KVPY Government Jobs Exams

Suppose a, b, c are real numbers, and each of the equations x^2 + 2ax + b2= 0 and x^2 + 2bx + c2 = 0 has two distinct
real roots. Then the equation x2 + 2cx + a2 =0 has–
(A) Two distinct positive real roots (B) Two equal roots
(C) One positive and one negative root (D) No real roots

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KVPY Other Question
The pKa of a weak acid is 5.85. The concentrations of the acid and its conjugate base are equal at a pH of –
(A) 6.85 (B) 5.85 (C) 4.85 (D) 7.85 Let (x, y) be a variable point on the curve 4x^2 + 9y^2 – 8x –36y + 15 = 0. Then
min(x ^2 − 2x + y^2 − 4y + 5) + max(x^ 2 − 2x + y^2 − 4y + 5) is