Mathematics-7.Unit-4.02-Quadratic Equation Test

1. 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, ∞) 2. The number of real roots of (6 –x)4 + (8 –x)4 = 16 is (A) 0 (B) 2 (C) 4 (D) none of these 3. Given that tan A and tan B are the roots of x2 –px + q = 0, then the value of sin2 (A + B) is (A) (B) (C) (D) 4. If p, q ∈ {1, 2, 3, 4}, the number of equations of the form px2 + qx + 1 = 0 having real roots is (A) 15 (B) 9 (C) 7 (D) 8 5. The harmonic mean of the roots of the equation (4 + )x2 – (4 + )x+ (8 +2 )=0 is (A) 2 (B) 4 (C) 7 (D) 8 6. In a triangle PQR, ∠R = . If tan and tan are the roots of the equation ax2 + bx + c (a ≠ 0). Then (A) a + b = c (B) b + c = 0 (C) a + c = b (D) b = c 7. If the roots of the equation x2 –2ax + a2 + a –3 = 0 are real and less than 3, then (A) a < 2 (B) 2 ≤ a ≤ 3 (C) 3 < a ≤ 4 (D) a > 4 8. If α and β are the roots of the equation, 2x2 –3x –6 =0, then equation whose roots are α2+2, β2 +2 is (A) 4x2+ 49x +118 = 0 (B) 4x2- 49x +118 = 0 (C) 4x2- 49x –118 = 0 (D) x2- 49x +118 = 0 9. If the roots of the equation x2 –px + q = 0 differ by unity then (A) p2 = 1- 4q (B) p2 = 1+ 4q (C) q2 = 1- 4p (D) q2 = 1+ 4p 10. If p and q are the roots of the equation x2 +px +q = 0 , then (A) p =1, q = -2 (B) p =0 , q = 1 (C) p = –2, q = 0 (D) p = –2, q = 1 11. 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 + c , b + c 12. x4 –4x –1 = 0 has (A) atmost one positive real root (B) atmost one negative real root (C) atmost two real roots (D) none of these . 13. If x2 +ax +b is an integer for every integer x then (A) ‘a’ is always an integer but ‘b’ need not be an integer (B) ‘b’ is always an integer but ‘a’ need not be an integer (C) a + b is always an integer (D) a and b are always integers. 14. Sum of the real roots of the equation x2 +5|x| +6 = 0 (A) equals to 5 (B) equals to 10 (C) equals to –5 (D) does not exit. 15. If c > 0 and 4a +c < 2b then ax2 –bx +c = 0 has a root in the interval (A) (0, 2) (B) (2, 4) (C) (0, 1) (D) (-2, 0) 16. The largest negative integer which satisfies is (A) – 4 (B) –3 (C) –1 (D) –2 17. If x2 –4x +log1/2a = 0 does not have two distinct real roots, then maximum value of a is (A) 1/4 (B) 1/ 16 (C) –1/4 (D) none of these 18. If |x-2|+|x-9|=7, then the set values of x is (A) {2, 9} (B) (2, 7) (C) {2} (D) [2, 9] 19. 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) –3 (C) 2 (D) None of these 20. If ax2 +bx + 6 =0 does not have two distinct real roots, then the least value of 3a+b is (A) 2 (B) –2 (C) 1 (D) –1 21. If the quadratic equation αx2 +βx+a2+b2+c2 – ab – bc – ca = 0 has imaginary roots, then (A) 2 (α - β) +(a - b)2 +(b - c)2 + (c - a)2 > 0 (B) 2 (α - β) +(a - b)2 +(b - c)2 + (c - a)2 < 0 (C) 2 (α - β) +(a - b)2 +(b - c)2 + (c - a)2 = 0 (D) none of these . 22. The roots α and β of the quadratic equation ax2 +bx +c = 0 are real and of opposite sign. Then the roots of the equation α(x-β)2 + β(x-α)2 =0 are (A) positive (B) negative (C) Real and of opposite sign (D) imaginary 23. If p, q, r are real and p ≠q, then the roots of the equation (p -q)x2+5(p +q)x -2(p -q)=0 are (A) real and equal (B) complex (C) real and unequal (D) none of these. 24. The real roots of the equation |x|3 – 3x2 +3|x| - 2 =0 are (A) 0, 2 (B) ± 1 (C) ± 2 (D) 1, 2 25. If the equations x2+ax+b=0 and x2+bx+a=0 have exactly one common root, then the numerical value of a + b is (A) 1 (B) –1 (C) 0 (D) none of these 26. If r be the ratio of the roots of the equation ax2 + bx + c = 0, then is equal to (A) (B) (C) (D) none of these 27. The set of values of a for which the inequality x2+ax+a2+6a<0 is satisfied for all x ∈ (1, 2) lies in the interval (A) (1, 2) (B) [1, 2] (C) [-7, 4] (D) None of these 28. If the product of the roots of the equation x2-3kx+2e2lnk-1=0 is 7, then for real roots the value of k is equal to (A) 1 (B) 2 (C) 3 (D) 4 29. If 2 + i√3 is a root of x2 + px + q = 0, where p, q are real, then (p, q) is equal to (A) (-4, 7) (B) (4, - 7) (C) (-7 4) (D) (4, 7) 30. The equations ax2 + bx + a = 0, x3 – 2x2 + 2x – 1 = 0 have two roots in common. Then a + b must be equal to (A) 1 (B) –1 (C) 0 (D) none of these