Question

If $f\left(x\right)=\left|\begin{array}{ccccc}{x}^n& & \sin x& & \cos x\\ n!& & \sin \frac{n\pi }{2}& & \cos \frac{n\pi }{2}\\ a& & a^2& & a^3\end{array}\right|$, then the value of $\frac{d^n}{d{x}^n}\left(f\right(x\left)\right)$ at $x=0$ for $n=2m+1$ is

• A. dependent on $n$
• B. $0$
• C. $1$
• D. dependent on $a$

Answer 1

The correct option is B $0$

We have,
$f\left(x\right)=\left|\begin{array}{ccccc}{x}^n& & \sin x& & \cos x\\ n!& & \sin \frac{n\pi }{2}& & \cos \frac{n\pi }{2}\\ a& & a^2& & a^3\end{array}\right|$
$\Rightarrow \frac{d^n}{d{x}^n}\left(f\right(x\left)\right)=\left|\begin{array}{ccccc}n!& & \sin \left(\frac{n\pi }{2}+x\right)& & \cos \left(\frac{n\pi }{2}+x\right)\\ n!& & \sin \frac{n\pi }{2}& & \cos \frac{n\pi }{2}\\ a& & a^2& & a^3\end{array}\right|$
$\Rightarrow {\left\{\frac{d^n}{d{x}^n}\left(f\right(x\left)\right)\right\}}_{x=0}=\left|\begin{array}{ccccc}n!& & \sin \frac{n\pi }{2}& & \cos \frac{n\pi }{2}\\ n!& & \sin \frac{n\pi }{2}& & \cos \frac{n\pi }{2}\\ a& & a^2& & a^3\end{array}\right|=0$ (Since the two rows are the same.)