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Question Bank No: 2

1. If $x^{2} + 2 ax + 10 - 3 a > 0$ for all x then

a)		$a < - 5$
b)		$- 5 < a < 2$
c)		$a > 5$
d)		$2 < a < 5$

2. The equation $(\cos β - 1) x^{2} + (\cos β) x + \sin β = 0$ in the variable x has real roots, then $β$ is in the interval.

a)		$(0, 2 π)$
b)		$(- π, 0)$
c)		$(- \frac{π}{2}, \frac{π}{2})$
d)		$(0, π)$

3. Let $α_{1}$ , $α_{2}$ and $β_{1}$ , $β_{2}$ be the roots of a $x^{2} + bx + c = 0$ and p $x^{2} + qx + r = 0$ respectively. If the system of equations $α_{1} y + α_{2} z = 0, β_{1} y + β_{2} z = 0$ has nontrivial solution, then

a)		abc=pqr
b)		a $b^{2} c = p q^{2} r$
c)		$\frac{ac}{b} = \frac{pr}{q}$
d)		$\frac{ac}{b^{2}} = \frac{pr}{q^{2}}$

4. The set of all real numbers x for which $x^{2} - | x + 2 | + x > 0$ is

a)		$(- \infty, - 2) \cup (2, \infty)$
b)		$(- \infty, - \sqrt{2}) \cup (\sqrt{2}, \infty)$
c)		$(- \infty, - 1) \cup (1, \infty)$
d)		$(\sqrt{2}, \infty)$

5. For real values of x, the function $\frac{\sin x \cos 3 x}{\sin 3 x cosx}$ does not take values

a)		between -1 and 1
b)		between 0 and 2
c)		between $\frac{1}{3}$ and 3
d)		between 0 and $\frac{1}{3}$

6. The sum of all the real roots of the equation ${| x - 2 |}^{2} + | x - 2 | - 2 = 0$ is

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

7. If $α and β (α < β)$ are the roots of the equation $x^{2} + bx + c = 0,$ where $c < 0 < b,$ then

a)		$0 < α < β$
b)		$α < 0 < β < \| α \|$
c)		$α < β < 0$
d)		$α < 0 < \| α \| < β$

8. For the equation 3 $x^{2} + px + 3 = 0, p > 0,$ if one root is square of the other, then p=

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

9. If $b > a,$ then the equation $(x - a) (x - b) - 1 = 0$ has

a)		both roots in $(a, b)$
b)		both roots in $(- \infty, a)$
c)		both roots in $(b, - \infty)$
d)		one root in $(- \infty, a)$ and the other in $(b, \infty)$

10. Let $a, b, c$ be real. If a $x^{2} + bx + c = 0$ has two real roots $α and β,$ where $α < - 1$ and $β > 1,$ then

a)		$\| \frac{b}{a} \| + \frac{c}{a} < - 1$
b)		$\| \frac{b}{a} \| + \frac{c}{a} = - 1$
c)		$\| \frac{b}{a} \| + \frac{c}{a} > - 1$
d)		$\| \frac{b}{a} \| - \frac{c}{a} = - 1$

11. If the roots of the equation $x^{2} - 2 ax + a^{2} + a - 3 = 0$ are real and less than 3, then

a)		a< 2
b)		2 $\leq a \leq 3$
c)		3<a $\leq 4$
d)		a > 4

12. In a triangle $PQR, R = \frac{π}{2}$ . If tan $\frac{P}{2}$ and tan $\frac{Q}{2}$ are the roots of the equation $a x^{2} + bx + c = 0,$ then

a)		a=b+c
b)		b= c+ a
c)		c= a+b
d)		b=c

13. The number of solution of the equation $\sqrt{x + 1} - \sqrt{x - 1} = \sqrt{4 x - 1}$ is

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

14. Let $α, β$ be the roots of the equation $(x - a) (x - b) = c \neq 0$ , then the roots of the equation $(x - α) (x - β) + c = 0$ are

a)		a, c
b)		b, c
c)		a, b
d)		a+c, b +c

15. Let $a, b, c \in R, a \neq 0.$ If $α$ is a root of ${a^{2} x}^{2} + bx + c = 0, β$ is a root of ${a^{2} x}^{2} - bx - c = 0 and α < β,$ then the equation $a^{2}$ $x^{2} + 2 bx + 2 c = 0$ has a root $γ$ such that

a)		$γ = \frac{α + β}{2}$
b)		$γ = α$
c)		$γ = α + \frac{β}{2}$
d)		$α < γ < β$

16. If $α, β$ are the roots of $x^{2} + px + q = 0 and$ $α^{4}, β^{4}$ are the roots of $x^{2} - rx + s = 0,$ then the equation $x^{2} - 4 qx + 2 q^{2} - r = 0$ has

a)		two real roots
b)		two negative roots
c)		four positive roots
d)		one positive and one negtive roots

