Question

Let $A=\begin{bmatrix} cos \alpha & sin ⁡ \alpha \\ -sin ⁡ \alpha & cos ⁡ \alpha \end{bmatrix}$ and matrix $B$ is defined such that $B=A+3A^{2}+3A^{3}+A^{4}$ . If $\left|B\right|=8$ , then the number of values of $\alpha $ in $\left[0 , \, 10 \pi \right]$ is

• A. $10$
• B. $12$
• C. $5$
• D. $3$

Answer 1

Answer: (A)

$B=A\left(I + 3 A + 3 A^{2} + A^{3}\right)=A\left(I + A\right)^{3}$
$\left|B\right|=\left|A\right| \, \left(\left|I + A\right|\right)^{3}=8\ldots \left(i\right)$
$A=\begin{vmatrix} cos \alpha & sin⁡\alpha \\ -sin⁡\alpha & cos⁡\alpha \end{vmatrix}=cos^{2}\alpha +sin^{2}\alpha =1$
$\left|I + A\right|=\begin{vmatrix} 1+cos \alpha & sin ⁡ \alpha \\ -sin ⁡ \alpha & 1+cos ⁡ \alpha \end{vmatrix}=\left(1 + cos ⁡ \alpha \right)^{2}+\left(s i n\right)^{2}\alpha \, =2+2cos ⁡ \alpha $
From $\left(i\right)$
$1\left(2 + 2 cos \alpha \right)^{3}=8\Rightarrow 2+2cos ⁡ \alpha =2$
$\Rightarrow cos \alpha =0$
$\alpha =\frac{\pi }{2}, \, \frac{3 \pi }{2}, \, \frac{5 \pi }{2}, \, \frac{7 \pi }{2}, \, \frac{9 \pi }{2}, \, \frac{11 \pi }{2}, \, \frac{13 \pi }{2},\frac{15 \pi }{2},\frac{17 \pi }{2}, \, \frac{19 \pi }{2}$
Total number of values $=10$