Question

If $\alpha=\sin \left(\cot ^{-1}\left(\tan \left(\cos ^{-1} \frac{2}{3}\right)\right)\right)$ and $\beta=\sin \left(\operatorname{cosec}^{-1}\left(\cot \left(\tan ^{-1} \frac{1}{3}\right)\right)\right)$ are the roots of the quadratic equation $ax ^2+ bx + c =0$ where $a , b , c$ are integers and $c$ is prime then the value of $( a + b + c )$ equals

• A. 1
• B. 2
• C. 3
• D. 9

Answer 1

Answer: (B)

$\alpha=\frac{2}{3} ; \beta=\frac{1}{3}$.
Hence the quadratic equation whose roots are $\alpha$ and $\beta$.
$x ^2- x +\frac{2}{9}=0 \Rightarrow 9 x ^2-9 x +2=0 $
$a + b + c =2 $