If y- cos ax- then y y1 y2 y3 y4 y5 y6 y7 y8 - where yn-dnydxn

Given, y=cos (ax) Then, y_1= a sin(ax) = a cos ( π2+ax ) y_2= a^2 cos(ax) = a^2 cos ( 2π2+ax ) y_3=+a^3 sin(ax) = a^3 cos ( 3π2+ax ) . . y_n= a^n cos ( nπ2+a ...

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Standard XII

Mathematics

Equations Involving sinx + cosx and sinx.cosx

If y= cos ax,...

Question

If $y = cos a x, then ∣ ∣ ∣ ∣ \begin{matrix} y & y_{1} & y_{2} y_{3} & y_{4} & y_{5} y_{6} & y_{7} & y_{8} \end{matrix} ∣ ∣ ∣ ∣ =$ (where $y_{n} = \frac{d^{n} y}{d x^{n}}$ )

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Solution

The correct option is A $0$
Given, $y = cos (a x)$
Then, $y_{1} = - a sin (a x) = a cos (\frac{π}{2} + a x)$
$y_{2} = - a^{2} cos (a x) = a^{2} cos (\frac{2 π}{2} + a x)$
$y_{3} = + a^{3} sin (a x) = a^{3} cos (\frac{3 π}{2} + a x)$
.
.
$y_{n} = a^{n} cos (\frac{n π}{2} + a x)$

$∴$ The determinant $Δ (x) = ∣ ∣ ∣ ∣ \begin{matrix} y & y_{1} & y_{2} y_{3} & y_{4} & y_{5} y_{6} & y_{7} & y_{8} \end{matrix} ∣ ∣ ∣ ∣$

$Δ (x) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} cos a x & a cos (\frac{π}{2} + a x) & a^{2} cos (\frac{2 π}{2} + a x) a^{3} cos (\frac{3 π}{2} + a x) & a^{4} cos (\frac{4 π}{2} + a x) & a^{5} cos (\frac{5 π}{2} + a x) a^{6} cos (\frac{6 π}{2} + a x) & a^{7} cos (\frac{7 π}{2} + a x) & a^{8} cos (\frac{8 π}{2} + a x) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$

$Δ (x) = a^{3} \times a^{6} ∣ ∣ ∣ ∣ ∣ \begin{matrix} cos (a x) & - a sin (a x) & - a^{2} cos (a x) sin (a x) & a cos (a x) & - a^{2} sin (a x) - cos (a x) & a sin (a x) & a^{2} cos (a x) \end{matrix} ∣ ∣ ∣ ∣ ∣$

$R_{1} \to R_{1} + R_{3}$
This gives $Δ (x) = a^{9} \times (0)$

Hence, $Δ (x) = 0$

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If $y = cos a x, then ∣ ∣ ∣ ∣ \begin{matrix} y & y_{1} & y_{2} y_{3} & y_{4} & y_{5} y_{6} & y_{7} & y_{8} \end{matrix} ∣ ∣ ∣ ∣ =$ (where $y_{n} = \frac{d^{n} y}{d x^{n}}$ )