Question

Let $\alpha$ and $\beta$ be 2 real numbers which satisfy the equations $ \cot ^2 \alpha-\operatorname{cosec}^2 \beta=\frac{2 k }{3}-5 \text { and }-\operatorname{cosec}^2 \alpha+\cot ^2 \beta=\frac{ k ^2}{2}, $ then sum of all possible value(s) of $k$ is equal to

• A. $-\frac{4}{3}$
• B. -6
• C. -1
• D. None

Answer 1

Answer: (A)

Adding the 2 equations, we get
$-2=\frac{2 k}{3}-5+\frac{k^2}{2} $
$\frac{2 k}{3}+\frac{k^2}{2}-3=0$
$3 k^2+4 k-18=0$
Hence sum of all values of $k=\frac{-4}{3}$