17. $\frac{2 x}{{2 x}^{2} + 5 x + 2} > \frac{1}{x + 1}$ if

a)		x < 2
b)		$- 1 < x < 1$
c)		$- 2 < x < - 1$
d)		x < -2

18. The sum of the values of x satisfying the equation $| x^{2} + 4 x + 3 | + 2 x + 5 = 0$ is

a)		5+ $\sqrt{3}$
b)		5 $- \sqrt{3}$
c)		$- 5 + \sqrt{3}$
d)		$- 5 - \sqrt{3}$

19. The product of all the value of x satisfying the equation ${(5 + 2 \sqrt{6})}^{x^{2} - 3} + {(5 - 2 \sqrt{6})}^{x^{2} - 3} = 10$ is

a)		4
b)		6
c)		8
d)		19

20. If a, b, c are positive, the minimum value of $(a + b + c)$ $(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$ is

a)		1
b)		3
c)		6
d)		9

21. If a, b, c are the sides of a scalene triangle, then the expression $(b + c - a)$ $(c + a - b) (a + b - c) - abc$ is

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

22. For $a \leq 0$ , the roots of the equation $x^{2} - 2 a | x - a | - 3 a^{2} = 0$ are

a)		a $(1 + \sqrt{2}), a (1 + \sqrt{6})$
b)		a $(\sqrt{2} - 1), a (\sqrt{6} - 1)$
c)		a $(1 - \sqrt{2}), a (\sqrt{6} - 1)$
d)		a $(1 - \sqrt{2}), a (1 - \sqrt{6})$

23. If the quadratic equations $x^{2} + ax + b = 0 and x^{2} + bx + a = 0 (a \neq b)$ have a common root, then a + b =

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

24. If $P (x) = a x^{2} + bx + c$ and $Q (x) = - a x^{2} + dx + c, ac \neq 0$ , then the equation $P (x)$ $\cdot Q (x)$ =0 has

a)		four real roots
b)		exactly two real roots
c)		atleast two real roots
d)		atmost two real roots

25. If a, b, c are in G.P and the equations ${ax}^{2} + 2 bx + c = 0$ and ${dx}^{2} + 2 ex + f = 0$ have a common root, then $\frac{a}{d}, \frac{b}{e}$ , $\frac{c}{b}$ are in

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

26. The values of m for which the system of equations 3x+ my=m, 2x-5y=20 has solution satisfying x >0, y>0 are given by

a)		m < 0
b)		m=5
c)		m > 0
d)		m > 30

27. Both the roots of the equation $(x - b) (x - c) + (x - c) (x - a) + (x - a) (x - b) = 0$ are

a)		positive
b)		negative
c)		real
d)		none of these

28. Let $a > o, b > 0, c > 0.$ Then both the roots of the equation $a x^{2} + bx + c = 0$

a)		are real and negative
b)		have imaginary
c)		have positive real parts
d)		none of these

29. If the product of the roots of the equation $x^{2} - 3 kx + 2 e^{2 \ln k} - 1 = 0$ is 7, then the roots are

a)		integers
b)		rational
c)		irrational
d)		imaginary

30. If $a < b < c < d$ , then the equation (x $- a) (x - c) + 2 (x - b) (x - d) = 0$ has

a)		imaginary roots
b)		equal roots
c)		distinct real roots
d)		none of these

31. The equation $x -$ $\frac{2}{x - 1}$ $= 1 - \frac{2}{x - 1}$ has

a)		no root
b)		one root
c)		two equal roots
d)		infinitely many roots

32. If $x^{2} - 3 x + 2 > 0$ and $x^{2} - 3 x - 4 \leq 0$ , then

a)		$\| x \| \leq - 2$
b)		2 $\leq x \leq 4$
c)		$x < 1$
d)		$2 < x \leq 4$

33. The equation $2 x^{2} + 3 x + 1 = 0$ has

a)		imaginary roots
b)		irrational roots
c)		rational roots
d)		integer root

34. If $a + b + c = 0,$ then the equation 3a $x^{2} + 2 bx + c = 0$ has

a)		imaginary roots
b)		one root in $[- 2, - 1]$ and the other in $[2, 3]$
c)		atleast in $[0, 1]$
d)		none of these

