CUET UG Mathematics — Calculus previous year questions with solutions.
Let $y = \sin(\cos x^2)$, then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to
The general solution of the differential equation $x\left(\frac{dy}{dx}\right) = y + x \tan\left(\frac{y}{x}\right)$ is
The value of $\int_{-5}^{5} |x + 3| dx$ is
The value of $\int_{-1}^{1} |x^3 - x| dx$ is
Match List-I with List-II | List-I | List-II | | --- | --- | | Definite integral | Value | | --- | --- | | (A) $\int_{1}^{e} \frac{\log x}{x} dx$ | (I) 4 | | (B) $\int_{-2}^{2} x^3(1 - x^2) dx$ | (II) $\frac{1}{2}$ | | (C) $\int_{1}^{2} x \, dx$ | (III) 0 | | (D) $\int_{-2}^{2} \vert x\vert dx$ | (IV) $\frac{3}{2}$ | Choose the correct answer from the options given below:
For $x > y > 0$, if $x^5 y^6 = (x + y)^{11}$, then $\frac{d^2y}{dx^2}$ is
The function $f(x) = x^4 - 2x^2$ is increasing on
The solution of the differential equation $\frac{dy}{dx} = \sqrt\frac{{y}}{x}$ is
A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(5x+7)$. Then the rate of change of its volume with respect to x is
$\int_{-\frac{5}{2}}^{\frac{5}{2}} |x| dx$ is equal to
Match List-I with List-II | List-I | List-II | |---|---| | **Differential equation** | **Degree** | | (A) $\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} - y = 0$ | (I) 6 | | (B) $\sqrt{\frac{d^3y}{dx^3}} - \sqrt[12]{\frac{d^2y}{dx^2}} = 0$ | (II) Not defined | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^{\frac{dx}{dx}} = x^2$ | (III) 3 | | (D) $\sqrt[3]{\frac{dy}{dx}} - \frac{d^2y}{dx^2} = e^x$ | (IV) 2 | Choose the correct answer from the options given below:
Which of the following are NOT correct regarding the equation of tangent and normal to the curve $y = \frac{x-11}{(x-2)(x-3)}$ at the point, where it cuts the $x$-axis? (A) The point of contact is (11, 0). (B) The equation of tangent is $x - 72y - 11 = 0$ (C) The equation of normal is $72x + y - 11 = 0$ (D) The slope of the tangent at the given point of contact is $\frac{1}{88}$ Choose the **correct** answer from the options given below:
If $y = \frac{1}{\sqrt[3]{1-x^3}}$ then $\frac{dy}{dx}$ is equal to
Let $[x]$ denote the greatest integer $\leq t$ and $a \mathbb{Z} = [ax: x \in \mathbb{Z}, a \in \mathbb{R}]$ (where $\mathbb{Z}$ is set of integer and $\mathbb{R}$ is set of real number). The set of points of discontinuity of the function $f(x) = [2x]$ is given by
$\int e^{-x}(\cot x + \cosec^2 x)dx =$
If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{dy}{dx}$ at $ \theta = \frac{\pi}{3}$ is
The demand function for a certain product is represented by the equation: $p = 20 + 5x - 3x^2$, where $x$ is the number of units demanded and $p$ is the price per unit (in Rs.), then the marginal revenue when 2 units are sold is:
If $x^m y^n = (x + y)^{m+n}$, then $\frac{d^2y}{dx^2}$ is equal to:
The area bounded by the curve $y = 4 + 3x - x^2$ and $x$-axis is equal to
Curd is at 80° F, five minutes later it came down at 60°F. After another 5 minutes, its temperature became 50° F. Given that the rate of change of temperature is proportional to (T - S), where S is temperature of the surroundings and T is temperature of the curd at any time t. Then the temperature of the surroundings is :
The function $f(x) = kx^3 + 6kx^2 + 18x + 17$ is increasing on $\mathbb{R}$(set of real numbers) if:
If $y = ax^2 + bx$ has minima at $x = 2$ and the minimum value is -12, then which of the following are correct? (A) $a = 3$ (B) $a = -3$ (C) $b = 12$ (D) $b = -12$ Choose the correct answer from the options given below:
The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is:
If the interval in which f(x) = $\frac{x}{4}$ + $\frac{4}{x}$, x ≠ 0 is strictly increasing is (-∞, a) ∪ (b, ∞), then