Question

If $y=\sin mx$, then the value of the determinant $\left|\begin{array}{ccc}y& y_1& y_2\\ y_3& y_4& y_5\\ y_6& y_7& y_8\end{array}\right|$ where $y_n=\frac{d^{n}y}{d{x}^n}$, is

• A. $m^9$
• B. $m^2$
• C. $m^3$
• D. none of these

Answer 1

The correct option is D none of these

we have $y=\sin mx$, therefore $y_1=m\cos mx$, $y_2=-m^2\sin mx$, etc

$\Delta =\left|\begin{array}{ccc}y& y_1& y_2\\ y_3& y_4& y_5\\ y_6& y_7& y_8\end{array}\right|$

$=\left|\begin{array}{ccc}\sin mx& m\cos x& -m^2\sin mx\\ -m^3\cos x& m^4\sin mx& m^5\cos mx\\ -m^3\sin mx& -m^7\cos x& m^8\sin mx\end{array}\right|$

$=m^{12}\left|\begin{array}{ccc}\sin mx& \cos x& -\sin mx\\ -\cos x& \sin mx& \cos mx\\ -\sin mx& -\cos x& \sin mx\end{array}\right|=0$