Question

Consider $P ( x )= x ^2+(\sin \phi-1) x -\frac{1}{2} \cos ^2 \phi$. Let $\alpha$ and $\beta$ be unequal real roots of the equation $P(x)=0$ such that $\left(\alpha^2+\beta^2\right)$ has the maximum value, then non-negative difference of the roots of $P ( x )=0$, is

• A. 1
• B. 2
• C. 0
• D. 4

Answer 1

Answer: (B)

$\Theta$ Equation has unequal real roots
$\therefore D >0 \Rightarrow(\sin \phi-1)^2-4 \cdot 1\left(\frac{-1}{2} \cos ^2 \phi\right)>0 $
$\Rightarrow \sin ^2 \phi-2 \sin \phi+1+2\left(1-\sin ^2 \phi\right)>0 $
$\Rightarrow-\sin ^2 \phi-2 \sin \phi+3>0$
$\Rightarrow(\sin \phi+3)(\sin \phi-1)<0$
$\Rightarrow \sin \phi \neq 1$
$\Theta \alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta=(\sin \phi-1)^2-2 \cdot 1 \cdot\left(\frac{-1}{2} \cos ^2 \phi\right)=2-2 \sin \phi$
$\therefore$ maximum value $=4$ when $\sin \phi=-1$
$\therefore \cos \phi=0$
$\therefore P ( x )= x ^2-2 x = x ( x -2)$
$\therefore$ Difference of roots $=2$