QUADRATIC EQUATION OBJECTIVE QUESTIONS 1. The set of values of p for which the roots of the equation 3x2 + 2x + p(p – 1) = 0 are of opposite signs is (a) ( – ∞, 0) (b) (0, 1) (c) (1, ∞) (d) (0, ∞). 2. The inequality ⏐2x – 3⏐ < 1 is valid when x lies in the interval (a) (3, 4) (b) (1, 2) (c) – 1, 2) (d) ( – 4, 3). 3. Let α and β be the roots of the equation x2+x+1 = 0. The equation whose roots are α19 , β7 is (a) x2 – x – 1 = 0 (b) x2 – x + 1 = 0 (c) x2 + x – 1 = 0 (d) x2 + x + 1 = 0. 4. If p and q are the roots of the equation x2+px+q = 0, then (a) p = 1 (b) p = 1 or 0 (c) p = – 2 (d) p = – 2 or 0. 5. If p, q, r are positive and are in A.P., the roots of the quadratic equations px2 + qx + r = 0 are real or (a) p = 1 (b) p = 1 or 0 (c) p = – 2 (d) p = – 2 or 0. 6. If p, q, r are positive and are in A.P., the roots of the quadratic equations px2 + qx + r = 0 are real for (a) 73rp−≥ (b) 743rp−< (c) all p and r (d) no p and r 7. If two equations a1 x2 + b1 x + c1 = 0 and a2 x2 + b2 x + c2 = 0 have a common root, then the value of (a1b2 – a2b1).(b1c2 – b2c1) is (a) – (a1c2 – a2c1)2 (b) (a1a2 – c1c2)2 (c) (a1c1 – a2c2)2 (d) (a1c2 – c1a2)2 8. The value of p for which the difference between the roots of the equation x2 + px + 8 = 0 is 2 are (a) ± 2 (b) ± 4 (c) ± 6 (d) ± 8 9. If f(x) = 2x3 + mx2 – 13x + n and 2,3 are roots of the equation f(x) = 0, then the values of m and n are (a) – 5, – 30 (b) – 5, 30 (c) 5, 30 (d) None of these 10. If the roots of the equation a(b – c) x2 + b(c – a)x + c(a – b) = 0 are equal, then a,b,c are in (a) HP (b) GP (c) AP (d) None of these 11. If 7 log7 (x2 – 4x + 5) = x – 1, x may have values (a) 2,3 (b) 7 (c) – 2, – 3 (d) 2, – 3. 12. If the roots of x2 + x + a = 0 exceed a, then Mathematics Online 1 QUADRATIC EQUATION (a) 2 < a < 3 (b) a > 3 (c) – 3 < a < 3 (d) a < – 2 13. If the ratio of the roots of x2 + bx + c = 0 and x2 + qx + r = 0 be the same, then (a) r2c = b2q (b) r2b = c2q (c) rb2 = cq2 (d) rc2 = bq2 14. If x = √7 + 4 √3, x + 1/x = (a) 4 (b) 6 (c) 3 (d) 2 15. If α, β are the roots of ax2 + bx + c = 0 the equation whose roots are 2 + α, 2 + β is (a) ax2 + x(4a – b) + 4a – 2b + c = 0 (b) ax2 + x(4a – b) + 4a + 2b + c = 0 (c) ax2 + x(b – 4a) + 4a + 2b + c = 0 (d) ax2 + x(b – 4a) + 4a – 2b + c = 0 16. 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 17. Two students while solving a quadratic equation in x, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of x2 correctively as – 6 and 1 respectively. The correct roots are (a) 3, – 2 (b) – 3, 2 (c) – 6, –1 (d) 6, – 1 18. The set of values of x which satisfy 5 x + 2 < 3 x + 8 and (x + 2)/(x – 1) < 4 is (a) (2, 3) (b) (– ∞, 1) ∪ (2, 3) (c) (– ∞, 1) (d) (1, 3) 19. If 8, 2 are the roots of x2 + ax + b = 0, and 3, 3 are the roots of x2 + αx + b = 0, then the roots of x2 + ax + b = 0 are (a) 8, – 1 (b) – 9, 2 (c) – 8, – 2 (d) 9, 1 20. If one root of x2 – x – k = 0 is square of the other, then k = (a) 2 ± √13 (b) 3 ± √2 (c) 2 ± √5 (d) 5 ± √2 21. If 3 + 4i is a root of the equation x2 + px + q = 0, then (a) p = 6, q = 25 (b) p = 6, q = 1 (c) p = – 6, q = – 7 (d) p = – 6, q = 25 22. The number of values of k for which the equation x2 – 3x + k = 0 has two distinct roots lying in the interval (0, 1) are (a) 0 (b) 2 (c) 3 (d) infinitely many 23. The product of the roots of the equation x2 + 6x + (2m – 1) = 0 is – 1. Then m = (a) 1 (b) 1/3 (c) 1 (d) – 1/3 24. If A = {x⏐ f (x) = 0} and B = { x ⏐ g(x) = 0}, then A Ι B will be the set of roots of the equation (a) [f (x) ]2 + [g(x)]2 = 0 (b) f (x) /g (x) (c) g(x) /f (x) (d) none of these Mathematics Online 2 QUADRATIC EQUATION25. Let x1, x2 be the roots of the equation x2 – 3x + p = 0 and let x3, x4 b25. be the roots of the equation x2 – 12x + q = 0. If the numbers x1, x2, x3, x4 (in order) form an increasing GP, then (a) p = 2, q = 16 (b) p = 2, q = 32 (c) p = 4, q = 16 (d) p = 4, q = 32 26. The roots of the equation ⏐x2 – x – 6 ⏐ = x + 2 are (a) – 2, 1, 4 (b) 0, 2, 4 (c) 0, 1, 4 (d) – 2, 2, 4 27. If the equation x3 – 3x + a = 0 has distinct roots between 0 and 1, then the value of a is (a) 2 (b) 1/2 (c) 3 (d) none of these 28. If f(x) = ax2 + bx + c, g(x)=ax2 + bx + c where ac ≠ 0, then f(x) g(x) = 0 has (a) at least three real roots (b) no real roots Theequationxxxxx221022222cossin,=+≤≤π The equation x x x 2 x 2 1 0 2 2 2 22cos sin = + , ≤ ≤ x π (c) at least two real roots (d) two real roots and two imaginary roots 29. The equation 2 cos2 x/2 sin2 x = x2 + 1/x2, 0 x ≤ x ≤ π/2 (a) no real solution (b) one real solution (c) more than one real solution (d) none of these. 30. The number of real roots of the equation (x – 1)2 + (x – 2)2 + (x – 3)2 = 0 is (a) 1 (b) 2 (c) 3 (d) none of these 31. The roots of the equation log2 (x2 – 4x + 5) = (x – 2) are (a) 4, 5 (b) 2, – 3 (c) 2, 3 (d) 3, 5 32. Let a, b and c be real nos. such that 4a + 2b + c = 0 and ab > 0. Then the equation ax2 + bx + c = 0 has (a) real roots (b) complex roots (c) exactly one root (d) none of these 33. The value of k for which the equation 3x2 + 2x (k2 + 1) + k2 – 3k + 2 = 0 has roots of opposite signs, lies in the interval (a) (– ∞, 0) (b) (– ∞, – 1) (c) (1, 2) (d) (3/2, 2) 34. If p and q are roots of the quadratic equation x2 + mx + m2 + a = 0, then value of p2 + q2 + pq is (a) 0 (b) a (c) – a (d) ± m2 35. The condition that one root of the equation ax2 + bx + c = 0 may be double of the other, is (a) b2 = 9ac (b) 2b2 = 9ac (c) 2b2 = ac (d) b2 = ac 36. If ecos x – e – cos x = 4, then the value of cos x is (a) log (2 + √5) (b) – log (2 + √5) (c) log (– 2 + √5) (d) none of these 37. If a, b, c are positive real numbers, then the roots of the equation ax2 + bx +c = 0 (a) are real and positive (b) real and negative (c) have negative real part (d)have positive rea part. 38. The quadratic equation whose roots are reciprocal of the roots of the equation ax2 + bx + c = 0 is (a) cx2 + bx + a = 0 (b) bx2 + cx + a = 0 Mathematics Online 3 QUADRATIC EQUATION (c) cx2 + ax + b = 0 (d) bx2 + ax + c = 0 39. If one root of the equation 5x2 + 13x + k = 0 is