Untitled Document

Question Bank No: 1

1. If the equation ${ax}^{2}$ + 2bx - 3z = 0 has no real root and $\frac{3 c}{4}$ < a + b, then

a)		c < 0
b)		c > 0
c)		c $\geq$ 0
d)		c = 0

2. If a and b are the non-zero distinct roots of $x^{2}$ + ax + b = 0, then the least value of $x^{2}$ + ax + b is

a)		2/3
b)		9/4
c)		- 9/4
d)		1

3. The quadratic equation 8 $\sec^{2}$ x - 6 sec x + 1 = 0 has

a)		infinitely many roots
b)		exactly two roots
c)		exactly four roots
d)		no root

4. If $α$ = cos $\frac{2 π}{7}$ +i sin $\frac{2 π}{7}$ , p = $α$ + $α^{2}$ + $α^{4}$ and q = $α^{3}$ + $α^{5}$ + $α^{6}$ , then the equation whose roots are p, q is

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

5. If the roots of $x^{2}$ + 2ax + b = 0 differ from the roots $x^{2}$ + 2bx + a = 0 by a constant then a + b is equal to

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

6. 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 = 3.5

7. If p and q are the roots of the equation $x^{2}$ + pq = (p + 1)x, then q is equal to

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

8. The value of k (k > 0) for which the equation $x^{2}$ + kx + 64 = 0 and $x^{2}$ - 8x + k = 0 both will have real roots is

a)		8
b)		- 16
c)		- 64
d)		16

9. If ${ax}^{2}$ + bx + c = a(x - $α$ ) (x - $β$ ), then a(αx + 1) ( $β$ x + 1) is equal to

a)		${ax}^{2}$ + bx + c
b)		${cx}^{2}$ - bx + a
c)		${cx}^{2}$ - bx - a
d)		${cx}^{2}$ + bx + a

10. For the equation ${| x |}^{2} +$ $| x |$ - 6 = 0

a)		there is only one root
b)		there are only two distinct roots
c)		there are only three distinct roots
d)		there are four distinct roots

11. The number of roots of the equation x - $\frac{2}{x - 1}$ =1- $\frac{2}{x - 1}$ is

a)		1
b)		2
c)		0
d)		infinitely many

12. The number of real roots of the equation ${2 x}^{4} + {5 x}^{2}$ + 3 = 0 is

a)		4
b)		0
c)		2
d)		3

13. If $x^{2}$ - 3x + 2 is a factor of $x^{4} - {px}^{2}$ + q, then p,q are

a)		2, 3
b)		4, 5
c)		5, 4
d)		0, 0

14. If the sum of the roots of ${ax}^{2}$ + bx + c = 0 be equal to the sum of their squares, then

a)		2ac = ab + $b^{2}$
b)		2ab = bc + $c^{2}$
c)		2bc = ac + $c^{2}$
d)		None of these

15. The roots of the equation ${(b - c) x}^{2}$ + (c - a)x + (a - b) = 0 are

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

16. If the quardatic equations ${ax}^{2}$ + 2cx + b = 0 and ${ax}^{2}$ + 2bx + c = 0 (b $\neq$ c) have a common root, then a + 4b + 4c is equal to

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

17. The number of real roots of ${(6 - x)}^{4}$ + ${(8 - x)}^{4}$ = 16 is

a)		0
b)		2
c)		5
d)		4

18. The value of ‘a’ for which the sum of the squares of the roots of the equation $x^{2}$ - (a – 2) x - a - 1 = 0 assumes the least value is

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

19. The equation of the smallest degree with real coefficients having 2 + 3i as one of the roots is

a)		$x^{2}$ + 4x + 13 = 0
b)		$x^{2}$ - 4x + 13 = 0
c)		$x^{2}$ + 4x - 13 = 0
d)		$x^{2}$ - 4x - 13 = 0

20. The number of real roots of teh equation ${| x |}^{2} - 3 | x |$ +2=0 is

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

21. The smallest value of $x^{2} - 3 x + 3 in the interval (- 3, \frac{3}{2}) is$

a)		$\frac{3}{4}$
b)		5
c)		-15
d)		-20

22. Root of the equation $3^{x - 1} + 3^{1 - x} = 2$ is

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

23. For what value of P the difference of the roots of the equation $x^{2} - Px + 8 = 0$ is 2

a)		$\pm 2$
b)		$\pm 4$
c)		$\pm 6$
d)		$\pm 8$

24. One root of the equation ${5 x}^{2} + 13 x + K = 0$ is the reciprocal of the other, if

a)		K=0
b)		K=5
c)		K= $\frac{1}{6}$
d)		6

25. The values of x which satisfy both the equations $x^{2} - 1 \leq 0$ and $x^{2} - x - 2 \geq 0$ lie in

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

26. If the roots of the equation $\frac{a}{x - a} + \frac{b}{x - b} = 1$ are equal in magnitude and opposite in sign, then

a)		a-b=0
b)		a+b=1
c)		a-b=1
d)		a+b=0

27. If $α, β$ are the roots of the equation $x^{2} - 2 x + 2 = 0$ , then the value of $α^{2} + β^{2}$ is

a)		2
b)		0
c)		1
d)		4

28. If one root of the equation i $x^{2} - 2 (i + 1) x + (2 - i) = 0$ is 2-i, the other root is

a)		-I
b)		2+I
c)		I
d)		2-I

29. The value or values of P for which the equation 2 $x^{2} - \sqrt{2}$ Px+P=0 has equal roots is or are

a)		0
b)		4
c)		0,4
d)		none of these

30. If 3 is a root of $x^{2} + Kx - 24 = 0,$ it is also a root of

a)		$x^{2} + 5 x + K = 0$
b)		$x^{2} - 5 x + K = 0$
c)		$x^{2} - Kx + 6 = 0$
d)		$x^{2} + Kx + 24 = 0$

31. If the difference between the roots of the equation $x^{2} + ax + 1 = 0$ is less than $\sqrt{5}$ , then $a \in$

a)		$(- 3, \infty)$
b)		$(3, \infty)$
c)		$(- \infty, - 3)$
d)		$(- 3, 3)$

32. Let $α, β$ be the roots of the equation $x^{2} - px + r = 0$ and $\frac{α}{2}$ , 2 $β$ be the roots of the equation $x^{2} - qx + r = 0$ . Then $r =$

a)		$\frac{2}{9} (p - q) (2 q - p)$
b)		$\frac{2}{9} (q - p) (2 p - q)$
c)		$\frac{2}{9} (q - 2 p) (2 q - p)$
d)		$\frac{2}{9} (2 p - q) (2 q - p)$

33. If x is real, the maximum value of $\frac{3 x^{2} + 9 x + 17}{3 x^{2} + 9 x + 7}$ is

a)		1
b)		$\frac{17}{7}$
c)		$\frac{1}{4}$
d)		41

34. If the roots of $x^{2} + px + q = 0$ are $\tan 30^{\circ}$ and $\tan 15^{\circ}$ , then $2 + q - p =$

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

35. If both the roots of the equation $x^{2} - 2 mx + m^{2} - 1 = 0$ lie between $- 2 and 4,$ then $m \in$

a)		$(- 1, 3)$
b)		$(1, 4)$
c)		$(- 2, 0)$
d)		$(3, \infty)$

36. If both roots of the equation $x^{2} - 2 kx + k^{2} + k - 5 = 0$ are less than 5, then $k \in$

a)		$(6, \infty)$
b)		$(5, 6]$
c)		$[4, 5]$
d)		$(- \infty, 4]$

37. If the roots of the equation $x^{2} - bx + c = 0$ be two consecutive integers, then $b^{2} - 4 c =$

a)		3
b)		$- 2$
c)		1
d)		2

38. The value of $a$ for which the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - a - 1 = 0$ assumes the least value is

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

39. If one root of the equation $x^{2} + px + 2 = 0 is 4,$ while the equation $x^{2} + px + q = 0$ has equal roots, then $q =$

a)		$\frac{49}{4}$
b)		4
c)		3
d)		12

40. If $1 - p$ is a root of $x^{2} + px + 1 - p = 0,$ then its roots are

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

41. If one root of ${(a}^{2} - 5 a + x) x^{2} + (3 a - 1) x + 2 = 0$ is twice the other, then a=

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

42. The product of real roots of the equation $t^{2} x^{2} + | x | + 9 = 0$

a)		$> 0$
b)		$< 0$
c)		does not exist
d)		irrelevant

43. If $2 a + 3 b + 6 c = 0, a, b, c \in R,$ then the equation $a x^{2} + bx + c = 0$ has a root in

a)		(0, 1)
b)		(2, 3)
c)		(4, 5)
d)		none of these

44. If the difference between the roots of $x^{2} + ax + b = 0$ is same as that of $x^{2} + bx + a = 0, a \neq b,$ then

a)		a +b+4=0
b)		a+b-4=0
c)		a- b- 4=0
d)		a- b+4=0

45. If $α \neq β$ but $α^{2} = 5 α - 3$ and $β^{2} = 5 β - 3$ , then the equation with roots $\frac{α}{β}$ , $\frac{β}{α}$ is

a)		3 $x^{2} - 25 x + 3 = 0$
b)		$x^{2} + 5 x - 3 = 0$
c)		$x^{2} - 5 x + 3 = 0$
d)		3 $x^{2} - 19 x + 3 = 0$

46. Let a, b, c be the sides of a scalene triangle. If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (ab + bc + ca) = 0$ $λ \in R,$ are real, then

a)		$λ < \frac{4}{3}$
b)		$λ > \frac{5}{3}$
c)		$λ \in (\frac{1}{3}, \frac{5}{3})$
d)		$λ \in (\frac{4}{3}, \frac{5}{3})$

47. Let $α, β$ be the roots of ${ax}^{2} + bx + c = 0. If α + β, α^{2} + β^{2}$ , $α^{3} + β^{3}$ are in G.P and $Δ = b^{2} - 4 ac,$ then

a)		$Δ \neq 0$
b)		$b Δ = 0$
c)		$Δ = 0$
d)		$c Δ = 0$

48. If the minimum value $f (x) = x^{2} + 2 bx + 2 c^{2}$ is greater than the maximum value of $g (x) = - x^{2} - 2 cx + b^{2}$ , then

a)		$\| b \| < 2 \| c \|$
b)		$\| c \| < 2 \| b \|$
c)		$\| c \| > \sqrt{2} \| b \|$
d)		$\| b \| > \sqrt{2} \| c \|$

49. If $α^{2} + (a - b) x + 1 - a - b = 0, a, b \in R$ has unequal real roots foe all values of b then

a)		$- 1 < a < 1$
b)		$a > 1$
c)		$a < - 1$
d)		$0 < a < 2$

50. If one root is square of the root of the equation $x^{2} + px + q = 0,$ then

a)		$p^{3} - (3 p - 1) q + q^{2} = 0$
b)		$p^{3} - (3 p + 1) q + q^{2} = 0$
c)		$p^{3} + (3 p - 1) q + q^{2} = 0$
d)		$p^{3} + (3 p + 1) q + q^{2} = 0$