Statement-1: If all real values of x obtained from the equation 4x –

1.	Statement-1: If all real values of x obtained from the equation 4x – (a – 3) 2x + (a – 4) = 0 are non-positive, then a∈ (4, 5]Statement-2: If ax2 + bx + c is non-positive for all real values of x, then b2 – 4ac must be –ve or zero and ‘a’ must be –ve.(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1.(C) Statement – 1 is True, Statement – 2 is False. (D) Statement – 1 is False, Statement – 2 is True
Answer» Correct option (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1. Explanation: Let f(x) = (x – a) (x – c) + 2 (x – b) (x – d) Then f(a) = 2 (a – b) (a – d) > 0 f(b) = (b – a) (b – c) < 0 f(d) = (d – a) (d – b) > 0 Hence a root of f(x) = 0 lies between a & b and another root lies between (b & d). Hence the roots of the given equation are real and distinct.

Statement-1: If all real values of x obtained from the equation 4x – (a – 3) 2x + (a – 4) = 0 are non-positive, then a∈ (4, 5]Statement-2: If ax2 + bx + c is non-positive for all real values of x, then b2 – 4ac must be –ve or zero and ‘a’ must be –ve.(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1.(C) Statement – 1 is True, Statement – 2 is False. (D) Statement – 1 is False, Statement – 2 is True

Answer»

Correct option (B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1.

Explanation:

Let f(x) = (x – a) (x – c) + 2 (x – b) (x – d)

Then f(a) = 2 (a – b) (a – d) > 0

f(b) = (b – a) (b – c) < 0

f(d) = (d – a) (d – b) > 0

Hence a root of f(x) = 0 lies between a & b and another root lies between (b & d).

Hence the roots of the given equation are real and distinct.