Question

Assertion :The determinant $\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1\\ \sin \alpha & \cos \alpha & 1\\ cos(\alpha +\beta )& -\sin (\alpha +\beta )& 1\end{array}\right|$ is independent of $\alpha$ Reason: If $f\left(\alpha \right)=\lambda ,$ then $f\left(\alpha \right)$ is independent of $\alpha$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R} is false,
• D. (A)is false but (R} is true.

Answer 1

The correct option is B Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

Let $f(\alpha ,\beta )=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1\\ \sin \alpha & \cos \alpha & 1\\ cos(\alpha +\beta )& -\sin (\alpha +\beta )& 1\end{array}\right|$
Applying $R_3\to R_3-R_1\left(\cos \beta \right)+R_2\sin \beta$
$\therefore f(\alpha ,\beta )=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1\\ \sin \alpha & \cos \alpha & 1\\ 0& 0& 1+\sin \beta -\cos \beta \end{array}\right|$
$=(1+\sin \beta -\cos \beta )({\cos }^2\alpha +{\sin }^2\alpha )$
$f(\alpha ,\beta )=f\left(\beta \right)=1+\sin \beta -\cos \beta$ which is independent of $\alpha$
Also, $f\left(\alpha \right)=\lambda$ which means f is independent of $\alpha$
But reason is not the correct explanation of assertion