If A = begin{bmatrix} a & 0 & 0 0 & b & 0 0 | vedclass

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

Question

(C) The matrix $A$ is a diagonal matrix,so its determinant $|A|$ is the product of its diagonal elements: $|A| = a 	imes b 	imes c = 7^x 	imes 7^{7

Vedclass · Accepted Answer

$\frac{7^{7^{7^x}}}{(\log 7)^3} + k$,where $k$ is constant of integration

Vedclass · Answer

(C) The matrix $A$ is a diagonal matrix,so its determinant $|A|$ is the product of its diagonal elements: $|A| = a 	imes b 	imes c = 7^x 	imes 7^{7^x} 	imes 7^{7^{7^x}}$. Using the property of exponents $a^m 	imes a^n = a^{m+n}$,we have: $|A| = 7^{x + 7^x + 7^{7^x}}$. We need to evaluate the integral $I = \int 7^{x + 7^x + 7^{7^x}} \, dx$. This integral does not have a standard elementary form. However,checking the derivative of the function $f(x) = 7^{7^{7^x}}$,we use the chain rule: $\frac{d}{dx} (7^{7^{7^x}}) = 7^{7^{7^x}} \cdot \log 7 \cdot \frac{d}{dx} (7^{7^x}) = 7^{7^{7^x}} \cdot \log 7 \cdot 7^{7^x} \cdot \log 7 \cdot \frac{d}{dx} (7^x) = 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x \cdot (\log 7)^3$. Therefore,$\frac{d}{dx} \left( \frac{7^{7^{7^x}}}{(\log 7)^3} ight) = 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x = |A|$. Thus,$\int |A| \, dx = \frac{7^{7^{7^x}}}{(\log 7)^3} + k$.

If $A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}$ where $a = 7^x$,$b = 7^{7^x}$,$c = 7^{7^{7^x}}$,then $\int |A| \, dx$ (where $|A|$ is the determinant of the matrix $A$) is equal to:

Similar Questions

The determinant $\left| \begin{array}{ccc} \cos(\theta + \phi) & -\sin(\theta + \phi) & \cos 2\phi \\ \sin \theta & \cos \theta & \sin \phi \\ -\cos \theta & \sin \theta & \cos \phi \end{array} \right|$ is :

The determinant $\left| \begin{array}{ccc} 4 + x^2 & -6 & -2 \\ -6 & 9 + x^2 & 3 \\ -2 & 3 & 1 + x^2 \end{array} \right|$ for $x \neq 0$ is not divisible by:

Consider a homogeneous system of three linear equations in three unknowns represented by $AX=O$. If $X=\left[\begin{array}{c}l \\ m \\ 0\end{array}\right]$, where $l \neq 0, m \neq 0, l, m \in \mathbb{R}$, represents an infinite number of solutions of this system, then the rank of $A$ is:

If $A = \begin{vmatrix} x & 1 \\ 1 & x \end{vmatrix}$ and $B = \begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix}$,then $\frac{dB}{dx}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module