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Question Bank No: 2

1. The natural numbers are arranged in rows as follows :
$\begin{matrix} 1 \\ 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 & 10 \end{matrix}$
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The sum of the numbers in the 20th row is

a)		4010
b)		4020
c)		4120
d)		4210

2. The A.M of the roots of a quadratic equation is A and G.M of its roots is G. The quadratic equation is

a)		$x^{2}$ + Ax + $G^{2}$ = 0
b)		$x^{2}$ + 2Ax + $G^{2}$ = 0
c)		$x^{2}$ - Ax + $G^{2}$ = 0
d)		$x^{2}$ - 2Ax + $G^{2}$ = 0

3. If a, b, c are in G.P and $a^{x}$ = $b^{y}$ = $c^{z}$ , then x, y, z are in

a)		A.P
b)		G.P
c)		H.P
d)		None of these

4. Let S be the sum, P, the product and R, the sum of reciprocal of n terms of a G.P. Then

a)		R = S $\cdot$ $P^{\frac{1}{n}}$
b)		R = S $\cdot$ $P^{\frac{2}{n}}$
c)		S = R $\cdot$ $P^{\frac{1}{n}}$
d)		S = R $\cdot$ $P^{\frac{2}{n}}$

5. The sum of the first p terms of an A.P is q and the sum of the first q terms is p. The sum of first (p+q) terms is

a)		0
b)		(p+q)
c)		- (p+q)
d)		-2 (p+q)

6. Let 1 be the A.M and 2 be the G.M of two numbers a and b. Then their H.M is

a)		3
b)		4
c)		$\frac{10}{3}$
d)		$\frac{16}{5}$

7. Let $| a |$ $<$ 1, $| b |$ $<$ 1 and consider the infinite ${G . P .}^{s}$ , x= 1+a+ $a^{2}$ +...., y= 1+b+ $b^{2}$ +...., and z=1+ab+ ${(ab)}^{2}$ +...., then z=

a)		$\frac{x + y}{xy - 1}$
b)		$\frac{x y}{x + y + 1}$
c)		$\frac{x y}{x + y - 1}$
d)		$\frac{x + y}{xy + 1}$

8. The natural numbers are divided into rows as follows :
$\begin{matrix} 1 \\ 2 & 3 & 4 \\ 5 & 6 & 7 & 8 & 9 \end{matrix}$
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The sum of the numbers in the 10th row is

a)		1720
b)		1721
c)		1723
d)		1729

9. If the $r^{th}$ term of the series is 1+x+ $x^{2}$ +.... + $x^{r - 1}$ , then the sum of the first n terms is

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

10. If p, q, r and s are integers. If the A.M of the roots of $x^{2}$ - px + $q^{2}$ = 0 and G.M. of the roots $x^{2}$ - rx + $s^{2}$ = 0 are equal, then

a)		q is an odd interger
b)		r is an even integer
c)		p is an even integer
d)		s is an odd integer

11. If a < b, then

a)		$\frac{a + b}{2}$ < b
b)		$\frac{a + b}{2}$ > b
c)		$\frac{a + b}{2}$ < a
d)		$\frac{a + b}{2}$ > a

12. Numbers of even prime natural numbers;

a)		1
b)		2
c)		infinite
d)		6

13. If x = 2.352, the x is equal to

a)		$\frac{2352}{999}$
b)		$\frac{2329}{999}$
c)		$\frac{2352}{990}$
d)		$\frac{2352}{990}$

14. The smallest positive integer which can be expressed as sum of two different cubes in two different ways is

a)		1000
b)		729
c)		1729
d)		1728

15. If m is a root of the equation (1 - ab) $x^{2}$ - ( $a^{2}$ + $b^{2})$ x- (1 + ab) = 0 and m harmonic means are inserted between a and b, then the difference between the last and the first of the means equals

a)		b - a
b)		ab (b - a)
c)		a (b - a)
d)		ab (a - b)