reciprocal of other, then the value of k is (a) 0 (b) 5 (c) 1/6 (d) 6 40. Both the roots of the equation (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are always (a) positive (b) negative (c) real (d) none of these 41. If the equations k (6x2 + 3) + rx + 2x2 – 1 = 0 and 6k (2x2 + 1) + px + 4x2 – 2 = 0 have both roots common, then the value of (2r – p) is (a) 0 (b) 1/2 (c) 1 (d) none of these 42. If x = 2 + 22/3 + 21/3 then the value of x3 – 6x2 + 6x is (a) 3 (b) 2 (c) 1 (d) none of these 43. The number of quadratic equations which are unchanged by squaring their roots is (a) 2 (b) 4 (c) 6 (d) none of these 44. If the equation x2 – 3kx + 2e2 log k – 1 = 0 has real roots such that the product of roots is 7, then the value of k is (a) ± 1 (b) ± 2 (c) ± 3 (d) none of these 45. If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of q is (a) 49/4 (b) 4/49 (c) 4 (d) none of these 46. If the roots of the equation x2 – px + q = 0 differ by unity, then (a) p2 = 49 (b) p2 = 4q + 1 (c) p2 = 4q – 1 (d) none of these 47. If α, β are roots of the equation ax2 + bx + c = 0 then the equation whose roots are 2 α + 3 β and 3 α + 2 β is (a) ab x2 – (a + b) cx + (a + b)2 = 0 (b) ac x2 – (a + c) bx + (a + c)2 = 0 (c) ac x2 + (a + c) bx – (a + c) bx – (a + c)2 = 0 (d) none of these 48. The number of roots of the equation 22111xxx−=−−− (a) 1 (b) 2 (c) 0 (d) infinitely many 49. The number of real roots of the equation ⏐ x⏐2 – 3⏐x⏐ + 2 = 0 is (a) 4 (b) 3 (c) 2 (d) 1 50. If the equation abxaxb+−− = 1 has roots equation magnitude but opposite in sign, then the value of of (a) – 1 (b) 0 (c) 1 (d) none of these Mathematics Online 4 QUADRATIC EQUATION 51. The equation 1141xxxhas+−−=+(a) no solution (b) one solution (c) two solutions (d) more than two solutions .52. The equation x34x1x86x11has+−−++−−= (a) no solution (b) one solution (c) two solutions (d) more than two solutions 52. The equation = 2.5 has 51logcosx5+ (a) no solution (b) one solution (c) two solutions (d) more than two solutions 53. The equation 125x + 45x = (2) 27x has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 54. The equation (0.4)x – 1 = (6.25)6x – 5 has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 55. The equation 1 + 3x/2 = 2x has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 56. The equation log3 (3x)+ + log3 (1 + x2) = 0 has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 57. If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 has (a) at least one root in (– 1, 0) (b) at least one root in (1, 2) (c) at least one root in (0, 1) (d) none of these 58. If for the quadratic equation x2 – kx + 1 = 0 (k > 0) one of the roots is A such that tan A = 2√5 – 1, then the other root is (a) greater than 1 (b) greater than 2 (c) less than 1 (d) none of these 59. If both the roots of the equation x2 – ax + 2 = 0 lie in the interval (0, 3), then a lies in (a) (– 11/3, – 2√2) (b) (– 1, – 3) (c) (– 2, 7) (d) none of these 60. If two roots of the equation x3 + mx2 + 11x – n = 0 are 2 and 3, then value of m + n is (a) – 1 (b) – 2 (c) – 3 (d) none of these 61. The value of m for which the equation given below has roots equal in magnitude but opposite in sign ab1xamxbm+=++++ (a) abab−+ (b) – 1 (c) 0 (d) abab+− 62. The equation ⎮x – x2 – 1⎮ = ⎮2x – 3 – x2⎮ has (a) infinite solutions (b) one solution (c) two solutions (d) no solution 63. If q ≠ 0 and the equation x3 + px2 + q = 0 has a root of multiplicity 2, then p and q are connected by (a) p2 – 2q = 0 (b) 4p3 + 27q + 1 = 0 Mathematics Online 5 QUADRATIC EQUATION (c) p2 + 2q = 0 (d) 4p3 + 27q = 0 64. If α, β, γ are such that α + β + γ = 2, α2 + β2 + γ2 = 6, α3 + β3 + γ3 = 8, then value of α5 + β5 + γ5 is (a) 10 (b) 12 (c) 18 (d) 32 65. If α, β, γ are the roots of the equation x3 – 13x + 11 = 0, then equation whose roots are β + γ, γ + α, α + β is (a) x3 – 13x – 11 = 0 (b) x3 – 13x + 11 = 0 (c) x3 + 13x – 12 = 0 (d) x3 + 26x – 22 = 0 66. If α, β are the roots of ax2 + bx + c = 0, then the roots of equation ax2 – bx(x – 1) + c(x – 1)2 = 0 are (a) α – 1 , β – 1 (b) ,11αβα+β+ (c) 11,α+β+αβ (d) none of these 67. If α is a real root of the equation 2x3 – 3x2 + 6x + 6 = 0, then value of [α], where [ ] is the greatest integer function is (a) – 1 (b) 0 (c) 1 (d) 3 68. The sum of the real roots of the equation ⎪x – 2⎪2 + ⎪x – 2⎪ – 2 = 0 is (a) 4 (b) 3 (c) 2 (d) 10 69. The product of real roots of the equation ⎪2x + 3⎪2 – 3⎪2x + 3⎪ + 2 = 0 is (a) 5/4 (b) 5/2 (c) 5 (d) 2 70. Sum of the roots of the equation 32log(logx)2229logx(logx) 1 =−+ is (a) 8 (b) 6 (c) 4 (d) 2 71. Let c1 and c2 be defined as follows : c1 : b2 – 4ac ≥ 0 & c2 : a, – b, c are of the same sign The roots of ax2 + bx + c = 0 are real and positive if (a) both c1 and c2 are satisfied (b) only c2 is satisfied (c) only c1 is satisfied (d) none of these 72. The number of roots of the equation x2 + x + 3 + 2 sin x = 0, x ∈ [– π, π] is (a) 2 (b) 3 (c) 4 (d) none of these 73. The number of roots of the equation x3 + x2 + 2x + sin x = 0, in [– 2π, 2π] is (a) 1 (b) 2 (c) 3 (d) none of these 74. The number of real roots of ⎥ x2 + 4⎥ x⎥ + 3⎥ + 2x – 11 = 0 is (a) 1 (b) 2 (c) 3 (d) none of these 75. The set of values of a for which the equation (a + 4)x2 + (a + 3)x – 1 = 0 has at least one positive root is (a) (4, ∞) (b) (– 4, ∞) (c) (– 3, ∞) (d) (3, ∞) 76. If two roots of x3 – px2 + qx – r = 0 are equal in magnitude but opposite in sign then (a) pq = r (b) p2 = qr (c) p + qr = 0 (d) none of these Mathematics Online 6 QUADRATIC EQUATION77. If the roots of x5 – 40x4 + Px3 + Qx2 + Rx + S = 0 are in G.P. and sum of their reciprocals is 10, then ⎥ S⎥ is (a) 4 (b) – 4 (c) 8 (d) none of these 78. Suppose a, b, c > 0, then the number of real roots of the equation ax2 + b⎥ x⎥ + c = 0 is (a) 1 (b) 4 (c) 1 (d) none of these 79. Suppose the roots α, β, γ of x3 – 3ax2 + 3bx – c = 0 are in H.P., then (a) β = 1/α (b) β = b (c) β = c/b (d) β = b/c 80. The numerical difference of the roots of x2 – 6x + 6 = 0 is (a) 0 (b) √6 (c) √(12) (d) √(18) 81. If the roots of the equation ax2 + bx + c = 0 are reciprocal to each other, then (a) a + c = 0 (b) b = 0 (c) a – c = 0 (d) None of these 82. Roots of ax2 + b = 0 are real and distinct if (a) ab > 0 (b) ab < 0 (c) a, b > 0 (d) a, b < 0 83. If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, the value of q is (a) 49/4 (b) 4/49 (c) 4 (d) None of these 84. The value of k for which the quadratic equation kx2 + 1 = kx + 3x – 11x2 has real and equal roots are (a) { – 11, – 3 } (b) { 5, 7 } (c) { 5, – 7 } (d) None of these 85. If α and β are roots of the equation ax2 + bx + c = 0, then (1 + α + α2) (1 + β + β2) = (a) 0 (b) + ve (c) – ve (d) None of these 86. If p > 0, q > 0 then roots of x2 – px – q = 0 are (a) Imaginary (b) Real and of opposite sign (c) Real and positive (d) Real and negative 87. If the equations x2 + 2x + 3λ = 0 and 2x2 + 3x + 5λ = 0 have a non–zero common root, then λ = (a) 1 (b) – 1 (c) 3 (d) None of these 88. If the equation 2xbxmaxcm1−=−+−1 has roots equal in magnitude but opposite in sign. Then m = (a) abab+− (b) abab−+ (c) abab−+ (d) None of these 89. The number of roots of the quadratic equation 8 sec2 θ – 6 sec θ + 1 = 0 is (a) Infinite (b) 1 (c) 2 (d) 0 90. Let α, β be the roots of the equation (x – a) (x – b) = c, c ≠ 0, Then the roots of the equation (x – α) (x – β) + c = 0 are Mathematics Online 7 QUADRATIC EQUATION(a) a, c (b) b, c (c) a, b (d) a + c, b + c 91. Then equation (cos p – 1) x2 + (cos p) x + sin p = 0 in the variable x, has real roots. Then p can take any value in the interval (a) (0, 2π) (b) ( – π, 0) (c) ( – ½ π, ½ π) (d) [0, π] 92. If one root of x2 – x – k = 0 is square of the other, then k = (a) 2 ± √3 (b) 3 ± √2 (c) 2 ± √5 (d) 5 ± √2 93. The value of p for which one root of the equation x2 – 30 x + p = 0 is the square of the other, are (a) 125 only (b) 125 and – 216 (c) 125 and 215 (d) 216 only 94. If the roots of ax2 + bx + c = 0 are in the ratio m : n, then (a) mna2 = (m + n)c2 (b) mnb2 = (m + n)ac (c) mnb2 = (m + n)2ac (b) None of these 95. If the ratio of the roots of ax2 + 2bx + c = 0 is same as the ratio of the roots of px2 + 2qx + r = 0, then (a) bqacpr= (b) 22bqacpr= (c) 22bqacpr= (b) None of these 96. If 8, 2 are the roots of x2 + ax + β = 0, and 3, 3 are the roots of x2 + αx + b = 0 then the roots of x2 + ax + b = 0 are (a) 8, – 1 (b) – 9, 2 (c) – 8 , – 2 (d) 9, 1 97. If α and β are the roots of 4x2 + 3x + 7 = 0 then the value of 3311+=αβ (a) 2764− (b) 6316 (c) 225343 (d) none of these 98. If p and q are the roots of the equation x2 + pq = (p + 1) x, then q = (a) – 1 (b) 1 (c) 2 (d) None of these 99. If the equations k(6x2 + 3) + rx + 2x2 – 1 = 0 and 6k(2x2 + 1) + px + 4x2 – 2 = 0 have both the roots common then the value of 2r – p is (a) 0 (b) 1 (c) ac (d) – ac 100. If (1 + m2) x2 + 2mcx + (c2 – a2) = 0 has equal roots, then (a) c2 = a2 ( 1 – m2) (b) c2 = a2 ( 1 – m) (c) a2 = c2 (m2 + 1) (d) c2 = a2 (1 + m2) 101. Both roots of the equation (x – b) (x – c) + (x – a) (x – c) + (x – a) (x – b) = 0 are always (a) + ve (b) – ve (c) Real (d) None of these 102. The set of values of p for which the roots of the equation 3x2+2x+p(p – 1)=0 are of opposite sign is (a) ( – ∞, 0) (b) (0, 1) (c) (1, ∞) (d) (0, ∞) 103. Let α and β be the roots of the equation x2+x+1=0. The equation whose roots are α19, β7 is (a) x2 – x – 1 = 0 (b) x2 – x + 1 = 0 (c) x2 + x – 1 = 0 (d) x2 + x + 1 = 0 Mathematics Online 8 QUADRATIC EQUATION104. If 3 + 4i is a root of the equation x2 + px + q = 0, then (a) p = 6, q = 25 (b) p = 6, q = 1 (c) p = – 6, q = – 7 (d) p = – 6, q = 25 105. If α and β are the roots of the equation ax2 + bx + c = 0, then the equation whose roots are 11,(ab)(ab)α+β+ (a) acx2 + bx + 1 = 0 (b) acx2 – bx + 1 = 0 (c) acx2 + bx – 1 = 0 (d) acx2 – bx – 1 = 0 106. The set of values of k for which both the roots of the equation 4x2 – 20kx + (25k2 + 15k – 66) = 0 are less than 2, is given by (a) (2, ∞) (b) (4/5, 2) (c) ( – ∞, – 1) (d) None of these 107. If the two equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have a common root then the value of (a1b2 – a2b1)(b1c2 – b2c1) is (a) – (a1c2 – c1a2)2 (b) (a1a2 – c1c2)2 (c) (a1c1 – a2c2)2 (d) (a1c2 – c1a2)2 108. In a quadratic equation with leading coefficient 1, a student reads the coefficient 16 of x wrongly as 19 and obtain the roots as – 15 and – 4. The correct roots are (a) 6, 10 (b) – 6 , – 10 (c) – 7 , – 9 (d) None of these 109. The ratio of the roots of x2 + bx + c = 0 is the same as that of the roots of x2 + qx + r = 0. Then (a) br2 = qc2 (b) cr2 = qb2 (c) rc2 = bq2 (d) rb2 = cq2 110. The roots of the equation a(b – c) x2 + b (c – a) x + c(a – b) = 0 are equal, then a, b, c are in (a) H.P. (b) G.P. (c) A.P. (d) None of these 111. The equation x – 2/(x – 1) = 1 – 2/(x – 1) has (a) no root (b) one root (c) two roots (d) infinitely many roots 112. The number of values of k for which the equation x2 – 3x + k = 0 has two distinct roots lying in the interval (0, 1) are (a) 0 (b) 2 (c) 3 (d) infinitely many 113. Let a,b,c be real numbers, a ≠ 0. If α is a root of a2x2 + bx + c = 0, β is a root of a2x2 – bx – c = 0 and 0 < α < β, then the equation a2x2 + 2bx + 2c = 0 has a root γ that always satisfies (a) γ = ½ (α + β) (b) γ = α + ½β (c) γ = α (d) α < γ < β 114. If the sum of the roots of ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then bc2, ca2, ab2 are in (a) A.P. (b) G.P. (c) H.P. (d) None of these 115. The value of p for which the difference between the roots of the equation x2 + px + 8 = 0 is 2 are (a) ± 2 (b) ± 4 (c) ± 6 (d) ± 8 116. If the sum of the roots of the equation ax2+bx+c=0 is equal to the sum of the squares of their reciprocals, then a/c, b/a, c/b are in Mathematics Online 9 QUADRATIC EQUATION (a) A.P. (b) G.P. (c) H.P. (d) None of these 117. Two students while solving a quadratic equation in x one copied the constant tem incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of x2 correctly as – 6 and 1 respectively. The correct roots are (a) 3, – 2 (b) – 3, 2 (c) – 6, – 1 (d) 6, – 1 118. If x is real and k = (x2 – x + 1)/(x2 + x + 1), then (a) 1/3 ≤ k ≤ 3 (b) k ≥ 3 (c) k ≤ 1/3 (d) None of these 119. If the equations ax2 + 2cx + b = 0 and ax2 + 2bx + c = 0 (b ≠ c) have a common root, then a + 4b + 4c = (a) 0 (b) 1 (c) – 1 (d) None of these 120. If f(x) = 2x3 + mx2 – 13x + n and 2,3 are roots of the equation f(x) = 0, then the values of m and n are (a) – 5, – 30 (b) – 5, 30 (c) 5, 30 (d) None of these 121. If a2 + b2 + c2 = 1, then ab + bc + ca lies in (a) [ 1/2, 2] (b) [ – 1, 2] (c) [ – ½, 1] (d) [1, ½] 122. The real roots of the equation x2 + 5 ⏐x⏐+ 4 = 0 are (a) (– 1, – 4) (b) (1, 4) (c) (– 4, 4) (d) None of these 123. If y = f(x) = tan x cot 3x, then (a) 1/3 < y < 1 (b) – ∞ < y < 1/3 or 3 < y < ∞ (c) 1/3 < y < ∞ (d) – ∞ < y < 1 124. The number of solutions of the equation ⏐x2⏐ – 3⏐ x ⏐ + 2 = 0 is (a) 4 (b) 1 (c) 3 (d) 2 125. = x – 1, x may have values 2(x4x5)7log−+ (a) 2, 3 (b) 7 (c) – 2, – 3 (d) 2, – 3 126. Solution of the in equation x2 + 2 ⏐x⏐– 15 ≥ 0 is given by (a) x ≤ – √3 or x ≥ √3 (b) x ≤ – 3 or x ≥ 3 (c) – 3 ≤ x ≤ 3 (d) None of these 127. If – x2 + 3x + 4 > 0, then (a) – 1 < x < 4 (b) x < –1 and x > 4 (c) – 1 ≤ x ≤ 4 (d) x ≤ – 1 or x ≥ 4 128. The set of values for which x3 + 1 ≥ x2 + x is (a) x ≤ 0 (b) x ≥ 0 (c) x ≥ – 1 (d) – 1 ≤ x ≤ 1 129. If x2 + 6x – 27 > 0, – x2 + 3x + 4 > 0, then x lies in the interval (a) (3, 4) (b) [3, 4] (c) (– ∞, 3] ∪ [4, ∞) (d) (– 9, 4) 130. Let x1, x2 be the roots of the equation x2 – 3x + p = 0 and let x3, x4 be the roots of the equation x2 – 12x + q = 0. If the numbers x1, x2, x3, x4 (in order) form an increasing GP, then (a) p = 2, q = 16 (b) p = 2, q = 32 (c) p = 4, q = 16 (d) p = 4, q = 32 Mathematics Online 10 QUADRATIC EQUATION131. The roots of the equation log2 (x2 – 4x + 5) = (x – 2) are (a) 4, 5 (b) 2, – 3 (c) 2, 3 (d) 3, 5 132. The value of k for which the equation 3x2 + 2x (k2 + 1) + k2 – 3k + 2 = 0 has roots of opposite signs, lies in the interval (a) (– ∞, 0) (b) (– ∞, – 1) (c) (1, 2) (d) (3/2, 2) 133. The condition that one root of the equation ax2 + bx + c = 0 may be double of the other, is (a) b2 = 9ac (b) 2b2 = 9ac (c) 2b2 = ac (d) b2 = ac 134. If ecos x – e– cos x = 4, then the value of cos x is (a) log (2 + √5) (b) – log (2 + √5) (c) log (– 2 + √5) (d) none of these 135. If a, b, c are positive real numbers, then the roots of the equation ax2 + bx + c = 0 (a) are real and positive (b) real and negative (c) have negative real part (d) have positive real part 136. The quadratic equation whose roots are reciprocal of the roots of the equation ax2 + bx + c = 0 is (a) cx2 + bx + a = 0 (b) bx2 + cx + a = 0 (c) cx2 + ax + b = 0 (d) bx2 + ax + c = 0 137. If x = 2 + 22/3 + 21/3, then the value of x3 – 6x2 + 6x is (a) 3 (b) 2 (c) 1 (d) none of these 138. If the equation x2 – 3kx + 2e2 log k – 1 = 0 has real roots such that the product of roots is 7, then the value of k is (a) ± 1 (b) + 2 (c) ± 3 (d) none of these 139. If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then value of q is (a) 49/4 (b) 4/49 (c) 4 (d) none of these 140. If sin α and cos α are roots of the equation px2 + qx + r = 0, then (a) p2 – q2 + 2pr = 0 (b) (p + r)2 = q2 – r2 (c) p2 + q2 – 2pr = 0 (d) (p – r)2 = q2 + r2 141. If the ratio of the roots of the equation x2 + px + q = 0 be equal to the ratio of the roots of x2 + lx + m = 0, then (a) p2m = q2l (b) pm2 = q2l (c) p2l = q2m (d) p2m = l2q 142. If a, b, c are in GP, then the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root if d/a, e/b, f/c are in (a) AP (b) GP (c) HP (d) none of these 143. If the equations x2 + ax + b= 0 and x2 + bx + a = 0 have a common root, then the numerical value of a + b is (a) 1 (b) 0 (c) – 1 (d) none of these Mathematics Online 11 QUADRATIC EQUATION144. If the sum of the roots of the equation (a + 1)x2 + (2a + 3)x + (3a + 4) = 0 is – 1, then the product of the roots is (a) 0 (b) 1 (c) 2 (d) 3 145. If one root of the equation 8x2 – 6x – a – 3 = 0 is the square of the other, then the values of a are (a) 4, – 24 (b) 4, 24 (c) – 4, – 24 (d) – 4, 24 146. The set of values of p for which the roots of the equation 3x2 + 2x + p (p – 1) = 0 are of opposite signs is (a) (– ∞, 0) (b) (0, 1) (c) (1, ∞) (d) (0, ∞) 147. If p and q are the roots of the equation x2 + px + q = 0, then (a) p = 1 (b) p = 1 or 0 (c) p = – 2 (d) p = – 2 or 0 148. If two equations a1 x2 + b1 x + c1 = 0 and a2 x2 + b2 x + c2 = 0 have a common root, then the value of (a1 b2 – a2 b1) . (b1 c2 – b2 c1) is (a) – (a1 c2 – a2 c1)2 (b) (a1 a2 – c1 c2)2 (c) (a1 c1 – a2 c2)2 (d) (a1 c2 – c1 a2)2 149. The value of p for which the difference between the roots of the equation x2 + px + 8 = 0 is 2 are (a) ± 2 (b) ± 4 (c) ± 6 (d) ± 8 SUBJECTIVE QUESTIONS Find all the values of the parameter a for which takes all real values for real values of x Show that tan x cot 3x (x ∈ R) cannot lie in the interval (1/3, 1). 150. If α, β are the roots of the equation ax2 + bx + c = 0, then the value of α3 + β3 is (a) 333abcba+ (b) 33ab3ab+ (c) 333abcba− (d) 33b3abca− 151. If the ratio of the roots of the equations x2 + bx + c = 0 is the same as that of x2 + qx + r = 0, then (a) r2 b = qc2 (b) r2 c = qb2 (c) c2 r = q2 b (d) b2 r = q2 c 152. If α, β are roots of x2 + px + q = 0 then – 11,−αβ are the roots of the equation (a) qx2 – px + 1 = 0 (b) qx2 + px + 1 = 0 (c) x2 + px + q = 0 (d) x2 – px + q = 0 153. If α, β are roots of zx2 + px + q = 0 then – 11,−αβ are the roots of the equation (a) cx2 + ax + b = 0 (b) bx2 + ax + a = 0 (c) cx2 + bx + a = 0 (d) ax2 + cx + b = 0 154. For what value of ‘p’ the difference of the roots of the equation x2 – px + 8 = 0 is 2 ? (a) ± 2 (b) ± 4 (c) ± 6 (d) ± 8 Mathematics Online 12 QUADRATIC EQUATION155. One root of the equation 5x2 + 13x + K = 0 is the reciprocal of the other, if (a) K = 0 (b) K = 5 (c) K = 1/6 (d) 6 156. If α ≠ β, but a2 = 5α – 3,β2 = 5β – 3, then the equation whose roots are andαββα is (a) x2 – 5x – 3 = 0 (b) 3x2 – 19x + 3 = 0 (c) 3x2 + 12x + 3 = 0 (d) none of these 157. The conditions that the equation ax2 + bx + c = 0 has both the roots positive is that (a) a, b and