If Delta (x) = left| begin{array}{ccc} x^n & | vedclass

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

Question

(B) We are given the determinant $\Delta (x) = \left| \begin{array}{ccc} x^n & \sin x & \cos x \ n! & \sin \frac{n\pi}{2} & \cos \frac{n\pi}{2} \ a

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(B) We are given the determinant $\Delta (x) = \left| \begin{array}{ccc} x^n & \sin x & \cos x \ n! & \sin \frac{n\pi}{2} & \cos \frac{n\pi}{2} \ a & a^2 & a^3 \end{array} ight|$.To find the $n^{th}$ derivative of the determinant,we differentiate the rows (or columns) one by one. Since only the first row contains $x$,we differentiate the first row $n$ times.$\frac{d^n}{dx^n}[\Delta (x)] = \left| \begin{array}{ccc} \frac{d^n}{dx^n}(x^n) & \frac{d^n}{dx^n}(\sin x) & \frac{d^n}{dx^n}(\cos x) \ n! & \sin \frac{n\pi}{2} & \cos \frac{n\pi}{2} \ a & a^2 & a^3 \end{array} ight|$.We know that $\frac{d^n}{dx^n}(x^n) = n!$,$\frac{d^n}{dx^n}(\sin x) = \sin(x + \frac{n\pi}{2})$,and $\frac{d^n}{dx^n}(\cos x) = \cos(x + \frac{n\pi}{2})$.Substituting these,we get:$\frac{d^n}{dx^n}[\Delta (x)] = \left| \begin{array}{ccc} n! & \sin(x + \frac{n\pi}{2}) & \cos(x + \frac{n\pi}{2}) \ n! & \sin \frac{n\pi}{2} & \cos \frac{n\pi}{2} \ a & a^2 & a^3 \end{array} ight|$.Now,evaluating at $x = 0$:$[\Delta^n(x)]_{x=0} = \left| \begin{array}{ccc} n! & \sin \frac{n\pi}{2} & \cos \frac{n\pi}{2} \ n! & \sin \frac{n\pi}{2} & \cos \frac{n\pi}{2} \ a & a^2 & a^3 \end{array} ight|$.Since the first row $(R_1)$ and the second row $(R_2)$ are identical,the value of the determinant is $0$.

If $\Delta (x) = \left| \begin{array}{ccc} x^n & \sin x & \cos x \\ n! & \sin \frac{n\pi}{2} & \cos \frac{n\pi}{2} \\ a & a^2 & a^3 \end{array} \right|$,then the value of $\frac{d^n}{dx^n}[\Delta (x)]$ at $x = 0$ is

Similar Questions

The rank of $\left[\begin{array}{ccc}2 & 1 & 1 \\ 0 & 3 & -1 \\ 1 & -1 & 1\end{array}\right]$ is

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 the value of $f'(x)$ at $x = 0$ is equal to

If the matrix $A = \begin{bmatrix} 1 & 2 & 3 & 0 \\ 2 & 4 & 3 & 2 \\ 3 & 2 & 1 & 3 \\ 6 & 8 & 7 & \alpha \end{bmatrix}$ is of rank $3$, then $\alpha$ equals to

If ${D_p} = \begin{vmatrix} p & 15 & 8 \\ p^2 & 35 & 9 \\ p^3 & 25 & 10 \end{vmatrix}$,then ${D_1} + {D_2} + {D_3} + {D_4} + {D_5} = $

If $A = \begin{bmatrix} 0 & 1 & 2 & 3 & 4 \\ 0 & 2 & 4 & 8 & 12 \\ 0 & 0 & 0 & 4 & 8 \end{bmatrix}$,then the rank of $A$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module