Question

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

• A. $-1$
• B. $0$
• C. $a^6$
• D. independent of $a$

Answer 1

Answer: (B), (D)

$0$
independent of $a$

$f\left(x\right)=\left|\begin{array}{ccc}{x}^n& \sin x& -\cos x\\ n!& \sin \left(\frac{n\pi }{2}\right)& \cos \left(\frac{n\pi }{2}\right)\\ a& a^2& a^3\end{array}\right|$
$\Rightarrow f^n\left(x\right)=\left|\begin{array}{ccc}n!& \sin (x+\frac{n\pi }{2})& -\cos (x+\frac{n\pi }{2})\\ n!& \sin \left(\frac{n\pi }{2}\right)& \cos \left(\frac{n\pi }{2}\right)\\ a& a^2& a^3\end{array}\right|$
$\Rightarrow f^n\left(0\right)=\left|\begin{array}{ccc}n!& \sin \left(\frac{n\pi }{2}\right)& -\cos \left(\frac{n\pi }{2}\right)\\ n!& \sin \left(\frac{n\pi }{2}\right)& \cos \left(\frac{n\pi }{2}\right)\\ a& a^2& a^3\end{array}\right|$
$f^n\left(0\right)=\left|\begin{array}{ccc}n!& \sin \left(\frac{n\pi }{2}\right)& 0\\ n!& \sin \left(\frac{n\pi }{2}\right)& 0\\ a& a^2& a^3\end{array}\right|$ ($\because \cos \left(\frac{n\pi }{2}\right)=0$ for $n=2m+1$)
$=0$