Question

Let $f(x)=\begin{vmatrix} \sin 3 x & 1 & 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^{2} \\ \cos 3 x & -1 & 2\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right) \\ \tan 3 x & 4 & 1+2 \tan 3 x \end{vmatrix}$
Then, the value of $f'(x)$ at $x=(2 n+1) \pi, n \in I$ (the set of integers) is equal to

• A. $(-1)^{n}$
• B. $(-1)^{n+1}$
• C. $3$
• D. $9$

Answer 1

Answer: (C)

Given, $f(x)=\begin{vmatrix}
\sin 3 x & 1 & 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^{2} \\
\cos 3 x & -1 & 2\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right) \\
\tan 3 x & 4 & 1+2 \tan 3 x
\end{vmatrix}$
On differentiating w.r.t. $x$, we get
$f'(x)=\begin{vmatrix}\frac{d}{d x}(\sin 3 x) & 1 & 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^{2} \\ \frac{d}{d x}(\cos 3 x) & -1 & 2\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right) \\ \frac{d}{d x}(\tan 3 x) & 4 & 1+2 \tan 3 x\end{vmatrix}$
$+\begin{vmatrix}\sin 3 x & \frac{d}{d x}(1) & 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^{2} \\ \cos 3 x & \frac{d}{d x}(-1) & 2\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right) \\ \tan 3 x & \frac{d}{d x}(4) & 1+2 \tan 3 x\end{vmatrix}$
$+\begin{vmatrix}\sin 3 x & 1 & 2 \frac{d}{d x}\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^{2} \\ \cos 3 x & -1 & 2 \frac{d}{d x}\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right) \\ \tan 3 x & 4 & \frac{d}{d x}(1+2 \tan 3 x)\end{vmatrix}$
$=\begin{vmatrix}3 \cos 3 x & 1 & 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^{2} \\ 3 \sin 3 x & -1 & 2\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right) \\ 3 \sec ^{2} 3 x & 4 & 1+2 \tan 3 x\end{vmatrix}$
$+\begin{vmatrix}\sin 3 x & 0 & 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right)^{2} \\ \cos 3 x & 0 & 2\left(\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right) \\ \tan 3 x & 0 & 1+2 \tan 3 x\end{vmatrix}$
$+\begin{vmatrix}\sin 3 x & 1 & 2 \times 2\left(\cos \frac{3 x}{2}+\sin \frac{3 x}{2}\right) \times\left(\frac{-3}{2} \sin \frac{3 x}{2}+\frac{3}{2} \cos \frac{3 x}{2}\right) \\ \cos 3 x & -1 & 2\left(-2 \cos \frac{3 x}{2} \times \frac{3}{2} \sin \frac{3 x}{2}-2 \sin \frac{3 x}{2} \times \frac{3}{2} \cos \frac{3 x}{2}\right) \\ \tan 3 x & 4 & \left(0+2 \times 3 \sec ^{2} 3 x\right)\end{vmatrix}$
At $x=(2 n+1) \pi$
$f'(x)=\begin{vmatrix}3(-1) & 1 & 2(1) \\ 0 & -1 & 2(-1) \\ 3 & 4 & 1+0\end{vmatrix}+0$
$+\begin{vmatrix}0 & 1 & 4(0-1) \times\left[-\frac{3}{2}(-1)+\frac{3}{2} \times 0\right] \\ -1 & -1 & 0 \\ 0 & 4 & 0\end{vmatrix}$
$=[-3(-1+8)-1(0+6)+2(0+3)]+[0-1(0-0)-6(-4)]$
$=-21+24=3$