Question

Consider the quadratic equation $x^2+2x\sin a+\cos a+1=0,a\in R.$ If
$D$ denotes the discriminant of the given quadratic equation, then which of the following is/are correct?

• A. Maximum value of $D$ is $8$
• B. Minimum value of $D$ is $-8$
• C. Maximum value of $D$ occurs when $\cos a=0$
• D. Maximum value of $D$ occurs when $\cos a=-\frac{1}{2}$

Answer 1

Answer: (B), (D)

The correct options are
B Minimum value of $D$ is $-8$
D Maximum value of $D$ occurs when $\cos a=-\frac{1}{2}$

$x^2+2x\sin a+\cos a+1=0$
$D=4{\sin }^{2}a-4\cos a-4=4-4{\cos }^{2}a-4\cos a-4=1-4({\cos }^{2}a+\cos a+\frac{1}{4})=1-4{(\cos a+\frac{1}{2})}^2$

Now,
$-1\le \cos a\le 1\Rightarrow -\frac{1}{2}\le \cos a\le \frac{3}{2}\Rightarrow \frac{9}{4}\ge {(\cos a+\frac{1}{2})}^2\ge 0\Rightarrow 9\ge 4{(\cos a+\frac{1}{2})}^2\ge 0\Rightarrow -8\le 1-4{(\cos a+\frac{1}{2})}^2\le 1\Rightarrow -8\le D\le 1$

So, $D_{min}=-8$, when $\cos a=1$

So, $D_{max}=1$, when $\cos a=-\frac{1}{2}$