16. The 10th common terms between the series 3 + 7 + 11 + ----and 1 + 6 + 11 + ----is

a)		191
b)		193
c)		211
d)		none of these

17. Non zero equal numbers are in

a)		A.P. only
b)		A.P & G.P only
c)		A.P, G.P & H.P
d)		A.P & H.P

18. The next term of the sequence 1, 2, 4, 7, 11, ----is

a)		15
b)		16
c)		17
d)		18

19. If the third term of an A.P is 12 and the seventh term is 24, then the $10^{th}$ term is

a)		36
b)		39
c)		30
d)		33

20. An A.P. consists of n (odd terms) and its middle term is m. Then the sum of the A.P.

a)		2 mn
b)		1/2 mn
c)		mn
d)		${mn}^{2}$

21. 5th term of a G.P. is 2, then the product of first 9 terms is

a)		256
b)		128
c)		512
d)		none of thes

22. The third term of a G.P is 4. The product of first five terms is

a)		$4^{3}$
b)		$4^{5}$
c)		$4^{2}$
d)		none of these

23. If a, b, c are in A.P as well as in G.P, then

a)		a = b $\neq$ c
b)		a $\neq$ b = c
c)		a $\neq$ b $\neq$ c
d)		a = b = c

24. If a > 0, b > 0, c > 0 are in G.P., then log a, log b, log c are in

a)		H.P
b)		A.P
c)		G.P
d)		none of these

25. If a, b, c are in G.P., then 1/a, 1/b, 1/c are in

a)		A.P
b)		G.P
c)		H.P
d)		none of these

26. The A.M, G.M and H.M between two positive numbers a and b are equal to

a)		a = b
b)		ab = 1
c)		a > b
d)		a < b

27. The A.M., of two integers is 34 and G.M. is 16, the numbers are

a)		2 and 64
b)		64 and 3
c)		64 and 4
d)		none of these

28. The A.M., H.M. and G.M. between two numbers are $\frac{114}{15}$ 15 and 12, but 15 not necessarily in this order. Then H.M., G.M. and A.M respectively are

a)		15, 12, $\frac{144}{15}$
b)		12, 15, $\frac{144}{15}$
c)		$\frac{144}{15},$ 12, 15
d)		$\frac{144}{15}$ , 15, 12

29. In an A.P of 81 terms, the $41^{st}$ term is 10. Then the sum of the series is

a)		41 x 81
b)		10 x 81
c)		10 x 41
d)		10 x 41

30. The value of n, for which $\frac{a^{n + 1} + b^{n + 1}}{a^{n} + b^{n}}$ is the A.M. between a an b is

a)		0
b)		1
c)		- $\frac{1}{2}$
d)		-1

31. If x, 2x + 2, 3x + 3 are in G.P., then the fourth term is

a)		27
b)		- 27
c)		27/2
d)		- 27/2

32. 3 + 3a + ${3 a}^{2}$ + ---- $\infty$ is equal to 45/8, a > 0 then a is

a)		15/23
b)		7/15
c)		7/8
d)		15/7

33. If 3 + 5 + 7 + ----.n terms = 7, the value of n is 5 + 8 + 11 + ----10 terms

a)		35
b)		36
c)		37
d)		40

34. The nth term of the series 3, $\sqrt{3}$ , 1----- is 1/243, then n is

a)		2
b)		1
c)		14
d)		15

35. 9th term of the series 2 + 7 + 14 + 23 + 34 + ----- is

a)		9998
b)		9999
c)		10000
d)		none of these

36. If P, Q, R be the A.M., G.M., H.M. respectively between any two rational numbers a and b, the P - Q is

a)		$\frac{a - b}{a}$
b)		$\frac{a + b}{2}$
c)		$\frac{2 ab}{2 + b}$
d)		$\frac{\sqrt{a} - \sqrt{b})^{2}}{\sqrt{2}}$

37. Given that n A.M.'s are inserted between two sets of numbers a, 2b and 2 a,b, where a, b $ϵ$ R. Suppose further that mth mean between these sets of numbers is same, then the ratio, a : b equals

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

38. $11^{3}$ + $12^{3}$ + $13^{3}$ + ---- + $20^{3}$ is

a)		an even integer
b)		an odd integer divisible by 5
c)		multiple of 10
d)		an odd integer but not a multiple of 5

39. pth term of an H.P. is qr and qth term is pr then the rth term of the H.P

a)		pqr
b)		1
c)		pq
d)		${pqr}^{2}$

40. The 10th term of the sequence $\sqrt{3}$ $\sqrt{12}$ , $\sqrt{27}$ ,---is

a)		$\sqrt{243}$
b)		$\sqrt{300}$
c)		$\sqrt{363}$
d)		$\sqrt{432}$

41. If a, b, c, d are in H.P. then ab + bc + cd is

a)		3 ad
b)		(a + b) (c + d)
c)		3 ac
d)		3 bd

42. The product of the terms equidistant from the beginning and end in a G.P (finite) is equal to

a)		last term
b)		first term
c)		product of the first and last term
d)		none

43. The sum of the first 10 terms of a G.P. is equal to 244 times the sum of first five terms. Then the common ratio is

a)		7
b)		3
c)		4
d)		5