# Quizzes : Quadratic Equations & Expressions

QUIZ-II

31. The number of real solutions of the equation $\large sine^x = 5^x + 5^{-x}$  is

(A) 0

(B) 1

(C) 2

(D) infinitely many

Ans: (A)

32. The equation (x – 3)9 + (x – 32)9 + …..(x – 39)9 = 0 has

(A) all real equation

(B) real root namely 3, 32, …..39

(C) one real & of imaginary roots

(D) None of these

Ans: (C)

33. The set of all real numbers x for which x2 – |x + 2| + x > 0 is

(A) (–∞, –2)∪ (2, ∞)

(B) (–∞, –√2)∪ (√2, ∞)

(C) (–∞, –1) ∪ (1, ∞)

(D) (√2, ∞)

Ans: (B)

34. If f(x) = x2 + 2bx + 2c2 and g(x) = -x2 – 2cx + b2, such that minimum f(x) > maximum g(x), then the relation b and c, is

(A) no real value of b and c

(B) 0 < c < b√2

(C) |c| > |b| √2

(D) |c| < |b|√2

Ans: (C)

35. If one of the roots of the equation 2x2 – 6x + k = 0 is (α + 5i ) / 2 , then the values of α and k are

(A) α = 3 , k = 8

(B) α = 3/2 , k = 17

(C) α = –3 , k = –17

(D) α = 3 , k = 17

Ans: (D)

36. The set of all x in the interval [0 , π] for which 2sin2 x – 3 sin x + 1 ≥ o is

(A) {π/2}

(B) φ

(C) [0 , π/4]

(D) [0 , π/6 ] ∪ [5π/6 , 0] ∪ {π/2}

Ans: (D)

37. Let a > 0, b > 0 then both roots of the equation ax2 + bx + c = 0

(A) are real and negative

(B) have negative real parts

(C) have positive real parts

(D) none of these

Ans: (B)

38. The number of positive integral solutions of x4 – y4 = 3789108 is

(A) 0

(B) 1

(C) 2

(D) 4

Ans: (A)

39. The set of values for which x3 + 1 ≥ x2 + x is

(A) x ≤ 0

(B) x ≥ 0

(C) x ≥ –1

(D) –1 ≤ x ≤ 1

Ans: (C)

40. If a ≤ 0, then the roots of x2 – 2a|x – a| – 3a2 = 0 is

(A) (-1+√6)a , a(1 –√2 )

(B) (√6-1) , (√2 – 1)

(C) a

(D) none of these

Ans: (A)

41. If the roots of the equation x2 – 2ax + a2 + a -3 = 0 are less than 3 then

(A) a < 2

(B) 2 ≤ a ≤ 3

(C) 3 < a ≤ 4

(D) a > 4

Ans: (A)

42. If a1 , a2 , a3 (a1 > 0) are in G. P. with common ratio r, then the value of r, for which the inequality 9a1 + 5 a3 > 14 a2 holds, can not lie in the interval

(A) [1, ∞)

(B) [1, 9/5]

(C) [4/5, 1]

(D) [5/ 9, 1]

Ans: (B)

43. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having 3√7 + 3√(49) as one of its roots. Then the product of all the roots of p(x) = 0 is

(A) 7

(B) 49

(C) 56

(D) 63

Ans: (C)

44. If α and β are the roots of x2 – 3px + p2 = 0 such that α2 + β2 = 7/4 then values of p are

(A) 2, 1

(B) 2, 1/2

(C) 1/2 , 1

(D) 1/2 , –1/2

Ans: (D)

45. For all ‘x’, x2 + 2ax + 10 – 3a > 0, then the interval in which ‘a’ lies is

(A) a < – 5

(B) – 5 < a < 2

(C) a > 5

(D) 2 < a < 5

Ans: (B)

46. If (m2 -3) x2 + 3mx + 3m + 1= 0 has roots which are reciprocals of each other, then the value of m equals to

(A) 4

(B) 1

(C) 2

(D) None of these

Ans: (A)

47. If the two equation x2 – cx + d = 0 and x2 – ax + b = 0 have one common root and the second has equal roots then 2(b + d) is equal to

(A) 0

(B) a + c

(C) ac

(D) –ac

Ans: (C)

48. If the inequality $\large \frac{mx^2 + 3x+4}{x^2 + 2x + 2} < 5$  is satisfied for all x ∈ R , then

(A) 1 < m < 5

(B) -1 < m < 5

(C) 1< m < 6

(D) m < 71/24

Ans: (D)

49. If the roots of the equation $\large \frac{a}{x+a + k} + \frac{b}{x+b + k} = 2$  are equal in magnitude but opposite in sign, then the value of k is

(A) – (a+b)/4

(B) (a+b)/4

(C) (a+b)/2

(D) 0

Ans: (A)

50. The number of solutions of the equation n-|x| |m – |x|| = 1
(where m, n > 1 and n > m) is

(A) 0

(B) 1

(C) 2

(D) 4