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 $\alpha =\alpha \left(\theta \right)$ and $\beta =\beta \left(\theta \right)$ are real number, and $I$ is the $2\times 2$ identity matrix.

If $\alpha *$ is the minimum of the set $\left\{\alpha \left(\theta \right):\theta \in [0,2\pi ]\right\}$ and $\beta *$ is the minimum of the set $\left\{\beta \left(\theta \right):\theta \in [0,2\pi ]\right\}$ then the value of $\alpha *+\beta *$ is

• A. $-\frac{37}{16}$
• B. $-\frac{29}{16}$
• C. $-\frac{31}{16}$
• D. $-\frac{17}{16}$

Answer 1

The correct option is B

$-\frac{29}{16}$

Explanation for the correct option:

Step 1. Find the value of $\alpha *+\beta *$:

Given, $M=\left[\begin{array}{cc}{\sin }^4\theta & -1-{\sin }^2\theta \\ 1+{\cos }^2\theta & {\cos }^4\theta \end{array}\right]=\alpha I+\beta M^{-1}$

$\begin{array}{rcl}\therefore \left|M\right|& =& {\sin }^4\theta .{\cos }^4\theta +{\sin }^2\theta {\cos }^2\theta +2\\ & =& {\left({\sin }^2\theta {\cos }^2\theta +\frac{1}{2}\right)}^2+\frac{7}{4}\end{array}$

Now,

$\left[\begin{array}{cc}{\sin }^4\theta & -1-{\sin }^2\theta \\ 1+{\cos }^2\theta & {\cos }^4\theta \end{array}\right]=\left[\begin{array}{cc}\alpha & 0\\ 0& \alpha \end{array}\right]+\frac{\beta }{\left|M\right|}\left[\begin{array}{cc}{\cos }^4\theta & 1+{\sin }^2\theta \\ -1-{\cos }^2\theta & {\sin }^4\theta \end{array}\right]$

$\left[\mathbf{\because }{\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\left|A\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\mathbf{;}\mathbf{}\mathit{i}\mathit{f}\mathbf{}\mathit{A}\mathbf{}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right]$

Step 2. Compare the corresponding elements, we get

$\begin{array}{rcl}\therefore \alpha & =& {\cos }^4\theta +\hspace{0.17em}{\sin }^4\theta \\ & =& 1-2{\sin }^2\theta {\cos }^2\theta \\ & =& 1-\frac{1}{2}\left({\sin }^{2}2\theta \right)\end{array}$

$\Rightarrow$$\alpha *=\frac{1}{2}$

and

$\begin{array}{rcl}\beta & =& -\left|M\right|\\ & =& -\left({\sin }^4\theta {\cos }^4\theta +{\sin }^2\theta {\cos }^2\theta +2\right)\\ & =& -{\left({\sin }^2\theta {\cos }^2\theta +\frac{1}{2}\right)}^2-\frac{7}{4}\end{array}$

$\Rightarrow$$\beta *=-\frac{37}{16}$

$\begin{array}{rcl}\therefore \alpha *+\beta *& =& \frac{1}{2}-\frac{37}{16}\\ & =& \mathbf{-}\frac{\mathbf{29}}{\mathbf{16}}\end{array}$

Hence, Option ‘B’ is Correct.