CUET UG 2025 — Calculus

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

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:

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

A cylindrical drum of radius 7 cm and height 2 m is being kept in a vertical position filled with milk. If the milk is leaking at 14 cm³/sec from its lower base, then the rate of decrease in the level of milk is: [Take $\pi = \frac{22}{7}$]

A integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x^2}$, $(x > 0)$ is equal to

A spherical ice ball is melting at the rate of 100 $\pi$ cm³/min. The rate at which its radius is decreasing when its radius is 15 cm, is

A square board of side 36cm is made into a box without top by cutting a square from each corner and folding up the flaps to form a box then maximum volume of the box is

Area (in sq. units) of the region bounded by curves $y^2 = x$ and $x = 4$ is

Area (in sq. units) of the region bounded by the curve $y^2 = 4x$, $y$-axis and the line $y = 3$ is

Area (in sq. units) of the region bounded by the curves $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is

Area (in sq. units) of the region bounded by the curve $y^2 = 4x$, y-axis and the line $y = 3$ is

Area of region bounded by the curves $x = y^3$, $x = 0$ between $y = -1$ and $y = 2$ is:

Area of the bounded region between the curve $y = |x - 2|$ and the line $y = 2$ is:

Area of the region bounded by $y = x^2$ and the line $y = 16$ is

Area of the region bounded by the curve $y = \sin x$ and x-axis between $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ is

Area of the region bounded by the curve $y^2 = 4x$, $y$-axis and the line $y = 3$ is equal to

Area of the region bounded by the curve $y = \sqrt{x}$ and lines $x + y = 2$, $y = 0$ is

Consider a closed cylinder of radius $r$ with a fixed surface area. The volume of the cylinder is maximum when its height is

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$. If it passes through the point $(0,0)$, then which of the following is/are true? (A) it is a straight line. (B) it is a parabola. (C) it also passes through the point $(-1, \frac{1}{\sqrt{e}} - 1)$ (D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$ Choose the **correct** answer from the options given below:

Consider the differential equation $\frac{dy}{dx} + y \tan x = \sec x$, then which of the following statements are correct? (A) It is homogeneous (B) It has $\sec x$ as its integrating factor (C) It's general solution is $y \sec x = \tan x + c$, where c is arbitary constant. (D) It's degree is not defined 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:

Consider the differential equation, $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $\log_e\left(\frac{y}{x}\right) = Cx$, where C is constant of integration (D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$, where C is constant of integration (E) If $y(1) = 1$, then its particular solution is $y = x$ Choose the correct answer from the options given below:

Consider the differential equation $xdy = (x + y) dx$. Which of the following are true? (A) It is a homogenous differential equation (B) It is a differential equation of order 2 (C) The general solution of the differential equation contains 2 arbitrary constants (D) Integrating factor of differential equation is $\frac{1}{x}$ (E) Degree of the differential equation is not defined Choose the correct answer from the options given below:

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:

Consider the function $f(x) = x^3 - 3x$. Then Match List-I with List-II | List-I | List-II | |---|---| | (A) Point of local Maxima | (I) 1 | | (B) Point of local Minima | (II) -1 | | (C) Local maximum value | (III) 2 | | (D) Local minimum value | (IV) -2 | Choose the correct answer from the options given below:

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:

The value of ∫₀¹ x·eˣ dx is:

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 :

Derivative of $x^x$ with respect to $x\log x$ is

Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

Differentiation of $\log[\log(\log x^5)]$ with respect to $x$ is

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

$\int_{1}^{2} \frac{1}{x(x+1)} dx, x > 0$ equals

$$\frac{d^2}{dx^2} \left\{ \det \begin{bmatrix} x^3 & x \\ 2 & e^x \end{bmatrix} \right\}$$ equals

$\int \frac{\sin 2x \, dx}{\sqrt{9 - \cos^4 x}}$ equals

$\int \tan^{-1}\sqrt{x} $ $dx$ equals to: (Here C is an arbitrary constant)

$\int_2^5 |x - 3|dx$ equals

$\int \frac{(x-1)e^x}{x^2} dx, x > 0$ equals (where C is an arbitrary constant)

For $x > 1$, $\int \frac{e^{7\log x} - e^{5\log x}}{e^{5\log x} - e^{4\log x}} dx$ equals.

