Question

Let $\alpha =\frac{cos 85 ^\circ sin 55 ^\circ sin 65 ^\circ }{cos 5 ^\circ sin 35 ^\circ sin 25 ^\circ }$ is a root of the quadratic equation $2x^{2}-px+q=0$ , where $p,q\in Q$ , then the value of $\left(p + q\right)$ is

• A. $2$
• B. $5$
• C. $8$
• D. $10$

Answer 1

Answer: (D)

$\alpha =\frac{cos 85 ^\circ sin 55 ^\circ sin 65 ^\circ }{cos 5 ^\circ sin 35 ^\circ sin 25 ^\circ }=\frac{sin 5 ^\circ sin 55 ^\circ sin 65 ^\circ }{cos 5 ^\circ cos 55 ^\circ cos 65 ^\circ }$
$=\frac{\frac{1}{4} sin 15 ^\circ }{\frac{1}{4} cos 15 ^\circ }=tan15^\circ $
$=2-\sqrt{3}$
Other root will be $2+\sqrt{3}\left[p , q \in Q\right]$
So, required equation is
$x^{2}-\left(4 x\right)+1=0$
$2x^{2}-8x+2=0$
$\therefore p+q=10$