c are of the same sign (b) a and b are of the same sign (c) b and c have the same sign opposite to that of a (d) a and c have the same sign opposite to that of b 158. If x2 – 11x + a and x2 – 14x + 2a have common factor then a equals (a) 0, 8 (b) 5, 24 (c) 0, 24 (d) 1, 2 159. If 2 +3 is a root of x2 + px + q = 0 where p and q real then (p, q) is (a) (7, 4) (b) (– 4, 7) (c) (–4, 7i) (d) (4i, – 7) 160. The real roots of the equation 2log7(x4x5)7−+ = x – 1, are (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5 161. If one root of the equation (x – 1) (7 – x) = m is three times the other, then m equals : (a) – 5 (b) 0 (c) 2 (d) 5 162. If α, β are the roots of the equation x2 + px + p2 + q = 0, then the value of α2 + αβ + β2 + q is equal to : (a) 0 (b) 1 (c) q (d) 2 q 163. If α, β are the roots of the equation x2 – px + 36 = 0 and α2 + β2 = 9, then the value of p are : (a) ± 3 (b) ± 6 (c) ± 8 (d) ± 9 164. If one root of the equation x2 = px + q is reciprocal of the other, then the correct relationship is : (a) q = – 1 (b) q = 1 (c) pq = – 1 (d) pq = 1 165. Let α, β be the roots of the equation (x – a) (x – b) = c, c ≠ 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, d 166. If x is real, then the expression 22x34x71x2x7+−+− can have no value between : (a) 3 and 7 (b) 4 and 8 (c) 5 and 9 (d) 6 and 10 Mathematics Online 13 QUADRATIC EQUATION167. If the roots of the equation 2xbxaxc1−λ−=−λ+1 are such that α + β = 0, then the value of λ is (a) abab−+ (b) c (c) 1c (d) abab+− 168. If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the square of their reciprocals, then 22bbcaca+= (a) 2 (b) – 2 (c) 1 (d) – 1 168. A lad was asked his age by his friend. The lad said, “The number you get when you subtract 25 times my age from twice the square of my age will be thrice your age.” If the friend’s age is 14, then the age of the lad is (a) 21 (b) 28 (c) 14 (d) 25 169. If the roots of the equation x2 + px – q = 0 differ by 1, then (a) p2 = 4q (b) p2 = 4q + 1 (c) p2 = 4q – 1 (d) none of these 170. The roots of the quadratic equation ax2 + bx + c = 0 will be reciprocal to each other if (a) a = 1/c (b) a = c (c) b = ac (d) a = b 171. Let α, β be the roots of x2 – x + p = 0 and γ, δ be roots of x2 – 4x + q = 0. If α, β, γ, δ are in G.P., then the integral values of p and q respectively are (a) – 2, – 32 (b) – 2, 3 (c) – 6, 3 (d) – 6, – 32 172. The number of real solutions of | x |2 + 3 | x | + 2 = 0 is (a) 0 (b) 2 (c) 4 (d) 1 173. The equation : (b – c)x2 + (c – a)x + (a – b) = 0 has (a) equal roots (b) irrational roots (c) rational roots (d) none of these 174. The expression ax2 + bx + c, a > 0 is positive for all real x only if (a) b2 – 4ac = 0 (b) b2 – 4ac ≠ 0 (c) b2 – 4ac < 0 (d) b2 – 4ac > 0 175. If the product of the roots of the equation mx2 + 6x + (2m – 1) = 0 is – 1, then the value of m is (a) 1 (b) – 1 (c) 1/3 (d) – 1/3 176. If the sum of the roots of the equation (m + 1)x2 + 2mx + 3 = 0 is 1, then the value of m is (a) 1/2 (b) – 1/2 (c) 1/3 (d) – 1/3 177. If the equation x2 – (k + 4) x + (4k + 1) = 0 has equal roots, then (a) k = 6 (b) k = 2 (c) k = 6, 2 (d) k = ± 6 178. If one root of the equation ax2 + bx + c = 0, a ≠ 0, be reciprocal of the other, then (a) b = c (b) b = c (c) a = 0 (d) b = 0 Mathematics Online 14 QUADRATIC EQUATION179. The complete set of values of k for which the quadratic equation x2 – kx + k + 2 = 0 has equal roots is given by (a) 21+ 2 (b) 212,212+− (c) 212− (d) –2–12 180. If α and β are the roots of 4x2 + 3x + 7 = 0, then the value of 11+αβ is (a) 34− (b) 37 (c) 3–7 (d) 47 181. If the roots of ax2 + bx + c = 0 are equal in magnitude but opposite in sign, then (a) a = 0 (b) b = 0 (c) c = 0 (d) none of these 182. If a, b are the roots of ax2 + bx + c = 0 then αβ2 + α2β + αβ equals (a) 2c(ab)a− (b) 0 (c) 2bca− (d) none of these 183. If α, β are the roots of the equation x2 – 2x + 2 = 0, then the value of α2 + β2 is (a) 2 (b) 0 (c) 1 (d) 4 184. For a given value of k, the product of the roots of x2 – 2kx + 3k2 – 4 = 0 is 5. The roots may be characterized as (a) integral (b) rational but not integral (c) irrational (d) imaginary 185. If one root of the equation x2 + px + 12 = 0 is 4 while the equation x2 + px + q = 0 has equal roots, the value of q is (a) 49/4 (b) 4/49 (c) 4 (d) none of these 186. The numerical difference of the roots of x2 – 7x – 9 = 0 is (a) 5 (b) 2 √85 (c) 9 √7 (d) √85 187. If x = 666........+++∞, then (a) x is an irrational number (b) 2 < x < 3 (c) x = 3 (d) none of these 188. If α and β are the roots of the equation x2 – p(x + 1 ) – q = 0, then the value of 222221212q2qα+α+β+β++α+α+β+β+, then (a) 2 (b) 3 (c) 0 (d) 1 189. The number of real roots of equation (x – 1)2 + (x – 2)2 + (x – 3)2 = 0 is (a) 2 (b) 1 (c) 0 (d) 3 190. If α, β are the roots of the equation ax2 + bx + c = 0, then the value of α3 + β3 is (a) 333abcba+ (b) 33ab3ab+ (c) 333abcba− (d) 33b3abca− Mathematics Online 15 QUADRATIC EQUATION191. If the ratio of the roots of the equations x2 + bx + c = 0 is the same as that of x2 + qx + r = 0, then (a) r2b = qc2 (b) r2c = qb2 (c) c2r = q2b (d) b2r = q2c 192. One root of the equation (x + 1) (x + 3) (x + 2) (x + 4) = 120 is (a) – 1 (b) 2 (c) 1 (d) 0 193. The solution of the equation |3 + 1/x| = 2 are (a) 0, – 1, – 1/5 (b) 2, – 1 (c) – 1, – 1/5 (d) none of these 194. If α, β are roots of ax2 + bx + c = 0, then 1/α, 1/β are the roots of (a) cx2 + ax + b = 0 (b) bx2 + ax + a = 0 (c) cx2 + bx + a = 0 (d) ax2 + cx + b = 0 195. If f(x) = 2x3 + mx2 – 13x + n and 2, 3 are roots of the equation f(x) = 0, then the value of m and n are (a) – 5, – 30 (b) – 5, 30 (c) 5, 30 (d) none of these 196. If the equation x2 – (2 + m)x + (m2 – 4m + 4) = 0 has coincident roots, then (a) m = 0, m = 1 (b) m = 0, m = 2 (c) m = 2/3, m = 6 (d) m = 2/3, m = 1 197. For what value of ‘p’ the difference of the roots of the equation x2 – px + 8 = 0 is 2 ? (a) ± 2 (b) ± 4 (c) ± 6 (d) ± 8 198. One root of the equation 5x2 + 13x + K = 0 is the reciprocal of the other, if (a) K = 0 (b) K = 5 (c) K = 1/6 (d) 6 