CUET UG Mathematics — Calculus previous year questions with solutions.
Consider the function $f(x) = \sin x$ in the interval $[\pi, 2\pi]$ then which of the following statements are correct? (A) $x = \frac{3\pi}{2}$ is its stationary point. (B) Its maximum value is 1 (C) Its minimum value is -1 (D) It attains its maximum value at $\pi$ and $2\pi$ Choose the **correct** answer from the options given below:
If $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then $\frac{d^2y}{dx^2}$ is equal to
If $I_n = \int_{0}^{\pi/4} \tan^n x dx$ then $I_{2024} + I_{2026}$ is equal to:
The particular solution of the differential equation $\frac{dy}{dx} = 8yx$ when $y = 1$ at $x = 0$
The area (in square units) of the region bounded by the curves $3y^2 = ax$, $y = a$, $a > 0$ and $y$-axis is:
The area of the smaller region of the circle $x^2 + y^2 = 8$ cut off by the line $x = 2$ is
If $x = t^{1/2}$, $y = t^{3/2}$, then $\frac{dy}{dx}$ =
The nearest integral value of the shaded area shown below is: 
The function $f(x) = 6 - 6x - 2x^2$
If $I = \int \frac{x^4 + x^2 + 1}{x^2 - x + 1} dx = \alpha x + \beta x^2 + \gamma x^3 + \delta$, $\delta$ is constant of integration, then $(\alpha + 2\beta + 3\gamma)$ equals
Solution of the differential equation $\frac{dy}{dx} = \sqrt{1 + x^2 + y^2 + x^2y^2}$ is : (Here $C$ is an arbitrary constant)
A boat 10 m high floating at a uniform speed of 13 meters per minute(m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is:
If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:
For the function, $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$, which of the following statements are correct? (A) $x = -3$ and $x = -5$ are the only critical points of the given function. (B) $x = -3$ is a point of local minimum. (C) The local minimum value at $x = -3$ is 23.1. (D) $x = -5$ is a point of local maximum. Choose the correct answer from the options given below:
Match List-I with List-II | List-I (Curve) | List-II (Slope of tangent at $x = 4$) | |---|---| | (A) $y = \sqrt{x^3}$ | (I) -1 | | (B) $y = \sqrt{x}$ | (II) 1 | | (C) $y = x^3 - 47x$ | (III) 1/4 | | (D) $xy = 16$ | (IV) 3 | Choose the correct answer from the options given below:
Value of $\int_2^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5-x}}dx$ is
The function $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 163$ has a local maxima at
Area (in sq. units) of the region bounded by the curve $y^2 = 4x$, y-axis and the line $y = 3$ is
$\int_0^1 x e^x dx$ is equal to
The derivative of $(\log x)^x$ with respect to $\log x$ is
$\int \frac{dx}{(1+5\sin^2 x)}$ is equal to
The equation of the tangent line to the curve $y = x^2 - 2x + 5$ which is parallel to the line $4x - y + 1 = 0$ is
If $f(x) = x^3\log_e x$, Then $f''(e^2)$ is equal to
Which of the following statements is/are true? (A) $(\tan^{-1} y - x)dy = (1 + y^2)dx$ is a differential equation where variables are separable. (B) $(1 + x^2)dy + 2xydy = \cot x \ dx (x \neq 0)$ is a first order linear differential equation. (C) $(4x + 6y + 5)dy - (3y + 2x + 4)dx = 0$ is not a homogeneous differential equation. (D) $(xy)dx - (x + y^2)dy = 0$ is a homogeneous differential equation. Choose the correct answer from the options given below: