Given a , b , c ∈ R. Statement-1:If c

Question

Given a , b , c ∈ R. Statement-1:If c<0 b2-4 a c<0 then 4 a+c>2 b. Statement-2: If D <0, then ax 2+ bx + c takes either positive value only or negative value only where D is the discriminant of quadratic expression. (A) Statement-1 is true, statement- 2 is true and statement-2 is correct explanation for statement-1. (B) Statement-1 is true, statement-2 is true and statement- 2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement- 2 is false. (D) Statement-1 is false, statement- 2 is true.

Tardigrade Admin · Accepted Answer

Let f(x)=a x2+b x+c Now, f (0)= c = negative ( given) D = b 2-4 ac = negative ( given) ⇒ f ( x ) takes negative value only ∀ x ∈ R ; ∴ f (-2)<0 ⇒ 4 a-2 b+c<0 ; ∴ 4 a+c<2 b ⇒ Option (D) is correct

Q. Given $a, b, c \in R$ .
Statement-1:If $c < 0& b^{2} - 4 a c < 0$ then $4 a + c > 2 b$ .
Statement-2 : If $D < 0$ , then $a x^{2} + b x + c$ takes either positive value only or negative value only where $D$ is the discriminant of quadratic expression.

Statement-1 is true, statement- 2 is true and statement-2 is correct explanation for statement-1.

Statement-1 is true, statement-2 is true and statement- 2 is NOT the correct explanation for statement-1.

Statement-1 is true, statement- 2 is false.

Statement-1 is false, statement- 2 is true.

Let $f (x) = a x^{2} + b x + c$
Now, $f (0) = c =$ negative ( given) & $D = b^{2} - 4 a c =$ negative ( given)
$\Rightarrow f (x)$ takes negative value only $\forall x \in R; ∴ f (- 2) < 0$
$\Rightarrow 4 a - 2 b + c < 0; ∴ 4 a + c < 2 b \Rightarrow$ Option (D) is correct

Q. Given a,b,c∈R. Statement-1:If c<0&b2−4ac<0 then 4a+c>2b. Statement-2 : If D<0, then ax2+bx+c takes either positive value only or negative value only where D is the discriminant of quadratic expression.