JEE Math Practice Question

Let \[ M=\left[\begin{array}{cc} \sin ^{4} \theta & -1-\sin ^{2} \theta \\ 1+\cos ^{2} \theta & \cos ^{4} \theta \end{array}\right]=\alpha I+\beta M^{-1} \] where $\alpha=\alpha(\theta)$ and $\beta=\beta(\theta)$ are real numbers, and $I$ is the $2 \times 2$ identity matrix. If $\alpha^{*}$ is the minimum of the set $\{\alpha(\theta): \theta \in[0,2 \pi)\}$ and $\beta^{*}$ is the minimum of the set $\{\beta(\theta): \theta \in[0,2 \pi)\}$ then the value of $\alpha^{*}+\beta^{*}$ is

1. A.$-\frac{37}{16}$
2. B.$-\frac{31}{16}$
3. C.$-\frac{29}{16}$
4. D.$-\frac{17}{16}$