CUET UG Mathematics — Calculus previous year questions with solutions.
In which of the following interval the function $f(x) = x^x, x > 0$ is strictly increasing?
The solution of the differential equation $\frac{dr}{dt} = -rt, r(0) = r_0$ is
If under pure competition demand and supply functions are given by $p = \sqrt{10 - x}$ and $p = \frac{1}{2}(x-2)$ respectively, where $p$ is price per unit and $x$ is quantity, then the consumer surplus is:
If $f(x) = \begin{cases} ax - 1 & {if } x \ > 1\\ \ 2x + 1 & {if } x < 1 \end{cases}$ is continuous at $x = 1$, then $a$ equals
If the cost function of a product is given by $C(x) = \frac{3}{4}x^2 - 5x + 21$, then the marginal cost when $x = 10$ is
The point on the curve y = (x - 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4) is:
If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to
$\int \frac{1}{(x + 1)(x + 2)} dx$ is equal to
Let f be a function defined by $f(x) = 2x^3 - 3x^2 - 36x + 2$, then which of the following are correct? (A) The critical points of f(x) are -2 and 3. (B) The function f(x) increases in the interval $(3, \infty)$ (C) The function f(x) decreases in the interval (-2,3) (D) The function f(x) increases in the interval (-2,3) Choose the **correct** answer from the options given below:
$\int_{-1}^1 \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1}dx$ is equal to
The function $f(x) = tanx - x$
Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = |x| & \text{(I) Not differentiable at } x=-2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x+2| & \text{(II) Not differentiable at } x=0 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = |x^2-4| & \text{(III) Not differentiable at } x=2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x-2| & \text{(IV) Not differentiable at } x=2,-2 \text{ only} \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:
A car is moving along the curve $y = x^3 + 12$. The point(s) on the curve at which the rate of change of its y-coordinate at a certain time is 3 times the rate of change of its x-coordinate is/are
Shown below is the graph of parabola $y^2=x$, the area (in sq. units) of the shaded region is: 
The area (in sq. units) of the region in the first quadrant bounded by $y = 3\sqrt{1-x^2}$, $x \in [0,1]$ and the x-axis is equal to
The rate of change of area of a circle with respect to its circumference when radius in 6 cm, is
For the differential equation $ydx - (x + 3y^2)dy = 0$, which of the following statements are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $x = 3y^2 + Cy$ : $C$ is an arbitrary constant (D) If $y(0) = 1$, then its particular solution is $x = 3y^2 - 1$ Choose the correct answer from the options given below:
Match List-I with List-II | List-I | List-II | |---|---| | (A) $\int_0^1 \frac{x^2}{1 + x^3} dx$ | (I) 0 | | (B) $\int_0^\pi 3\sin x dx$ | (II) $2\log_e\left(\frac{3}{2}\right)$ | | (C) $\int_{-1}^1 \sin^5 x \cos^6 x dx$ | (III) 6 | | (D) $\int_2^3 \frac{4}{x^2 - 1} dx$ | (IV) $\frac{1}{3}\log_e 2$ | Choose the correct answer from the options given below:
The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is
The area (in sq.units) of the region bounded by the curve $y = \cos x$ between $x = -\frac{\pi}{2}, x = \frac{\pi}{2}$ and the x-axis is
$\int (x^4 + x^2 + 1)d(x^2)$ is equal to: (where c is an integration constant)
Area of the region bounded by the curve $y = \sqrt{x}$ and lines $x + y = 2$, $y = 0$ is
If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ is
The value of derivative of the function $\cot^{-1}\{(\cos 2x)^{1/2}\}$ at $x = \frac{\pi}{6}$ is