199. If a ≠ b, but α2 = 5 α – 3, β2 = 5 β – 3, then the equation whose roots are α/β and β/α is (a) x2 – 5x – 3 = 0 (b) 3x2 – 19x + 3 = 0 (c) 3x2 + 12x + 3 = 0 (d) none of these 200. The conditions that the equation ax2 + bx + c = 0 has both the roots positive is that (a) a, b and c are of the same sign (b) a and b are of the same sign (c) b and c have the same sign opposite to that of a (d) a and c have the same sign opposite to that of b 201. If x2 – 11x + a and x2 – 14x + 2a have common factor then a equals (a) 0, 8 (b) 5, 24 (c) 0, 24 (d) 1, 2 202. If 2 + i√3 is a root of x2 + px + q = 0 where p and q are real then (p, q) is (a) (7, 4) (b) (– 4, 7) (c) (– 4, 7i) (d) (4i, – 7) 203. The real roots of the equation 2log7(x4x5)7−+ = x – 1, are (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5 204. The equation x + 211x1x=+−−2, has (a) no real root (b) one real root (c) two equal roots (d) infinitely many roots Mathematics Online 16 QUADRATIC EQUATION205. If one root of the equation (x – 1) (7 – x) = m is three times the other, then m equals : (a) – 5 (b) 0 (c) 2 (d) 5 206. If α, β are the roots of the equation x2 + px + p2 + q = 0, then the value of α2 + αβ + β2 + q is equal to (a) 0 (b) 1 (c) q (d) 2 q 207. If α, β are the roots of the equation 8x2 – 3x + 27 = 0, then the value of 112233⎛⎞⎛⎞αβ+⎜⎟⎜⎟βα⎝⎠⎝⎠ (a) 1/3 (b) 1/4 (c) 7/2 (d) 4 208. If α, β are the roots of the equation x2 – px + 36 = 0 and α2 + β2 = 9, then the value of p are : (a) ± 3 (b) ± 6 (c) ± 8 (d) ± 9 209. If one root of the equation x2 = px + q is reciprocal of the other, then the correct relationship is : (a) q = – 1 (b) q = 1 (c) pq = – 1 (d) pq = 1 210. The quadratic equation whose roots are three times the roots of the equation 3 ax2 + 3 bx + c = 0 is (a) ax2 + bx + c = 0 (b) ax2 + 3bx + c = 0 (c) ax2 + bx + 3c = 0 (d) ax2 + 3bx + 3c = 0 211. If x is real, then the least value of the expression (ax2 + bx + c), a > 0 is : (a) b2a− (b) b2 – 4 ac (c) 24acb4a− (d) 224acb4a− 212. For all x ∈ R, if mx2 – 9 mx + 5m + 1 > 0, then m lies in the interval (a) 4,061⎛⎞−⎜⎟⎝⎠ (b) 40,61⎡⎟⎢⎣⎠⎞ (c) 461,614⎛⎞⎜⎟⎝⎠ (d) 61,04⎛⎤−⎜⎥⎝⎦ 213. The set of values of m for which both roots of the equation x2 – (m + 1) x + m + 4 = 0 are real and negative consists of all m such that (a) – 3 < m ≤ – 1 (b) – 4 < m ≤ – 3 (c) – 3 ≤ m ≤ 5 (d) – 3 ≥ m or m ≥ 5 214. If x is real, then the expression 22x34x71x2x7+−+− can have no value between: (a) 3 and 7 (b) 4 and 8 (c) 5 and 9 (d) 6 and 10 215. The expression ax2 + bx + c has the same sign as of a if (a) b2 – 4ac > 0 (b) b2 – 4ac = 0 (c) b2 – 4ac ≤ 0 (c) b and c have the same sign as a. 216. In the equation ax2 + bx + c = 0, if a ≠ 0 and b = 0, then the roots are (a) equal (b) equal in magnitude but opposite in sign (c) reciprocal to each other (d) reciprocal to each other but have same sign Mathematics Online 17 QUADRATIC EQUATION217. The value of m for which the equation x3 – mx2 + 3x – 2 = 0 has two roots equal in magnitude but opposite in sign, is (a) 1/2 (b) 2/3 (c) 3/4 (d) 4/5 218. If the roots of the equation 2xbx1axc1−λ−=−λ+ are such that α + β = 0, then the value of λ is (a) abab−+ (b) c (c) 1c (d) abab+− 219. If α, β are the roots of the equation ax2 + bx + c = 0, then the value of α3 + β3 is (a) 333abcba+ (b) 33ab3ab+ (c) 333abcba− (d) 3b3abca3− 220. If If the ratio of the roots of the equations x2 + bx + c = 0 is the same as that of x2 + qx + r = 0, then (a) r2 b = qc2 (b) r2 c = qb2 (c) c2 r = q2 b (d) b2 r = q2 c 221. If α, β are the roots of x2 + px + q = 0 then 11,−−αβ are the roots of the equation (a) qx2 – px + 1 = 0 (b) qx2 + px + 1 = 0 (c) x2 + px + q = 0 (d) x2 – px + q = 0 222. If α, β are the roots of ax2 + bx + c = 0, then 11,−αβ are the roots of (a) cx2 + ax + b = 0 (b) bx2 + ax + a = 0 (c) cx2 + bx + a = 0 (d) ax2 + cx + b = 0 223. For what value of ‘p’ the difference of the roots of the equation x2 – px + 8 = 0 is 2 ? (a) ± 2 (b) ± 4 (c) ± 6 (d) ± 8 224. One root of the equation 5x2 + 13x + K = 0 is the reciprocal of the other, if (a) K = 0 (b) K = 5 (c) K = 1/6 (d) 6 224. If α ≠ β, but α2 = 5α – 3, β2 = 5β – 3, then the equation whose roots are 1and1 αβ is (a) x2 – 5x – 3 = 0 (b) 3x2 – 19x + 3 = 0 (c) 3x2 + 12x + 3 = 0 (d) none of these 225. The conditions that the equation ax2 + bx + c = 0 has both the roots positive is that (a) a, b and c are of the same sign (b) a and b are of the same sign (c) b and c have the same sign opposite to that of a (d) a and c have the same sign opposite to that of b 226. If x2 – 11x + a and x2 – 14x + 2a have common factor then a equals (a) 0, 8 (b) 5, 24 (c) 0, 24 (d) 1, 2 Mathematics Online 18 QUADRATIC EQUATION227. If 2 + i3 is a root of x2 + px + q = 0 where p and q are real then (p, q) is (a) (7, 4) (b) (– 4, 7) (c) (–4, 7i) (d) (4i, – 7) 228. The real roots of the equation 2log7(x4x5)7−+ = x – 1, are (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5 229. If one root of the equation (x – 1) (7 – x) = m is three times the other, then m equals : (a) – 5 (b) 0 (c) 2 (d) 5 230. If α, β are the roots of the equation x2 + px + p2 + q = 0, then the value of α2 + αβ + β2 + q is equal to : (a) 0 (b) 1 (c) q (d) 2 q 231. If α, β are the roots of the equation x2 – px + 36 = 0 and α2 + β2 = 9, then the value of p are : (a) ± 3 (b) ± 6 (c) ± 8 (d) ± 9 232. If one root of the equation x2 = px + q is reciprocal of the other, then the correct relationship is : (a) q = – 1 (b) q = 1 (c) pq = – 1 (d) pq = 1 233. Let α, β be the roots of the equation (x – a) (x – b) = c, c ≠ 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, d 234. If x is real, then the expression 22x34x7x2x7+− 1 +−can have no value between : (a) 3 and 7 (b) 4 and 8 (c) 5 and 9 (d) 6 and 10 235. If the roots of the equation xbx1axc12−λ−=−λ+are such that α + β = 0, then the value of λ is (a) abab−+ (b) c (c) 1c (d) abab+− 236. If sum of roots of quadratic equation ax2 + bx + c = 0 is equal to sum of square of their reciprocals, then (a) 2 (b) – 2 (c) 1 (d) – 1 237. A lad was asked his age by his friend. The lad said, “The