Question

Suppose that for some angles x and y the equations ${\sin }^{2}x+{\cos }^{2}y=\frac{3a}{2}$ and ${\cos }^{2}x+{\sin }^{2}y=\frac{a^2}{2}$ hold simultaneously. The possible values of a is

• A. $\frac{1}{2}$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

${\sin }^{2}x+{\cos }^{2}y=\frac{3a}{2}$ ...(1)
${\cos }^{2}x+{\sin }^{2}y=\frac{a^2}{2}$ ...(2)
Adding (1) and (2)
${\sin }^{2}x+{\cos }^{2}y+{\cos }^{2}x+{\sin }^{2}y=\frac{3a}{2}+\frac{a^2}{2}$
$\Rightarrow 2=\frac{3a}{2}+\frac{a^2}{2}\Rightarrow a^2+3a-4=0$
$\Rightarrow a^2+4a-a-4=0\Rightarrow (a+4)(a-1)=0\Rightarrow a=1,a=-4$