JEE 2023 — Mathematics PYQ

Q: How do I solve JEE 2023 Mathematics questions?

Practice JEE 2023 Mathematics previous year questions with step-by-step solutions on Mathem Solvex. Each question includes a detailed explanation and video solution to help you master the concept quickly.

Q: What is the difficulty level of JEE Mathematics questions?

JEE Mathematics questions range from moderate to advanced difficulty. Consistent practice with official previous year papers and topic-wise drills on Mathem Solvex is the most effective preparation strategy.

Q: Are JEE previous year papers available for free?

Yes. Mathem Solvex provides free access to JEE previous year question papers and topic-wise PDFs. You can download them or attempt them as timed live mock tests directly on the platform.

Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$ , if $\int \left[ \left(\frac{x}{e}\right)^{2x} + \left(\frac{e}{x}\right)^{2x} \right] \ln x \, dx = \frac{1}{\alpha} \left(\frac{x}{e}\right)^{\beta x} - \frac{1}{\gamma} \left(\frac{e}{x}\right)^{\delta x} + C$ , then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to:

Choose the correct answer:

Explanation

Solution:

Maan lijiye $y = \left(\frac{x}{e}\right)^x$ . Dono side log lene par:

$\ln y = x \ln(\frac{x}{e}) = x(\ln x - \ln e) = x \ln x - x$

Iska derivative lene par: $\frac{1}{y} \frac{dy}{dx} = (x \cdot \frac{1}{x} + \ln x) - 1 = \ln x$

Toh, $dy = \left(\frac{x}{e}\right)^x \ln x \, dx$ .

Integral ko likha ja sakta hai:

\int \left[ \left(\frac{x}{e}\right)^{2x} + \frac{1}{(x/e)^{2x}} \right] \ln x \, dx

$y = (x/e)^x$ rakhne par integral banta hai:

\int (y^2 + y^{-2}) \frac{dy}{y} = \int (y + y^{-3}) \, dy = \frac{y^2}{2} + \frac{y^{-2}}{-2} + C

Maan wapas rakhne par:

= \frac{1}{2} \left(\frac{x}{e}\right)^{2x} - \frac{1}{2} \left(\frac{e}{x}\right)^{2x} + C

Compare karne par: $\alpha = 2, \beta = 2, \gamma = 2, \delta = 2$

Pucha gaya expression:

\alpha + 2\beta + 3\gamma - 4\delta = 2 + 2(2) + 3(2) - 4(2) = 2 + 4 + 6 - 8 = 4

Answer: (1) 4

Choose the correct answer:

Explore More