Question

If $\Delta \left(x\right)=\left|\begin{array}{ccc}{x}^n& sinx& cosx\\ n!& sin\frac{n\pi }{2}& cos\frac{n\pi }{2}\\ a& a^2& a^3\end{array}\right|$, then the value of $\frac{d^n}{d{x}^n}\left[\Delta \right(x\left)\right]atx=0$ is

• A. -1
• B. 1
• C. Dependent of a

Answer 1

$\frac{d^n}{d{x}^n}\left[\Delta \right(x\left)\right]=\left|\begin{array}{ccc}\frac{d^n}{d{x}^n}{x}^n& \frac{d^n}{d{x}^n}sinx& \frac{d^n}{d{x}^n}cosx\\ n!& sin\left(\frac{n\pi }{2}\right)& cos\left(\frac{n\pi }{2}\right)\\ a& a^2& a^3\end{array}\right|=\left|\begin{array}{ccc}n!& sin(x+\frac{n\pi }{2})& cos(x+\frac{n\pi }{2})\\ n!& sin\left(\frac{n\pi }{2}\right)& cos\left(\frac{n\pi }{2}\right)\\ a& a^2& a^3\end{array}\right|\Rightarrow \left[{\Delta }^n\right(x){]}_{x=0}=\left|\begin{array}{ccc}n!& sin(0+\frac{n\pi }{2})& cos(0+\frac{n\pi }{2})\\ n!& sin\left(\frac{n\pi }{2}\right)& cos\left(\frac{n\pi }{2}\right)\\ a& a^2& a^3\end{array}\right|=0\{Since{R}_1\equiv R_2\}.$