Question:
If \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]\) where \(a=7^x, b=7^{7^x}, c=7^{7^{7^x}}\) then \(\int|A| d x\), (Where \(|A|\) is the determinant of the matrix \(A\) ) is equal to
If \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right]\) where \(a=7^x, b=7^{7^x}, c=7^{7^{7^x}}\) then \(\int|A| d x\), (Where \(|A|\) is the determinant of the matrix \(A\) ) is equal to
Show Hint
The determinant of a diagonal matrix is the product of its diagonal entries, which is the derivative of a tower of exponentials.
Updated On: Nov 13, 2026
\(\frac{7^{7^{x}}}{(\mathrm{log}7)^{3}}+\mathrm{k}\), where k is constant of integration
\(\frac{7^{7^{7^{x}}}}{\mathrm{log}7}+\mathrm{k}\), where k is constant of integration
\(\frac{7^{7^{7^{x}}}}{(\mathrm{log}7)^{3}}+\mathrm{k}\), where k is constant of integration
\(7^{7^{7^{x}}}(\mathrm{log}7)^{3}+\mathrm{k}\), where k is constant of integration
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
For a diagonal matrix, \(|A|=abc=7^{x}\cdot7^{7^{x}}\cdot7^{7^{7^{x}}}\).
Step 2: Substitution
Let \(t=7^{7^{7^{x}}}\). Differentiating step by step, \(\dfrac{dt}{dx}=t\,(\log7)^{3}\,7^{7^{x}}\,7^{x}\).
Step 3: Match the integrand
So \(|A|\,dx=7^{x}\cdot7^{7^{x}}\cdot t\,dx=\dfrac{dt}{(\log7)^{3}}\).
Step 4: Calculation
\(\int|A|\,dx=\dfrac{t}{(\log7)^{3}}+k=\dfrac{7^{7^{7^{x}}}}{(\log7)^{3}}+k\).
Final Answer:
The integral is \(\frac{7^{7^{7^{x}}}}{(\log7)^{3}}+k\). \[ \boxed{\text{(C) }\dfrac{7^{7^{7^{x}}}}{(\log7)^{3}}+k} \]
For a diagonal matrix, \(|A|=abc=7^{x}\cdot7^{7^{x}}\cdot7^{7^{7^{x}}}\).
Step 2: Substitution
Let \(t=7^{7^{7^{x}}}\). Differentiating step by step, \(\dfrac{dt}{dx}=t\,(\log7)^{3}\,7^{7^{x}}\,7^{x}\).
Step 3: Match the integrand
So \(|A|\,dx=7^{x}\cdot7^{7^{x}}\cdot t\,dx=\dfrac{dt}{(\log7)^{3}}\).
Step 4: Calculation
\(\int|A|\,dx=\dfrac{t}{(\log7)^{3}}+k=\dfrac{7^{7^{7^{x}}}}{(\log7)^{3}}+k\).
Final Answer:
The integral is \(\frac{7^{7^{7^{x}}}}{(\log7)^{3}}+k\). \[ \boxed{\text{(C) }\dfrac{7^{7^{7^{x}}}}{(\log7)^{3}}+k} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Integration Questions
- Evaluate the integral: \[ \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx \]
- Evaluate the following integrals: \[ \int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx \] and \[ \int \frac{1}{x^4 + 5x^2 + 6} \, dx \]
- Evaluate the integral: \[ \int \frac{\sin x}{\sin\left(x-\frac{\pi}{4}\right)}\,dx \]
- Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{dx}{1 + \cos x} \]
- Evaluate the integral: \[ \int \left[ \frac{1 - \log x}{1 + (\log x)^2} \right]^2 dx \]
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions