CUET UG Mathematics — Calculus previous year questions with solutions.
The integral $\int \frac{2dx}{e^{2x}-1}$ is equal to:
The area of the region bounded by the line $y = 2x$ and the x-axis between $x = -2$ and $x = 2$ is
$\int \frac{f'(x)}{f(x) \log_e[f(x)]} dx$ is equal to
Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The maximum value of $f(x) = \sin(3x) + 6$ | (I) 2 | | (B) The maximum value of $f(x) = -\vert x + 2\vert + 4$ | (II) 5 | | (C) The minimum value of $f(x) = (3x + 1)^2 + 5$ | (III) 7 | | (D) The minimum value of $f(x) = 2 \cos x + 4$ | (IV) 4 | Choose the correct answer from the options given below:
Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Order and Degree** | | (A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$ | (I) order = 1 and degree = 2 | | (B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$ | (II) order = 2 and degree = 1 | | (C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$ | (III) order = 2 and degree = 2 | | (D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$ | (IV) order = 2 and degree = 4 | Choose the correct answer from the options given below:
$\int\limits_{\sqrt{log_e 2}}^{\sqrt{log_e 4}} xe^{x^2} dx$ is equal to
The area of the region bounded by the parabola $y^2 = 8x$ and its latus rectum in the first quadrant, is
For $x > e$, $\int \frac{dx}{x - \sqrt{x}}$ is equal to
The area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant, is
For the function $f(x) = x^x, x > 0$, which of the following are TRUE? (A) $f'(x) = x^x(1 + \log x)$ (B) $x = e$ is the critical point (C) $f$ is increasing in $(\frac{1}{e}, \infty)$ (D) $f$ is increasing in $(0, \infty)$ Choose the *correct* answer from the options given below:
$\int \frac{\cos 2x - \cos 2α}{\cos x - \cos α} dx$ is equal to
The area (in sq. units) of the region enclosed by the curve $9x^2 + 4y^2 = 36$ is
If $x = a\sin 2t(1 + \cos 2t)$ and $y = b\cos 2t(1 - \cos 2t)$, then $(\frac{dy}{dx})_{\text{at } x=\frac{\pi}{4}}$ is equal to
The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)
Consider the region bounded by the lines $y - 1 = x, x = -2, x = 3$ and $x$ - axis. Then (A) The area of the bounded region is given by $\int_{-2}^{3}(x + 1)dx$ (B) The numerical value of the area is $\frac{15}{2}$ sq. units (C) The numerical value of the area is 8 sq. units (D) The numerical value of the area is $\frac{17}{2}$ sq. units Choose the **correct** answer from the options given below:
Let $f(x)=\begin{cases} |x|+3 & \text{if } x\le -3 \\ -2x & \text{if } -3<x<3 \\ 6x+2 & \text{if } x\ge 3 \end{cases}$ Then, which of the following is true?
The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is
If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to
Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Points of discontinuity** | | (A) $f(x) = \frac{x^2 + 1}{x}$ | (I) $x = 4$ | | (B) $f(x) = \frac{\vert x - 1 \vert}{x - 1}$ | (II) $x = 2$ | | (C) $f(x) = \begin{cases} x - 1, & x < 2 \\ x + 1, & x \ge 2 \end{cases}$ | (III) $x = 0$ | | (D) $f(x) = \frac{1 - x}{(x - 4)}$ | (IV) $x = 1$ | Choose the **correct** answer from the options given below:
In which of the following intervals, the function $f(x) = -x^2 - 2x + 15$ is decreasing?
The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to
The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)
Value of $\int \left(\frac{1}{logx} - \frac{1}{(logx)^2}\right)dx$ is
$\int \frac{1}{x(x^5-1)} dx$ is equal to