Question

Assertion: $\left|\begin{array}{ccc}\cos (\theta +\alpha )& \cos (\theta +\beta )& \cos (\theta +\gamma )\\ \sin (\theta +\alpha )& \sin (\theta +\beta )& \sin (\theta +\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$ is independent of $\theta$
Reason: If $f(\theta )=c$, then $f(\theta )$ is independent of $\theta$.

• A. Assertion is correct, reason is correct; reason is a correct explanation for assertion.
• B. Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• C. Assertion is correct, reason is incorrect
• D. Assertion is incorrect, reason is correct.

Answer 1

Answer: (A)

Let $f(\theta )=\left|\begin{array}{ccc}\cos (\theta +\alpha )& \cos (\theta +\beta )& \cos (\theta +\gamma )\\ \sin (\theta +\alpha )& \sin (\theta +\beta )& \sin (\theta +\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$
$\therefore f^{\prime }(\theta )=\left|\begin{array}{ccc}-\sin (\theta +\alpha )& -\sin (\theta +\beta )& -\sin (\theta +\gamma )\\ \sin (\theta +\alpha )& \sin (\theta +\beta )& \sin (\theta +\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$
$+\left|\begin{array}{ccc}\cos (\theta +\alpha )& \cos (\theta +\beta )& \cos (\theta +\gamma )\\ \cos (\theta +\alpha )& \cos (\theta +\beta )& \cos (\theta +\gamma )\\ \sin (\beta -\gamma )& \sin (\gamma -\alpha )& \sin (\alpha -\beta )\end{array}\right|$
$+\left|\begin{array}{ccc}\cos (\theta +\alpha )& \cos (\theta +\beta )& \cos (\theta +\gamma )\\ \sin (\theta +\alpha )& \sin (\theta +\beta )& \sin (\theta +\gamma )\\ 0& 0& 0\end{array}\right|$
$=0+0+0=0$
$\Rightarrow f^{\prime }(\theta )=0$
$\Rightarrow f(\theta )=c$