Question

If the minimum value of the expression $E(x)=4 \operatorname{cosec}^2 x+\sin ^2 x$ lies between the roots of the quadratic equation $x^2+2 k x+k^2-9=0$, then least integral value of $k$ is

• A. -8
• B. -7
• C. -9
• D. -10

Answer 1

Answer: (B)

$E(x)=\operatorname{cosec}^2 x+\sin ^2 x-2+3 \operatorname{cosec}^2 x+2$
$=(\operatorname{cosec} x-\sin x)^2+3 \operatorname{cosec}^2 x+2 \Rightarrow E_{\min }=5$
$f(x)=x^2+2 k x+k^2-9$
$f(5)< 0 \Rightarrow k^2+10 k+16< 0 $
$k \in(-8,-2)$