Question

Let $\lambda$ and $\alpha$ be real. The set of all values of $\lambda$ for which the system of linear equations

$\lambda x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0x+\left(\sin \alpha \right)y-\left(\cos \alpha \right)z=0-x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0$

has a non-trivial solution is

• A. $[0,\sqrt{2}]$
• B. $[-\sqrt{2},0]$
• C. $[-\sqrt{2},\sqrt{2}]$
• D. None of these

Answer 1

Answer: (C)

$[-\sqrt{2},\sqrt{2}]$

Since the system has a non-trivial solution, therefore

$\left|\begin{array}{ccc}\lambda & \sin \alpha & \cos \alpha \\ 1& \cos \alpha & \sin \alpha \\ -1& \sin \alpha & -\cos \alpha \end{array}\right|=0$

$\Rightarrow \lambda (-{\cos }^2\alpha -{\sin }^2\alpha )-(-\sin \alpha \cos \alpha -\sin \alpha \cos \alpha )-({\sin }^2\alpha -{\cos }^2\alpha )=0$

$\Rightarrow -\lambda +\sin 2\alpha +\cos 2\alpha =0\Rightarrow \lambda =\sin 2\alpha +\cos 2\alpha$

$\Rightarrow \lambda =\sqrt{2}\cos (2\alpha -\frac{\pi }{4})\le 1\forall \alpha \in R$

$\therefore -\sqrt{2}\le \lambda \le \sqrt{2}\Rightarrow \lambda \in [-\sqrt{2},\sqrt{2}]$