For $x \in \mathbb{R}$, if $f(x) = -(x-1)^2 + 2$, then (A) $f$ is an increasing function on $(-\infty, 1]$ (B) $f$ has no critical points (C) $f$ has a maximum value at $x = 1$ (D) $f$ has a minimum value at $x = 1$ 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

For $|x| < 1$, if $x = \cos\left(\frac{1}{a}\log y\right)$, then

For $x \neq -1$, if $\int \frac{xe^x dx}{(1+x)^2} = \frac{ae^x}{(1+x)^b} + c$, where a, b are fixed numbers and c is the integration constant, then $a + b$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$ is equal to

For $x \in \mathbb{R} - \{-1,0,1\}$, $\int \frac{1}{x - x^5}dx$ is equal to

For $x > e$, $\int \frac{dx}{x - \sqrt{x}}$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{1}{\sin^2 x + \sin 2x} dx$ is equal to

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$ (A) Integrating factor is $2x$ (B) Integrating factor is $x^2$ (C) General Solution is $y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$ Where C is an arbitrary constant. (D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant. Choose the correct answer from the options given below:

For the differential equation $(x + y)dy + (x - y)dx = 0$, which of the following is/are correct? (A) Differential equation is homogeneous (B) Order of differential equation is 1 (C) Integrating factor of differential equation is $e^x$ (D) Degree of the equation is not defined Choose the **correct** answer from the options given below:

For the differential equation $x\frac{dy}{dx} + 3y = x^2\log_e x$, which of the following statements are TRUE? (A) Product of order and degree is 1 (B) Integrating factor is $x^3$ (C) Integrating factor is $3x$ (D) General solution is $y = \frac{x^3}{36}(6\log_e|x| - 1) + Cx^{-3}$, C is an arbitrary constant. Choose the correct answer from the options given below:

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:

For the function $f(x) = 2x^3 - 3x^2 - 12x + 5$, the difference of maximum and minimum value of $f(x)$ is

For the function $f(x) = e^{-2x}(2-x)^2$, the point of local maxima is:

For the function $f(x) = x^{1/x}$, $x > 0$, which of the following are correct? (A) $x = 0$ is the only point where extremum may occur. (B) The given function is maximum at $x = e$. (C) The function has no extreme value for $x > 0$. (D) The maximum value of the function $f(x)$ is $e^{1/e}$. Choose the correct answer from the options given below:

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:

For the function $f(x) = -2x^3 + 3x^2 + 36x - 10$, which of the following is/are true? (A) $f$ is increasing in $(-\infty, -2)$ (B) $f$ is increasing in $(-2, 3)$ (C) $f$ is decreasing in $(-\infty, -2)$ (D) $f$ is decreasing in $(3, \infty)$ Choose the correct answer from the options given below:

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:

For the function $f(x) = sinx + cosx, x \in [0, \pi]$, which one of the following is correct?

For $x > 0$, the minimum value of $\frac{x}{\log_e x}$ is

For $y \neq 0$, the particular solution of the differential equation $2ye^{x/y}dx + (y - 2xe^{x/y})dy = 0$ at the point (1, 1) is

For what value of $\alpha$, the function $f$ defined by $f(x) = \begin{cases} \alpha(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$?

For x ∈ ℝ - {0}, the function f(x) = $\frac{3}{x}$ + 7 is decreasing when

Function $f(x) = x^x, x > 0$ decreases on the interval

Function $f(x) = x^3 - 3x + 3$ is (A) Increasing in the interval $(-1, 1)$ (B) Increasing in the interval $(1, \infty)$ (C) Decreasing in the interval $(-1, 1)$ (D) Increasing in the interval $(-\infty, -1) \cup (1, \infty)$ Choose the correct answer from the options given below:

General solution of the differential equation $\frac{dy}{dx} = e^{\frac{x^2}{2}} + xy$ is

Given differential equation, (1 + y²)dx = (tan⁻¹y - x)dy, then which of the following is/are true? (A) Integrating factor = tan⁻¹x (B) Integrating factor = tan⁻¹y (C) Integrating factor = $e^{tan⁻¹y}$ (D) Degree = 1 Choose the correct answer from the options given below:

$\int \left(\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}\right) dx =$ (Given that $c$ is an arbitrary constant)

If a revenue function is given by $R(x) = 2027x - 1013x^2 - 675x^3$, then the marginal revenue function (MR) is:

If $f(x)$ and $g(x)$ are continuous functions in [0, a] such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = a$ then $\int_{0}^{a} f(x)g(x)dx =$

