rd, 4th terms in the expansion of ${(1 + x)}^{n} are in$ A.P., then x=

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

16. If n is an even integer and N= $\frac{3 n}{2}$ , then $\sum_{r = 1}^{N} {(- 3)}^{r - 1} (\begin{matrix} 3 n \\ 2 r - 1 \end{matrix}) =$

a)		N
b)		-2N
c)		1
d)		0

17. If $\sum_{r = 0}^{2 n} C_{r} {(x - 2)}^{r} = \sum_{r = 0}^{2 n} b_{r} {(x - 3)}^{r}$ and $a_{r} = 1 for all r \geq n, then b_{n}$ =

a)		$(\begin{matrix} 2 n \\ n - 1 \end{matrix})$
b)		$(\begin{matrix} 2 n \\ n \end{matrix})$
c)		$(\begin{matrix} 2 n + 1 \\ n \end{matrix})$
d)		$(\begin{matrix} 2 n - 1 \\ n \end{matrix})$

18. ${(x + \sqrt{x^{3} - 1})}^{5} + {(x - \sqrt{x^{3} - 1})}^{5}$ is a polynomial of degree

a)		5
b)		6
c)		7
d)		8

19. If $n \in N, then \frac{n^{7}}{7} + \frac{n^{5}}{5} + \frac{2 n^{3}}{3} - \frac{n}{105}$ is

a)		a negative natural number
b)		a positive natural number
c)		a positive rational number
d)		a negative rational number

20. If $n > 2$ and $C_{r} = (\begin{matrix} n \\ r \end{matrix})$ , then $1^{2} \cdot C_{0} - 2^{2} \cdot C_{1} + 3^{2} \cdot C_{2} - .......... =$

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

21. Let $R = {(5 \sqrt{5} + 11)}^{2 n + 1} and f = R - [R]$ , the fractional part. Then $R f =$

a)		$4^{2 n}$
b)		$4^{2 n - 1}$
c)		$4^{2 n + 1}$
d)		$2^{2 n + 1}$

22. If n is an even integer and $C_{r} = (\begin{matrix} n \\ r \end{matrix})$ , then $\frac{2 (\frac{n}{2})! (\frac{n}{2})!}{n!}$ $[{C_{0}}^{2} - 2 \cdot {C_{1}}^{2} + 3 \cdot {C_{2}}^{2} - ..... + (n + 1) {C_{n}}^{2}]$

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

23. $\sum_{r = 0}^{n}$ ${(- 1)}^{r} (\begin{matrix} n \\ r \end{matrix}) [\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + ..... to m terms]$ =

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

24. If $S_{n} = 1 + q + q^{2} + .... + q^{n}, q \neq 1$ and $S_{n} = 1 + \frac{q + 1}{2} + {(\frac{q + 1}{2})}^{2} + .... + {(\frac{q + 1}{2})}^{n}, then$ , $(\begin{matrix} n + 1 \\ 1 \end{matrix}) + (\begin{matrix} n + 1 \\ 2 \end{matrix})$ $S_{1} + (\begin{matrix} n + 1 \\ 3 \end{matrix}) S_{2} + ..... + (\begin{matrix} n + 1 \\ n + 1 \end{matrix}) S_{n}$ =

a)		$2^{n} \cdot S_{n}$
b)		$2^{n - 1} \cdot S_{n}$
c)		$2^{n} \cdot S_{n + 1}$
d)		$2^{n} \cdot S_{n - 1}$

25. If ${(1 + x)}^{n}$ = $\sum_{r = 0}^{n} {C_{r} x}^{r}$ , then the sum of the products of the $C_{r}$ taken two at a time represented by $\sum_{0 \leq i \leq j \leq n}$ $C_{i}$ $C_{i}$ is

a)		$2^{2 n - 1} - (\begin{matrix} 2 n - 1 \\ n \end{matrix})$
b)		$2^{2 n - 1} - (\begin{matrix} 2 n \\ n \end{matrix})$
c)		$2^{2 n} - (\begin{matrix} 2 n \\ n \end{matrix})$
d)		$2^{2 n} - (\begin{matrix} 2 n \\ n - 1 \end{matrix})$

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

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

27. If ${(1 + ax)}^{n}$ =1+8x+24 $x^{2} + ...,$ then a+n=

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

28. The sum of all the coefficients of the polynomial ${(1 + x - {3 x}^{2})}^{2}$ is

a)		0
b)		1
c)		-1
d)		3

29. If a= $99^{50}$ + $100^{50}$ and b= $101^{50}$ , then

a)		a=b
b)		a >b
c)		a <b
d)		none of these

30. The greatest integer m such that $5^{m}$ divides $7^{2 n} + 2^{3 n - 3} \cdot 3^{n - 1}$ for $n \in N, is$

a)		0
b)		1
c)		2
d)		3

31. $(\begin{matrix} 47 \\ 4 \end{matrix}) + \sum_{i = 1}^{5} (\begin{matrix} 52 - i \\ 3 \end{matrix})$ =

a)		$(\begin{matrix} 50 \\ 4 \end{matrix})$
b)		$(\begin{matrix} 51 \\ 3 \end{matrix})$
c)		$(\begin{matrix} 52 \\ 3 \end{matrix})$
d)		$(\begin{matrix} 52 \\ 4 \end{matrix})$

32. Given the positive integers $r > 1, n > 2$ and the coefficients of ${3 r}^{th}$ and ${r + 2}^{th}$ terms in the expansion of ${(1 + x)}^{2 n}$ are equal, then n=

a)		2r
b)		2r-1
c)		2r+1
d)		3r

33. If $C_{r} = (\begin{matrix} 2 n \\ r \end{matrix}),$ then $1 \cdot {C_{1}}^{2} - 2 \cdot {C_{2}}^{2} + 3 \cdot {C_{3}}^{2} - .......... - 2 n \cdot {C_{2 n}}^{2}$ =

a)		${(- 1)}^{n} (\begin{matrix} 2 n \\ n \end{matrix})$
b)		${(- 1)}^{n - 1} (\begin{matrix} 2 n - 1 \\ n \end{matrix})$
c)		${(- 1)}^{n} (\begin{matrix} 2 n \\ n - 1 \end{matrix})$
d)		${(- 1)}^{n - 1} \frac{(2 n)!}{n! (n - 1)!}$

34. If $C_{r} = (\begin{matrix} n \\ r \end{matrix}),$ then $\sum_{r = 1}^{n} r^{2} \cdot C_{r} =$

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

35. If $(\begin{matrix} n \\ r - 1 \end{matrix}) = 36, (\begin{matrix} n \\ r \end{matrix}) = 84, (\begin{matrix} n \\ r + 1 \end{matrix}) = 126$ then r=

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

36. ${(\begin{matrix} 2 n \\ 0 \end{matrix})}^{2} - {(\begin{matrix} 2 n \\ 1 \end{matrix})}^{2} + {(\begin{matrix} 2 n \\ 2 \end{matrix})}^{2} - ...... + {(\begin{matrix} 2 n \\ 2 n \end{matrix})}^{2} =$

a)		$(\begin{matrix} 2 n \\ n \end{matrix})$
b)		$(\begin{matrix} 2 n \\ n - 1 \end{matrix})$
c)		${(- 1)}^{n - 1}$ $(\begin{matrix} 2 n \\ n - 1 \end{matrix})$
d)		${(- 1)}^{n}$ $(\begin{matrix} 2 n \\ n \end{matrix})$

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

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

38. $C_{r} = (\begin{matrix} n \\ r \end{matrix}),$ then $\sum_{r = 0}^{n} C_{r} \cdot \frac{2^{r + 2}}{(r + 1) (r + 2)} =$

a)		$\frac{3^{n + 2} - 2 n - 1}{n (n + 1)}$
b)		$\frac{3^{n + 2} - 2 n - 3}{(n + 1) (n + 2)}$
c)		$\frac{3^{n + 2} - 2 n - 5}{(n + 1) (n + 2)}$
d)		none of these

39. If $n \in N,$ then the highest integer m such that $2^{m}$ divides $3^{2 n + 2} - 8 n - 9$ is

a)		3
b)		4
c)		6
d)		8

40. The least positive integer n such that $(\begin{matrix} n - 1 \\ 3 \end{matrix})$ + $(\begin{matrix} n - 1 \\ 4 \end{matrix}) > (\begin{matrix} n \\ 3 \end{matrix})$ is

a)		6
b)		7
c)		8
d)		9

41. If $A_{1}, A_{2}, A_{3}, A_{4}$ are any four consecutive binomial coefficients in the expansion of ${(1 + x)}^{n},$ then $\frac{A_{1}}{A_{1} + A_{2}} + \frac{A_{3}}{A_{3} + A_{4}} + \frac{{2 A}_{2}}{A_{2} + A_{3}}$ =

a)		n-2
b)		1
c)		-1
d)		0

42. If a is a constant and $n > 1$ , an integer, then $\sum_{n = 0}^{n} {(- 1)}^{r} (a - r) (\begin{matrix} n \\ r \end{matrix})$ =

a)		0
b)		n
c)		-a
d)		n-a

43. $If C_{r} = (\begin{matrix} n \\ r \end{matrix})$ , then $\sum_{r = 0}^{n} {(r + 1) C}_{r} =$

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

44. If $C_{r} = (\begin{matrix} 10 \\ r \end{matrix})$ , then $\sum_{r = 0}^{10} C_{r} \cdot \frac{2^{r + 1}}{r + 1}$ =

a)		$\frac{3^{11}}{11}$
b)		$\frac{2^{11}}{11}$
c)		$\frac{3^{11} - 1}{11}$
d)		$\frac{2^{11} - 1}{11}$

45. If $x = {(\sqrt{2} + 1)}^{6}$ , then the integer part $[x]$ is

a)		98
b)		197
c)		196
d)		198

46. Let 2n $\in N$ . If the coefficients of $2^{nd}, 3^{rd}, 4^{th}$ terms in the expression of ${(1 + x)}^{2 n}$ are equal, then n=

a)		14
b)		7
c)		$\frac{7}{2}$
d)		none of these

47. If $\sum_{r = 0}^{n} C_{r} x^{r} = {(1 + x)}^{n}, then$ $\sum_{r = 1}^{n} {r \cdot C}_{r} =$

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

48. If $(\begin{matrix} 15 \\ 3 r \end{matrix}) = (\begin{matrix} 15 \\ r + 3 \end{matrix})$ , then r=

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

49. If the coefficients of $x^{7}$ in ${(a x^{2} + \frac{1}{bx})}^{11} and$ $x^{- 7}$ in ${(ax - \frac{1}{b x^{2}})}^{11}$ are equal, then

a)		a+b=1
b)		a - b=1
c)		ab=-1
d)		ab=1

50. If ${(1 + x)}^{15} = \sum_{r = 0}^{15} C_{r} x^{r}, then \sum_{r = 2}^{15} {(r - 1) C}_{r} =$

a)		11 $\cdot 2^{14} - 1$
b)		13 $\cdot 2^{14} - 1$
c)		13 $\cdot 2^{14} + 1$
d)		11 $\cdot 2^{14} + 1$