35. If one root of ${ax}^{2} + bx + c = 0$ is equal to $n^{th}$ power of the other, then b=

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

36. If $2 + i \sqrt{3}$ is a root of the equation $x^{2} + px + q = 0,$ where p, q real, then (p,q)=

a)		$(4, - 7)$
b)		$(4, 7)$
c)		$(- 4, 7)$
d)		$(- 4, - 7)$

37. The number of solutions of the equation ${| x |}^{2} - 3 | x | + 2 = 0$ is

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

38. If $y = {(\frac{(x + 1) (x - 3)}{x - 2})}^{\frac{1}{2}}$ takes real values if

a)		$x \leq - 1$
b)		$- 1 < x \leq 2$
c)		$x \geq 3$
d)		$2 \leq x \leq 3$

39. If one root of the equation $(a - b) x^{2} + ax + 1 = 0$ isdouble the other and if a is real, then the greatest value of b is

a)		$\frac{7}{6}$
b)		$\frac{8}{7}$
c)		$\frac{9}{8}$
d)		$\frac{10}{9}$

40. If $α, β$ be the roots of $x^{2} + x + 2 = 0$ and $γ, δ$ be the roots of $x^{2} + 3 x + 4 = 0$ then $(α + γ) (α + δ) (β + γ) (β + δ)$ =

a)		$- 18$
b)		18
c)		24
d)		44

41. If $a > c, b > c, x > c,$ the minimum value of $\frac{(a + x) (b + x)}{c + x}$ is

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

42. The least value of $\frac{6 x^{2} - 18 x + 21}{6 x^{2} - 18 x + 17}$ is

a)		$\frac{7}{15}$
b)		$\frac{15}{7}$
c)		$\frac{7}{3}$
d)		1

43. If a, b are the roots of $x^{2} + px + 1 = 0$ and c, d are the roots of $x^{2} + qx + 1 = 0$ , then $(a - c) (b - c) (a + d) (b + d) =$

a)		2pq
b)		${(p + q)}^{2}$
c)		$p^{2} + q^{2}$
d)		$q^{2} - p^{2}$

44. In the equation $x^{2} + px + q = 0$ the ciefficient of x was incorrectly written as 17 instead of 13. Then the roots were found to be $- 2$ and $-$ 15. The correct root are

a)		$- 1$
b)		$- 8$
c)		$- 5$
d)		$- 10$

45. $\frac{8 x^{2} + 16 x - 51}{(2 x - 3) (x + 4)} > 3$ if

a)		$x < - 4$
b)		$x < \frac{5}{2}$
c)		$- 1 < x <$ 1
d)		$x > 4$

46. If the sum of the roots of the equation $a x^{2} + bx + c = 0$ is equal to the sum of the reciprocals of their squares, then $\frac{a}{c}$ , $\frac{b}{a}, \frac{c}{b}$ are in

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

47. Let $p, q, r, s$ be real numbers such that $pr = 2 (q + s) .$ Consider the quadratic equations : $x^{2} + px + q = 0$ and $x^{2} + rx + s = 0.$ Then

a)		none of these has real roots
b)		both have real roots
c)		atmost one equation has real roots
d)		at least one equation has real roots

48. If $x^{2} + px + q = 0$ and $x^{2} + p^{'} x + q^{'} = 0$ have one common root, then the common root is

a)		$\frac{q - q^{'}}{p^{'} - p}$
b)		$\frac{q - q^{'}}{p - p^{'}}$
c)		$\frac{p q^{'} - p^{'} q}{p - q^{'}}$
d)		$\frac{p q^{'} - p^{'} q}{q^{'} - q}$

49. The expression $a (x^{2} - y^{2}) - bxy$ admits of two linear factors for

a)		a+b=0
b)		a = b
c)		4a = $b^{2}$
d)		all a and b

50. The values of x fro which the inequalities $x^{2} + 6 x - 27 > 0 and - x^{2} + 3 x + 4 > 0$ hold simultaneously lie in

a)		$(- 1, 4)$
b)		$(- \infty, - 9)$ $\cup (3, \infty)$
c)		$(- 1, - 1)$
d)		$(3, 4)$