# Practice Zone Maths : Quadratic Equations & Expressions

11. The number of ordered pairs (a, b) such that the equations ax + by = 1 and xλ + yλ = 50 have all solutions integral is

(A) 72

(B) 66

(C) 84

(D) 36

12. If the roots of the equation (a2 + b2) x2 + 2x (ac + bd) + c2 + d2 = 0, are real, then these are equal. This statement is (a, b, c, d ∈ R)

(A) true

(B) false

(C) can’t say

(D) none of these

13. If equation x2 – (2 + m)x + (m2 – 4m + 4) = 0 has coincident roots then

(A) m = 0, 1

(B) m = 2/3 , 1

(C) m = 0, 2

(D) m = 2/3 , 6

14. A root of the equation, sinx + x – 1 = 0, lies in the interval

(A) (0, π/2)

(B) (- π/2, 0)

(C) (π/2, π)

(D) ( -π, -π/2)

15. If p, q, r ∈ R and the quadratic equation px2 + qx + r = 0 has no real root, then

(A) p(p + q + r) < 0

(B) p(p – 2q + 4r)

(C) p(p + 4q + 2r) < 0

(D) None of these

16. If x2 – 4x + log1/2 a = 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

17. The least value of |a| for which tan q and cot q are the roots of the equation x2 + ax + b = 0 is

(A) 2

(B) 1

(C) 1/2

(D) 0

18. If the equation x3 – 3ax2 + 3bx – c = 0 has positive and distinct roots, then

(A) a2 > b

(B) ab > c

(C) a3 > c

(D) a3 > b2 > c

19. The value of a for which exactly one root of the equation eax2 – e2ax + ea – 1 = 0 lies between 1 and 2 are given by

(A) (B) 0 < a < 100

(C) (D) None of these

20. If α, β be the roots of x2 – a(x – 1) – b = 0, then the value of is

(A) 4/(a+b)

(B) 1/(a+b)

(C) 0

(D) 1