Let f(x) = begin{vmatrix} cos x & sin x & cos | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given $f(x) = \begin{vmatrix} \cos x & \sin x & \cos x \ \cos 2x & \sin 2x & 2\cos 2x \ \cos 3x & \sin 3x & 3\cos 3x \end{vmatrix}$.Using the pr

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(D) Given $f(x) = \begin{vmatrix} \cos x & \sin x & \cos x \ \cos 2x & \sin 2x & 2\cos 2x \ \cos 3x & \sin 3x & 3\cos 3x \end{vmatrix}$.Using the property of differentiation of a determinant,$f'(x) = \Delta_1 + \Delta_2 + \Delta_3$,where $\Delta_i$ is the determinant obtained by differentiating the $i$-th column.$\Delta_1 = \begin{vmatrix} -\sin x & \sin x & \cos x \ -2\sin 2x & \sin 2x & 2\cos 2x \ -3\sin 3x & \sin 3x & 3\cos 3x \end{vmatrix} = 0$ (since $C_1$ and $C_2$ are proportional at $x = \pi/2$ is not directly obvious,but evaluating at $x = \pi/2$ gives $\Delta_1 = 0$).$\Delta_2 = \begin{vmatrix} \cos x & \cos x & \cos x \ \cos 2x & 2\cos 2x & 2\cos 2x \ \cos 3x & 3\cos 3x & 3\cos 3x \end{vmatrix} = 0$ (since $C_2$ and $C_3$ are proportional).$\Delta_3 = \begin{vmatrix} \cos x & \sin x & -\sin x \ \cos 2x & \sin 2x & -4\sin 2x \ \cos 3x & \sin 3x & -9\sin 3x \end{vmatrix}$.At $x = \pi/2$,$\cos(\pi/2) = 0, \sin(\pi/2) = 1, \cos(\pi) = -1, \sin(\pi) = 0, \cos(3\pi/2) = 0, \sin(3\pi/2) = -1$.$f'(\pi/2) = \Delta_3(\pi/2) = \begin{vmatrix} 0 & 1 & -1 \ -1 & 0 & 0 \ 0 & -1 & 9 \end{vmatrix} = -1 \begin{vmatrix} -1 & 0 \ 0 & 9 \end{vmatrix} - 1 \begin{vmatrix} -1 & 0 \ 0 & -1 \end{vmatrix} = -1(-9) - 1(1) = 9 - 1 = 8$. Wait,re-evaluating: $\Delta_3 = \begin{vmatrix} 0 & 1 & -1 \ -1 & 0 & 0 \ 0 & -1 & 9 \end{vmatrix} = 0 - 1(-9 - 0) + (-1)(1 - 0) = 9 - 1 = 8$. The correct value is $8$.

Let $f(x) = \begin{vmatrix} \cos x & \sin x & \cos x \\ \cos 2x & \sin 2x & 2\cos 2x \\ \cos 3x & \sin 3x & 3\cos 3x \end{vmatrix}$. Then $f'\left(\frac{\pi}{2}\right) = $

Similar Questions

Let $f(x) = \left|\begin{array}{ccc} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{array}\right|$,where $a \in R$. Then the sum of the squares of all the values of $a$ for which $2f'(10) - f'(5) + 100 = 0$ is:

If $f(x) = \begin{vmatrix} x^3-x & 2e^{2x} & \sin x^2 \\ \cos(2x) & x+x^2 & e^{-x} \\ \tan 3x & \ln(1-2x) & x^2+x+1 \end{vmatrix}$,then $f'(0)$ is equal to:

If $f(x) = \left| \begin{array}{ccc} 2 \cos x & 1 & 0 \\ x - \frac{\pi}{2} & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x \end{array} \right|$, then $f^{\prime}(\pi)$ is equal to

If $f(x) = \left| \begin{array}{ccc} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{array} \right|$,then find $\lim_{x \to 0} \frac{f'(x)}{x}$.

The rank of the matrix $\left[\begin{array}{cccc}3 & 2 & 1 & -4 \\ 2 & 3 & 0 & -1 \\ 1 & -6 & 3 & -8\end{array}\right]$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module