Aspire Faculty ID #18338 · Topic: AMU MCA 2017 · Just now
AMU MCA 2017

If $y = \sin(mx)$, then the value of the determinant $\begin{vmatrix} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{vmatrix}$ where $y_n = \dfrac{d^n y}{dx^n}$ is

Solution

$y = \sin(mx)$ $y_1 = m\cos(mx), \quad y_2 = -m^2\sin(mx)$ $y_3 = -m^3\cos(mx), \quad y_4 = m^4\sin(mx), \quad y_5 = m^5\cos(mx)$ $y_6 = -m^6\sin(mx), \quad y_7 = -m^7\cos(mx), \quad y_8 = m^8\sin(mx)$ $\Delta = \begin{vmatrix} \sin(mx) & m\cos(mx) & -m^2\sin(mx) \\ -m^3\cos(mx) & m^4\sin(mx) & m^5\cos(mx) \\ -m^6\sin(mx) & -m^7\cos(mx) & m^8\sin(mx) \end{vmatrix}$ $= -m^6 \begin{vmatrix} \sin(mx) & m\cos(mx) & -m^2\sin(mx) \\ -m^3\cos(mx) & m^4\sin(mx) & m^5\cos(mx) \\ \sin(mx) & m\cos(mx) & -m^2\sin(mx) \end{vmatrix}$ Since row 1 and row 3 are identical, $\Delta = 0$
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