Question

If $x=\alpha ,\beta$ satisfies both the equations $cos{}^{2}x+\alpha \cos x+b=0$ and $\sin {}^{2}x+p\sin x+q=0$ then the relation between $\alpha ,b,p$ and $q$ is

• A. $1+b+a^2=p^2-q-1$
• B. $a^2+b^2=p^2+q^2$
• C. $2(b+q)=a^2+p^2-2$
• D. none of these

Answer 1

The correct option is C $2(b+q)=a^2+p^2-2$

Since the first equation has roots $\cos \alpha$ and $\cos \beta$, and the second equation has roots $\sin \alpha$ and $\sin \beta$; we find the sum of the squares of roots in each case.
Then ${\cos }^2\alpha +{\cos }^2\beta =2-{\sin }^2\alpha -{\sin }^2\beta$
$\Rightarrow (-a{)}^2-2b=2-[(-p{)}^2-2q]$
$\Rightarrow a^2+p^2-2=2(b+q)$