If $x = -1$ and $x = -2$ are the extreme points of $f(x) = \alpha\log|x| + \beta x^2 + x$ then

If $x = e^t$ and $y = e^{2t}$ then $\frac{d^2y}{dx^2} =$

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is

If $x = a\cos\alpha + b\sin\alpha$ and $y = a\sin\alpha - b\cos\alpha$, then $\left(x\frac{dy}{dx} - y^2\frac{d^2y}{dx^2}\right)$ is equal to:

If $y = t - \frac{1}{t}$ and $x = t + \frac{1}{t}$, then $\frac{dy}{dx}$ is equal to

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

If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to

If $y = \left(\log\left(x + \sqrt{x^2+a^2}\right)\right)^2$ and $x \neq \frac{1-a^2}{2}$, then $(x^2+a^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to

If $x^2 - y^2 = t - \frac{1}{t}$, and $x^4 + y^4 = t^2 + \frac{1}{t^2}$, then which of the following is correct?

If $x = \frac{a}{1 + t}$ and $y = \frac{a}{(1 + t)^2}$ where $a > 0$ , then $\frac{d^2y}{dx^2}$ at $t = 1$ is

If $y = \log_e\left(\frac{e^2}{x^2}\right)$ for $x \neq 0$, then $\frac{d^2y}{dx^2}$ equals

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$, then

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:

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

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 $f(x) = \begin{cases}\frac{1- \tan x}{4x-\pi}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4}\end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of k is

If $f(x) = \begin{cases} \frac{\tan(\frac{\pi}{4} - x)}{\cot 2x} & , x ≠ \frac{\pi}{4} \\ 2K + 1 & , x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of K is equal to

If $f(x) = \begin{cases} mx + 1,\ x \geq \pi/2 \\sin x + n, x \leq \pi/2, & \end{cases}$ is continuous at $x = \pi/2$, where $m \in \mathbb{Z}$ (set of integers), then $\sin 2n =$

If $g(x) = \begin{cases} \frac{αx}{|x|}, & \text{if } x < 0 \\ 5, & \text{if } x ≥ 0 \end{cases}$ is continuous at x = 0, then the value of α is

If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

If it is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is:

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:

If maximum value of $f(x) = 2x^3 + 3x^2 - 6ax + 10$ occurs at $x = -3$, then the value of $\alpha$ is ____

If the area above x-axis, bounded by the curves $y = 3^{\beta x}$, $x = 0$ and $x = 3$ is $\frac{26}{\log_e 3}$, then the value of $\beta$ is:

If the area of an equilateral triangle is increasing at the rate of $4\sqrt{3}$ cm²/sec, then the rate of increase of its perimeter when the side is 4cm, is

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

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

If the function $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x} & : x \neq \frac{\pi}{2} \\ 3 & : x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then $k$ is equal to

If the function $f(x) = \begin{cases}\frac{\sin 3x}{x}, & \text{if } x \neq 0\\ \frac{3k}{2}, & \text{if } x = 0\end{cases}$ is continuous at $x = 0$, then the value of $k$ is

If the function $f(x) = \begin{cases} ax + 2, & x \leq 1 \\ x^2 + 3x + b, & x > 1 \end{cases}$ is differentiable at $x = 1$, then the value of $(2a + b)$ is

If the function $f(x) = 2x^3 + 9x^2 + 12x-1$ is given,then $f(x)$ have

If the function $f(x) = 2x^2 - kx + 7$, is increasing on $[1,2]$, then $k$ lies in the interval

If the integral $I = \int \frac{x^2}{\sqrt{1+x}} dx = \frac{1}{\alpha} (1+x)^\alpha - \frac{8\alpha}{15} (1+x)^{\alpha-1} + 2(1+x)^{\alpha-2} + C$, $C$ is constant of integration, then the value of $\alpha$ is:

If the integral $I = \int \left[log_e(log_e x)^2 + \frac{a}{log_e x}\right] dx = x log_e(log_e x)^2 + C$, where C is constant of integration. Then the value of $a$ is:

If the interval in which f(x) = $\frac{x}{4}$ + $\frac{4}{x}$, x ≠ 0 is strictly increasing is (-∞, a) ∪ (b, ∞), then

If the interval in which the function f(x) = $\frac{x}{x^2+1}$ is strictly increasing is (-a, a), then a is equal to

