Question

Let $\lambda$ and $\alpha$ be real. Find the set of all values of $\lambda$ for which the system of linear equations
$\lambda x+\left(sin\alpha \right)y+\left(cos\alpha \right)z=0$
$x+\left(cos\alpha \right)y+\left(sin\alpha \right)z=0$
$-x+\left(sin\alpha \right)y+\left(cos\alpha \right)z=0$
has a non-trivial solution. For $\lambda =1$, find all values of $\alpha$ which are possible

• A. $\pi /8$
• B. $7\pi /8$
• C. $15\pi /8$
• D. $9\pi /8$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\pi /8$
B $7\pi /8$
D $9\pi /8$

To find all values of $\alpha$ which are possible
For $\lambda =1$
$\left[\begin{array}{ccc}1& sin\alpha & cos\alpha \\ 1& cos\alpha & sin\alpha \\ -1& sin\alpha & cos\alpha \end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
On Solving this equation, we get
$2cos2\alpha =0$
$cos2\alpha =0$
$2\alpha =\frac{\pi }{2}$
Finally , we get
$\alpha =\frac{\pi }{8}$

$\alpha =\frac{7\pi }{8}$ and
$\alpha =\frac{9\pi }{8}$