Question

Assertion :The system of linear equations $\begin{array}{c}x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0\\ x+\left(\cos \alpha \right)y+\left(\sin \alpha \right)z=0\\ -x+\left(\sin \alpha \right)y-\left(\cos \alpha \right)z=0\end{array}$ has a non trivial solution for only one value of $\alpha$ lying between 0 and $\pi$. Reason: $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ has no solution in the interval $-\frac{\pi }{4}<x<\frac{\pi }{4}.$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given system of linear equations $\begin{array}{c}x+\left(\sin \alpha \right)y+\left(\cos \alpha \right)z=0\\ x+\left(\cos \alpha \right)y+\left(\sin \alpha \right)z=0\\ -x+\left(\sin \alpha \right)y-\left(\cos \alpha \right)z=0\end{array}$
For non-trivial solution,
$\left|\begin{array}{ccc}1& \sin \alpha & \cos \alpha \\ 1& \cos \alpha & \sin \alpha \\ -1& \sin \alpha & -\cos \alpha \end{array}\right|=0$
$\Rightarrow -2{\sin }^2\alpha +2\sin \alpha \cos \alpha =0$
$\Rightarrow$ either $\sin \alpha =0$ or $\tan \alpha =1$
$\Rightarrow \alpha =\frac{\pi }{4}(as0<\alpha <\pi )$
Hence, the statement 1 is true.
For statement 2, given $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$
$C_1\to C_1+C_2+C_3$
$(\sin x+2\cos x)\left|\begin{array}{ccc}1& \cos \alpha & \cos \alpha \\ 1& \sin \alpha & \cos \alpha \\ 1& \cos \alpha & \sin \alpha \end{array}\right|=0$
$\Rightarrow (\sin x+2\cos x)(\cos x-\sin x{)}^2=0$
$\Rightarrow (\sin x+2\cos x)(\cos x-\sin x{)}^2=0$
$\Rightarrow \tan x=-2,\tan x=-1$
Since, $-\frac{\pi }{4}<x<\frac{\pi }{4}$
$\Rightarrow -1<\tan x<1$
Hence, there is no solution for the given system of equations (determinant).
Hence, reason is also true.But reason is not the correct explanation for assertion.