CUET UG Mathematics — Calculus previous year questions with solutions.
The area of region bounded by the curve $y^2 = 4ax$ and the straight line $x = 2a$, $a > 0$ in the first quadrant is:
The value of $\int_0^1 \log_e\left(\frac{1}{x} - 1\right)dx$ is:
Area of the region bounded by $y = x^2$ and the line $y = 16$ is
The value of $\int_{-a}^a f(x)dx$ where $f(x) = \frac{7^x}{1+7^x}$ is:
The area bounded by the x-axis and the parabola $y = 3x-x^2$ is:
If maximum value of $f(x) = 2x^3 + 3x^2 - 6ax + 10$ occurs at $x = -3$, then the value of $\alpha$ is ____
$\int_0^1 \frac{dx}{\sqrt{1+x} - \sqrt{x}}$ is equal to
If $y = \sin^{-1}x$, then $(1-x^2)\frac{d^2y}{dx^2}$ is equal to
The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:
The value of $\int_{-\pi/2}^{\pi/2}(x^5 + x^3\cos x)dx$ is
If $x = -1$ and $x = -2$ are the extreme points of $f(x) = \alpha\log|x| + \beta x^2 + x$ then
The area (in sq.units) of the region bounded by the line $2y + x = 8$, the x-axis and the lines $x = 2$ and $x = 4$ is
If $x = a\left(\cos t + \log \tan\frac{t}{2}\right), y = a\sin t$, then value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$ is
The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is
The degree of the differential equation $\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}} = a^2 \frac{d^2y}{dx^2}$ is:
The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at
If m and n are respectively the order and degree of the differential equation $(\frac{d^2y}{dx^2})^{2} + (\frac{dy}{dx})^3 + y= 4x$, then the value of $m + n$ is:
The particular solution of the differential equation $xdy = (2x^2 + 1)dx, x \neq 0$, given that $y = 1$ when $x = 1$ is:
Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equation** | **Sum of order and degree** | | (A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$ | (I) 2 | | (B) $\frac{dy}{dx} = \sin(x + y)$ | (II) 3 | | (C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$ | (III) 4 | | (D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$ | (IV) 5 | Choose the **correct** answer from the options given below:
Consider the differential equation $xdy = (y + 2x^3)dx$. Then which of the following are TRUE? (A) It is a homogeneous differential equation. (B) Product of the order and degree of the differential equation in one. (C) Integrating factor is x. (D) General solution of the differential equation is $y = x^3 + Cx$, where C is an arbitary constant. Choose the *correct* answer from the options given below:
$\int \frac{\log_e x}{(1 + \log_e x)^2} dx$ is equal to
Match List-I with List-II Where $\mathbb{R}$ is set of real numbers | List-I | List-II | |---|---| | (A) $\sin x$ is continuous on: | (I) $\mathbb{R} - \{0\}$ | | (B) $ \tan x$ is continuous on: | (II) $\mathbb{R}$ | | (C) $\cot x$ is continuous on: | (III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$ | | (D) $x^{-n}, n \in \mathbb{N}$ is continuous on: | (IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$ | Choose the correct answer from the options given below:
Let the degree and order of the differential equation $2x^3\frac{dy}{dx}-5\left(\frac{d^2y}{dx^2}\right)^2=6\left(\frac{dy}{dx}\right)^3$ be $m$ and $n$ respectively. Then (A) m = 2 (B) n = 3 (C) m = 3 (D) mn = 4 Choose the correct answer from the options given below:
$\int \frac{x^3 - 1}{x^2} dx$ is equal to