number you get when you subtract 25 times my age from twice the square of my age will be thrice your age.” If the friend’s age is 14, then the age of the lad is (a) 21 (b) 28 (c) 14 (d) 25 238. If the roots of the equation x2 + px – q = 0 differ by 1, then (a) p2 = 4q (b) p2 = 4q + 1 (c) p2 = 4q – 1 (d) none of these 239. The roots of the quadratic equation ax2 + bx + c = 0 will be reciprocal to each other if (a) a = 1/c (b) a = c (c) b = ac (d) a = b Mathematics Online 19 QUADRATIC EQUATION240. Let α, β be the roots of x2 – x + p = 0 and γ, δ be roots of x2 – 4x + q = 0. If α, β, γ, δ are in G.P., then the integral values of p and q respectively are (a) – 2, – 32 (b) – 2, 3 (c) – 6, 3 (d) – 6, – 32 241. The number of real solutions of | x |2 + 3 | x | + 2 = 0 is (a) 0 (b) 2 (c) 4 (d) 1 242. The equation : (b – c)x2 + (c – a)x + (a – b) = 0 has (a) equal roots (b) irrational roots (c) rational roots (d) none of these 243. The expression ax2 + bx + c, a > 0 is positive for all real x only if (a) b2 – 4ac = 0 (b) b2 – 4ac ≠ 0 (c) b2 – 4ac < 0 (d) b2 – 4ac > 0 244. If the product of the roots of the equation mx2 + 6x + (2m – 1) = 0 is – 1, then the value of m is (a) 1 (b) – 1 (c) 13 (d) 13− 245. If the sum of the roots of the equation (m + 1)x2 + 2mx + 3 = 0 is 1, then the value of m is (a) 12 (b) 12− (c) 13 (d) 13− 246. If the equation x2 – (k + 4) x + (4k + 1) = 0 has equal roots, then (a) k = 6 (b) k = 2 (c) k = 6, 2 (d) k = ± 6 247. If one root of the equation ax2 + bx + c = 0, a ≠ 0, be reciprocal of the other, then (a) b = c (b) b = c (c) a = 0 (d) b = 0 248. The complete set of values of k for which the quadratic equation x2 – kx + k + 2 = 0 has equal roots is given by (a) 2+ 2 (b) 212,212+− (c)21− 2 (d) 212−− 249. If α and β are the roots of 4x2 + 3x + 7 = 0, then the value of 11+αβ is (a) – 3/4 (b) 3/7 (c) – 3/7 (d) 4/7 250. If the roots of ax2 + bx + c = 0 are equal in magnitude but opposite in sign, then (a) a = 0 (b) b = 0 (c) c = 0 (d) none of these 251. If α, β are the roots of ax2 + bx + c = 0 then αβ2 + α2β+ αβ equals (a)2c(ab)a− (b) 0 (c) 2bca− (d) none of these 252. If α, β are the roots of the equation x2 – 2x + 2 = 0, then the value of α2 + β2 is (a) 2 (b) 0 (c) 1 (d) 4 Mathematics Online 20 QUADRATIC EQUATION253. For a given value of k, the product of the roots of x2 – 2kx + 3k2 – 4 = 0 is 5. The roots may be characterized as (a) integral (b) rational but not integral (c) irrational (d) imaginary 254. If one root of the equation x2 + px + 12 = 0 is 4 while the equation x2 + px + q = 0 has equal roots, the value of q is (a) 49/4 (b) 4/49 (c) 4 (d) none of these 255. The numerical difference of the roots of x2 – 7x – 9 = 0 is (a) 5 (b) 2 √85 (c) 9 √7 (d) √85 256. If x = 666............+++∞, then (a) x is an irrational number (b) 2 < x < 3 (c) x = 3 (d) none of these 257. If α and β are the roots of the equation x2 – p(x + 1) – q = 0, then value of 222221212q2α+α+β+β++α+α+β+β+q (a) 2 (b) 3 (c) 0 (d) 1 258. The equation x1x14x1+−−=− has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 259. The equation x34x1x86x11+−−++−−= (a) no solution (b) one solution (c) two solutions (d) more than two solutions 260. The equation = 2.5 has 51logcosx5+ (a) no solution (b) one solution (c) two solutions (d) more than two solutions 261. The equation 125x + 45x = (2) 27x has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 262. The equation (0.4)x – 1 = (6.25)6x – 5 has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 263. The equation 1 + 3x/2 = 2x has (a) no solution (b) one solution (c) two solutions (d) more than two solutions Mathematics Online 21 QUADRATIC EQUATION 264. The equation log3 (3x)+ + log3 (1 + x2) = 0 has (a) no solution (b) one solution (c) two solutions (d) more than two solutions 265. If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 has (a) at least one root in (– 1, 0) (b) at least one root in (1, 2) (c) at least one root in (0, 1) (d) none of these 266. If for the quadratic equation x2 – kx + 1 = 0 (k > 0) one of the roots is A such that tan A = 2√5 – 1, then the other root is (a) greater than 1 (b) greater than 2 (c) less than 1 (d) none of these 267. If both the roots of the equation x2 – ax + 2 = 0 lie in the interval (0, 3), then a lies in (a) (– 11/3, – 2√2) (b) (– 1, – 3) (c) (– 2, 7) (d) none of these PRACTICE QUESTIONS Let x1, x2 be the roots of the equation x2 – 3x + p = 0 and let x3, x4 be the roots of the equation x2 – 12x + q = 0. If the numbers x1, x2, x3, x4 (in order) form an increasing GP, then (a) p = 2, q = 16 (b) p = 2, q = 32 (c) p = 4, q = 16 (d) p = 4, q = 32 268. The roots of the equation log2 (x2 – 4x + 5) = (x – 2) are (a) 4, 5 (b) 2, – 3 (c) 2, 3 (d) 3, 5 269. The value of k for which the equation 3x2 + 2x (k2 + 1) + k2 – 3k + 2 = 0 has roots of opposite signs, lies in the interval (a) (– ∞, 0) (b) (– ∞, – 1) (c) (1, 2) (d) (3/2, 2) 270. The condition that one root of the equation ax2 + bx + c = 0 may be double of the other, is (a) b2 = 9ac (b) 2b2 = 9ac (c) 2b2 = ac (d) b2 = ac 271. If ecos x – e– cos x = 4, then the value of cos x is (a) log (2 + √5) (b) – log (2 + √5) (c) log (– 2 + √5) (d) none of these 272. If a, b, c are positive real numbers, then the roots of the equation ax2 + bx + c = 0 (a) are real and positive (b) real and negative (c) have negative real part (d) have positive real part 273. The quadratic equation whose roots are reciprocal of the roots of the equation ax2 + bx + c = 0 is (a) cx2 + bx + a = 0 (b) bx2 + cx + a = 0 (c) cx2 + ax + b = 0 (d) bx2 + ax + c = 0 274. If x = 2 + 22/3 + 21/3, then the value of x3 – 6x2 + 6x is (a) 3 (b) 2 (c) 1 (d) none of these 275. If the equation x2 – 3kx + 2e2 log k – 1 = 0 has real roots such that the product of roots is 7, then the value of k is Mathematics Online 22 QUADRATIC EQUATION(a) ± 1 (b) + 2 (c) ± 3 (d) none of these 276. If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then value of q is (a) 49/4 (b) 4/49 (c) 4 (d) none of these 277. If sin α and cos α are roots of the equation px2 + qx + r = 0, then (a) p2 – q2 + 2pr = 0 (b) (p + r)2 = q2 – r2 (c) p2 + q2 – 2pr = 0 (d) (p – r)2 = q2 + r2 278. If the ratio of the roots of the equation x2 + px + q = 0 be equal to the ratio of the roots of x2 + lx + = 0, then (a) p2m = q2l (b) pm2 = q2l (c) p2l = q2m (d) p2m = l2q 279. If a, b, c are in GP, then the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root if d/a, e/b, f/c are in (a) AP (b) GP (c) HP (d) none of these 280. If the equations x2 + ax + b= 0 and x2 + bx + a = 0 have a common root, then the numerical value of a + b is (a) 1 (b) 0 (c) – 1 (d) none of these 281. If the sum of the roots of the equation (a + 1)x2 + (2a + 3)x + (3a + 4) = 0 is – 1, then the product of the roots is (a) 0 (b) 1 (c) 2 (d) 3 282. If one root of the equation 8x2 – 6x – a – 3 = 0 is the square of the other, then the values of a are (a) 4, – 24 (b) 4, 24 (c) – 4, – 24 (d) – 4, 24 283. The set of values of p for which the roots of the equation 3x2 + 2x + p (p – 1) = 0 are of opposite signs is (a) (– ∞, 0) (b) (0, 1) (c) (1, ∞) (d) (0, ∞) 284. If p and q are the roots of the equation x2 + px + q = 0, then (a) p = 1 (b) p = 1 or 0 (c) p = – 2 (d) p = – 2 or 0 285. If two equations a1x2 + b1x + c1 = 0 & a2x2 + b2x + c2 = 0 have a common root, then value of (a1b2 – a2b1) . (b1c2 – b2c1) is (a) – (a1 c2 – a2 c1)2 (b) (a1 a2 – c1 c2)2 (c) (a1 c1 – a2 c2)2 (d) (a1 c2 – c1 a2)2 286. The value of p for which the difference between the roots of the equation x2 + px + 8 = 0 is 2 are (a) ± 2 (b) ± 4 (c) ± 6 (d) ± 8 287. If two roots of the equation x3 + mx2 + 11x – n = 0 are 2 and 3, then value of m + n is (a) – 1 (b) – 2 (c) – 3 (d) none of these Mathematics Online 23 QUADRATIC EQUATION288. Value of m for which equation given below has roots equal in magnitude but opposite in sign abxamxbm+++++ = 1 is (a) abab−+ (b) – 1 (c) 0 (d) abab+− 289. The equation ⎮x – x2 – 1⎮ = ⎮2x – 3 – x2⎮ has (a) infinite solutions (b) one solution (c) two solutions (d) no solution If q ≠ 0 and the equation x3 + px2 + q = 0 has a root of multiplicity 2, then p and q are connected by (a) p2 – 2q = 0 (b) 4p3 + 27q + 1 = 0 (c) p2 + 2q = 0 (d) 4p3 + 27q = 0 290. If α, β, γ are such that α + β + γ = 2, α2 + β2 + γ2 = 6, α3 + β3 + γ3 = 8, then value of α5 + β5 + γ5 is (a) 10 (b) 12 (c) 18 (d) 32 291. If the roots of the equation a (b – c) x2 + b (c – a) x + c (a – b) = 0 are real and equal and α, β be the roots of equation ax2 + bx + c = 0, then H.M. of α and β is (a) 1 – αβ (b) 1 + αβ (c) αβ – 1 (d) – 1 – αβ 292. The number of solution of equation | x – 1 | = ex is (a) 0 (b) 1 (c) 2 (d) none of these 293. If sin θ and cos θ are the roots of the equation ax2 + bx + c = 0, then (a) (a – c)2 = b2 + c2 (b) (a + c)2 = b2 – c2 (c) a2 = b2 – 2ac (d) a2 + b2 – 2ac = 0 294. Number of roots of the equation sin x + cos x = x2 – 2x + √6 is (a) 0 (b) 2 (c) 4 (d) an odd number 295. For the equation 3x2 + px + 3 = 0, p > 0, if one of the roots is the square of the other, then p is equal to (a) 1/3 (b) 1 (c) 3 (d) 2/3 296. If α and β (α < β), are the roots of the equation x2 + 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 297. Let α, β be the roots of the equation (x – a) (x – b) = c, c ≠ 0. 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 298. In the equation x3 + 3Hx + G = 0, if G and H are real and G2 + 4H3 > 0, then the roots are (a) all real and equal (b) all real and distinct (c) one real and two imaginary (d) all real and two are equal 299. Let α, β be the roots of x2 – x + p = 0 and γ, δ be the roots of x2 – 4x + q = 0. If α, β, γ, δ are in G.P., then the integral values of p and q respectively are (a) – 2, – 32 (b) – 2, 3 (c) – 6, 3 (d) – 6, – 32 Mathematics Online 24 QUADRATIC EQUATION300. The no. of solutions of log4 (x – 1) = log2 (x – 3) is (a) 3 (b) 1 (c) 2 (d) 0 301. Both roots of the equation (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are always (a) positive (b) negative (c) real (d) none of these 302. The number of real solutions of the equation sin ex = 5x + 5– x is (a) 0 (b) 1 (c) 2 (d) infinitely many 303. If a, b, c are in G.P., then the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root if d/a, e/b, f/c are in (a) G.P. (b) A.P. (c) H.P (d) none of these 304. If α and β are the roots of a quadratic equation such that α + β = 2, α4 + β4 = 272, then the quadratic equation is (a) x2 – 2x – 16 = 0 (b) x2 – 2x – 8 = 0 (c) x2 – 2x + 8 = 0 (d) none of these 305. If the equation : (λ – 1) x2 + (λ + 1) x + (λ – 1) = 0 has real roots, then λ can have any value in the interval (a) 1,33⎛⎜⎝⎠⎞⎟(b) (–3, 3) (c) (0, 3) (d) 1,43⎛⎞⎜⎟⎝⎠ 306. The number of real solutions of x2 – 3 | x | + 2 = 0 is (a) 1 (b) 2 (c) 3 (d) 4 307. The number of real solutions of the equation x2 – 3 | x | – 4 = 0, is (a) 4 (b) 1 (c) 2 (d) infinite 308. The roots of the equation = x – 1 are 27log(x4x5)7−+ (a) 4, 5 (b) 2, – 3 (c) 2, 3 (d) 3, 5 309. The value of p for which one root of the equation x2 – 30x + p = 0 is the square of the other are (a) 125 only (b) 125 and – 216 (c) 125 and 215 (d) 216 only 310. If α and β are the roots of x2 – 3ax + a2 = 0 such that α2 + β2 = 1.75, then the values of a are (a) 1/2, 1 (b) 2, 1 (c) 2, 1/2 (d) 1/2, – 1/2 311. Let α and β be the roots of the equation x2 + x + 1 = 0. The quadratic equation whose roots are α19, β7 is (a) x2 – x + 1 = 0 (b) x2 – x + 1 = 0 (c) x2 + x – 1 = 0 (d) x2 + x + 1 = 0 312. If the roots of the quadratic equation ax2 + bx + c = 0 be in the ratio m : n, then (a) am + bn + c (b) abc0mn++= (c) ac (m + n)2 = b2mn (d) abcmn = 1 Mathematics Online 25 QUADRATIC EQUATION313. If y = tanxtan3x, then (a) y < 13 or y > 3 (b) 13≤ y ≤ 3 (c)y ≤ 13 or y ≥ 3 (d) 13 ≤ y ≤ 1 314. The equation sin x = x2 + x + 1 has (a) one real solution (b) no real solution (c) more than one real solution (d) two positive solutions 315. If p and q are odd integers, then the equation x2 + 2px + 2q = 0 (a) has no integral root (b) has no rational root (c) has no irrational root (d) has no imaginary root Mathematics Online 26 QUADRATIC EQUATION