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Question Bank No: 1

1. If $a_{1}$ , $a_{2}$ ......, $a_{n}$ are positive numbers such that $a_{1} . a_{2} .$ ....... $a_{n}$ = 1, then their sum is

a)		a positive integer
b)		divisible by n
c)		never less than n
d)		None of these

2. If 0 $<$ $θ$ $<$ $\frac{π}{2}$ , then the expression $\sqrt{x^{2} + x}$ + $\frac{\tan^{2} θ}{\sqrt{x^{2} + x}}$ is always greater than or equal to

a)		2 tan $θ$
b)		2
c)		1
d)		$\sec^{2}$ $θ$

3. Solution of x $(\log_{10} x)^{2}$ - 3 $\log_{10}$ x + 1 $>$ 1000 for x $\in$ R is

a)		(10, $\infty$ )
b)		(100, $\infty$ )
c)		(1000, $\infty$ )
d)		(1, $\infty$ )

4. Solution of (5x - 1) $<$ (x + 1)2 $<$ (7x - 3) is

a)		(1, 4)
b)		[2, 4]
c)		(2, 4)
d)		( $- \infty$ , 1) $\cup$ (2, $\infty$ )

5. For positive numbers a, b, c the least value of (a + b + c) $(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$ is

a)		3
b)		9
c)		$\frac{27}{4}$
d)		None of these

6. If C is an obtuse angle in a triangle, then

a)		tan A tan B $<$ 1
b)		tan A tan B $>$ 1
c)		tan A tan B = 1
d)		None of these

7. If the product of n positive numbers is 1, then their sum is

a)		A positive integer
b)		divisible by n
c)		equal to n + $\frac{1}{n}$
d)		never less than n

8. If a, b, c are different positive real numbers such that b + c - a, c + a - b and a + b - c are positive, then (b + c - a) (c + a - b) (a + b - c) - abc is

a)		Positive
b)		negative
c)		non-positive
d)		non-negative

9. Minimum value of $\frac{b + c}{a}$ + $\frac{c + a}{b}$ + $\frac{a + b}{c}$ , (for real positive numbers a, b, c) is

a)		1
b)		2
c)		4
d)		6

10. If $x^{2}$ + 6x - 27 $>$ 0 and $x^{2}$ - 3x - 4 $<$ 0, then

a)		x $>$ 3
b)		x $<$ 4
c)		3 $<$ x $<$ 4
d)		x = $\frac{7}{2}$

11. If x $>$ 0, $λ$ $>$ 0 and $λ$ x + 1x-1 is always non-negative, then the least value of $λ$ is

a)		1
b)		$\frac{1}{2}$
c)		0
d)		$\frac{1}{4}$

12. If x is real, the expression $\frac{x^{2} + 2 x - 11}{x - 3}$ takes all real values except those which lie between a and b, then a and b are

a)		-12, -4
b)		-12, 12
c)		4, 12
d)		-4, 4

13. Solution of $| x - 1 | + | x - 2 | + | x - 3 | \geq 6$ is

a)		[0, 4]
b)		(- $\infty$ , -2) $\cup$ (2, 3)
c)		(- $\infty$ , 0] $\cup$ [4, $\infty$ )
d)		None of these

14. Solution of $| x + \frac{1}{x} |$ $>$ 2 is

a)		R $-$ ${0}$
b)		R $-$ {-1, 0, 1}
c)		R $-$ ${1}$
d)		R $-$ ${- 1, 1}$

15. Solution of 2x – 1 = $| x + 7 |$ is

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

16. Solution of 1+ $\frac{3}{x}$ $>$ 2 is

a)		(0, 3]
b)		[-1, 0)
c)		(-1, 0) $\cup$ (0, 3)
d)		None of these

17. Solution of $| 3 x + 2 |$ $\geq$ 1 is

a)		$[\frac{1}{3}, 1]$
b)		$(\frac{1}{3}, 1)$
c)		${\frac{1}{3}, 1}$
d)		( $- \infty$ , $\frac{1}{3}]$ $\cup$ [1, $\infty$ )

18. Solution of $| 3 x + 2 |$ $<$ 1 is

a)		$[- 1, - \frac{1}{3}]$
b)		${- \frac{1}{3}, - 1}$
c)		$(- 1, - \frac{1}{3})$
d)		None of these

19. Solution of $\frac{x - 7}{x + 3}$ $>$ 2 is

a)		( $-$ 3, $\infty$ )
b)		( $-$ $\infty$ , $-$ 13)
c)		( $-$ 13, $-$ 3)
d)		None of these

20. $\frac{x + 4}{x - 3}$ $<$ 2 is satisfied when x satisfies

a)		( $-$ $\infty$ , 3) $\cup$ (10, $\infty$ )
b)		(3, 10)
c)		( $- \infty$ , 3) $\cup$ [10, $\infty$ )
d)		None of these