Question

Consider the system of equations $x{\cos }^{3}y+3x\cos y{\sin }^{2}y=14$ ; $x{\sin }^{3}y+3x{\cos }^{2}y\sin y=13$. The value of ${\sin }^{2}y+2{\cos }^{2}y$ is

• A. $\frac{4}{5}$
• B. $\frac{9}{5}$
• C. $2$
• D. none of these

Answer 1

The correct option is B $\frac{9}{5}$

$x{\cos }^{3}y+3x\cos y{\sin }^{2}y=14$ ...(1)
$x{\sin }^{3}y+3x{\cos }^{2}y\sin y=13$ ...(2)
Adding (1) and (2)
$x{(\cos y+\sin y)}^3=27$
Subtracting (2) from (1)
$x{(\cos y-\sin y)}^3=1$
$\frac{\cos y+\sin y}{\cos y-\sin y}=\frac{3}{1}$

Divide numerator and denominator by $\cos y$
$\Rightarrow \tan y=\frac{1}{2}\Rightarrow \sin y=\pm \frac{1}{\sqrt{5}}\Rightarrow \cos y=\pm \frac{2}{\sqrt{5}}$....{Take a triangle with $\tan y=\frac{1}{2}$ find other sides of the triangle we get $\sin y$ and $\cos y$}
$\therefore {\sin }^{2}y+2{\cos }^{2}y={(\pm \frac{1}{\sqrt{5}})}^2+2{(\pm \frac{2}{\sqrt{5}})}^2=\frac{9}{5}$