Question

Let $F\left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \sin x\\ \cos x& -\sin x& \cos x\\ x& 1& 1\end{array}\right|$. Which of the following statement hold true?

• A. Range of $F\left(x\right)$ is $(-\infty ,\infty )$
• B. $F^{\prime }\left(\frac{\pi }{2}\right)=0$
• C. $F\left(x\right)$ is bounded
• D. $F\left(x\right)$ is continuous and differentiable every where in its domain.

Answer 1

Answer: (A), (D)

The correct options are
A Range of $F\left(x\right)$ is $(-\infty ,\infty )$
D $F\left(x\right)$ is continuous and differentiable every where in its domain.

Expanding the given determinant along first column we get,
$F\left(x\right)=\sin x(\cos x+\sin x)-\cos x(\cos x-\sin x)+x({\cos }^{2}x+{\sin }^{2}x)$
$\Rightarrow F\left(x\right)={\sin }^{2}x-{\cos }^{2}x+2\sin x\cos x+x=x+\sin 2x-\cos 2x$
$\Rightarrow F^{\prime }\left(x\right)=1+2\cos 2x+2\sin 2x$
Clearly range of $F\left(x\right)$ is $(-\infty ,\infty )$. Hence it is not bounded.
Also $F^{\prime }\left(\pi /2\right)=-1\ne 0$