Calculus PYQ

If $x = \frac{a}{1 + t}$ and $y = \frac{a}{(1 + t)^2}$ where $a > 0$ , then $\frac{d^2y}{dx^2}$ at $t = 1$ is

For the function $f(x) = 2x^3 - 3x^2 - 12x + 5$, the difference of maximum and minimum value of $f(x)$ is

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

If $y = \sqrt{2024x + 2025}$, then which of the following is correct?

$\int \frac{e^{2x} - 1}{e^{2x} + 1} dx =$

Match List-I with List-II (where $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \tan x \, dx$ | (I) $\log\vert \sec x + \tan x\vert + c$ | | (B) $\int \cot x \, dx$ | (II) $\log\vert \sec x\vert + c$ | | (C) $\int \sec x \, dx$ | (III) $\log\vert \sin x\vert + c$ | | (D) $\int \cosec x \, dx$ | (IV) $\log\vert \cosec x - \cot x\vert + c$ | Choose the correct answer from the options given below:

The product of order and degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + x^2y\left(\frac{d^2y}{dx^2}\right)^3 = 2x^5$ is:

The integral I = $\int e^x\left(\frac{x - 1}{3x^2}\right) dx$ is equal to

The value of the integral $I = \int_0^1 \frac{1}{\sqrt{x^2 + 2x + 3}} dx$ is:

The volume of spherical balloon is increasing at the rate of $4 \text{ cm}^3/ \text{sec}$. The rate of increase of its surface area, when the radius is 3cm will be :-

The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

The value of the definite integral $\int_0^1 e^x \frac{(1-x)^2}{(1+x^2)^2}dx$ is:

If $\int \frac{2x - 5}{(2x - 3)^3} e^{2x} dx = \frac{\lambda e^{2x}}{(2x - 3)^2} + C$, where $C$ is an arbitrary constant then the value of $\lambda$ is

The area bounded by the curve $y = \log x, y = 0$ and $x = e$, is

The area (in sq. units) of the region bounded by $y = -1, y = 2, x = y^3$ and $x = 0$ is equal to

If $\int_0^a 3x^2dx = 8$, then the value of $a$ is:

The value of $\int_0^1 [\log x - \log(1-x)] dx$ is

$\int \frac{dx}{\sqrt{5 - 4x - x^2}}$ is equal to

The solution of the differential equation $\frac{dy}{dx} - \frac{y}{x} = 2\log_e x$

The value of $\int_2^4 \frac{x}{x^2 + 1} dx$ is equal to

Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals