If f(x) = left| begin{array}{ccc} 2 cos x & 1 | vedclass

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

Question

(B) Given $f(x) = \left| \begin{array}{ccc} 2 \cos x & 1 & 0 \ x - \frac{\pi}{2} & 2 \cos x & 1 \ 0 & 1 & 2 \cos x \end{array} ight|$.Using the pr

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) Given $f(x) = \left| \begin{array}{ccc} 2 \cos x & 1 & 0 \ x - \frac{\pi}{2} & 2 \cos x & 1 \ 0 & 1 & 2 \cos x \end{array} ight|$.Using the property of differentiation of a determinant,$f^{\prime}(x)$ is the sum of three determinants where each row is differentiated one at a time:$f^{\prime}(x) = \left| \begin{array}{ccc} -2 \sin x & 1 & 0 \ 1 & 2 \cos x & 1 \ 0 & 1 & 2 \cos x \end{array} ight| + \left| \begin{array}{ccc} 2 \cos x & 0 & 0 \ x - \frac{\pi}{2} & -2 \sin x & 1 \ 0 & 0 & 2 \cos x \end{array} ight| + \left| \begin{array}{ccc} 2 \cos x & 1 & 0 \ x - \frac{\pi}{2} & 2 \cos x & 0 \ 0 & 1 & -2 \sin x \end{array} ight|$.Now,substitute $x = \pi$ into the expression. Since $\sin \pi = 0$ and $\cos \pi = -1$:$f^{\prime}(\pi) = \left| \begin{array}{ccc} 0 & 1 & 0 \ 1 & -2 & 1 \ 0 & 1 & -2 \end{array} ight| + \left| \begin{array}{ccc} -2 & 0 & 0 \ \frac{\pi}{2} & 0 & 1 \ 0 & 0 & -2 \end{array} ight| + \left| \begin{array}{ccc} -2 & 1 & 0 \ \frac{\pi}{2} & -2 & 0 \ 0 & 1 & 0 \end{array} ight|$.Evaluating each determinant:$1$. $\left| \begin{array}{ccc} 0 & 1 & 0 \ 1 & -2 & 1 \ 0 & 1 & -2 \end{array} ight| = 0 - 1(-2 - 0) + 0 = 2$.$2$. $\left| \begin{array}{ccc} -2 & 0 & 0 \ \frac{\pi}{2} & 0 & 1 \ 0 & 0 & -2 \end{array} ight| = 0$ (since the second column is all zeros).$3$. $\left| \begin{array}{ccc} -2 & 1 & 0 \ \frac{\pi}{2} & -2 & 0 \ 0 & 1 & 0 \end{array} ight| = 0$ (since the third column is all zeros).Therefore,$f^{\prime}(\pi) = 2 + 0 + 0 = 2$.

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

Similar Questions

The structure of $IF_7$ is $......$ .

If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$,then the ratio of the final and initial fundamental frequencies is :

Which of the following is the most important cause driving animals and plants to extinction?

The dispersive powers of the materials of two lenses forming an achromatic combination are in the ratio of $4: 3$. If the effective focal length of the two lenses is $+60 ~cm$,then the focal lengths of the lenses should be:

Vedclass Test Series

Exam Paper Generator

Online Exam Module