If the interval in which the function $f(x) = 4x^3 - 6x^2 - 72x + 30$ is strictly decreasing, is (a,b) then a+b is equal to

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$, $x > 0$ occurs at $x = a$, then $a^2f''(a)$ is equal to

If the maximum value of the function $f(x) = \frac{2\log_e x}{x}$, $x > 0$ occurs at $x = e$, then $e^3 f''(e)$ is equal to

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

If the slope of the tangent to the curve $y = y(x)$ at any point $(x, y)$ is $\frac{2x}{y^2}$ and the curve passes through the point $\left(\frac{1}{\sqrt3}, 1\right)$, then equation of curve is

If $e^y(x + 1) = 1$, then

If $y = \sqrt{x + \sqrt{x + \sqrt{x + ...\ ...\ ...}}}$, then

If $f(x) = \sin x - \cos x$, $x \in [0, 2\pi]$ then (A) $f(x)$ is increasing in $(0, \frac{3\pi}{4})$ (B) $f(x)$ is decreasing in $(0, \frac{3\pi}{4})$ (C) $f(x)$ is decreasing in $(\frac{3\pi}{4}, \frac{7\pi}{4})$ (D) $f(x)$ is decreasing in $(\frac{7\pi}{4}, 2\pi)$ Choose the correct answer from the options given below:

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{dy}{dx}$ at $ \theta = \frac{\pi}{3}$ is

If $y = (x+1)(x^2+1)(x^4+1)(x^8+1)$ then $\frac{dy}{dx}$ at $x = -1$ is

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ =

If $y = \sqrt{ax + b}$ then $y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 =$

If $y = 5e^{2x} + 4e^{3x}$, then $\frac{d^2y}{dx^2}$ equals:

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ equals

If $x = e^{\cos 2t}$, $y = e^{\sin 2t}$, then $\frac{dy}{dx}$ equals to

If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots + \infty}}}$ then $\frac{dy}{dx}$ equals to

If $y = \frac{1}{1+x^{b-a}+x^{c-a}} + \frac{1}{1+x^{c-b}+x^{a-b}} + \frac{1}{1+x^{a-c}+x^{b-c}}$ then $\frac{d^2y}{dx^2}$ is

If $y = x\sin y$, then $\frac{dy}{dx}$ is:

If $y = (log x)^{(log x)}$, $x > 1$ then $\frac{dy}{dx}$ is equal to

If $y = \sin^{-1}x$, then $(1-x^2)\frac{d^2y}{dx^2}$ is equal to

If $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}, x \in (-1, 0)$, then $\frac{dy}{dx}$ is equal to

If $x^m y^n = (x + y)^{m+n}$, then $\frac{d^2y}{dx^2}$ is equal to:

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:

If $xy = e^{(x-y)}$, then $\frac{dy}{dx}$ is equal to:

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

If $f(x) = x^3\log_e x$, Then $f''(e^2)$ is equal to

If $xy + \frac{x^2}{y} = x^3y + y$, then $\frac{dy}{dx}$ is equal to

If $I_n = \int_{0}^{\pi/4} \tan^n x dx$ then $I_{2024} + I_{2026}$ is equal to:

If $y = e^{acos^{-1}x}, -1 < x < 1$, then $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx}$ is equal to

If $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$, then $\frac{dy}{dx}$ is equal to

If $\frac{d}{dx}[ax^3 + ax^2 + ax + 1] = 9x^2 + 6x + 3$, then $a$ is equal to

If $x = at^2, y = 2at$; then $\frac{d^2y}{dx^2}$ is equal to

If $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then $\frac{d^2y}{dx^2}$ is equal to

If $y = \frac{1}{\sqrt[3]{1-x^3}}$ then $\frac{dy}{dx}$ is equal to

If $f(a-x) = f(x)$, then $\int_0^a xf(x)dx$ is equal to

If $\sin y = x \cos(a + y)$, then $\frac{dy}{dx}$ is equal to

If $y = \sin^{-1} \sqrt\frac{x}{x+1} + \sec^{-1}\sqrt{\frac{x+1}{x}}$, then $\frac{dy}{dx}$ is

If $2f(x) + f\left(\frac{1}{x}\right) = x^2 + 1$, then $\int f(x) dx$ is: (Here C is an arbitrary constant)

If $y = \left(x + \sqrt{x^2+1}\right)^m$, then $\frac{dy}{dx}$ is

If $y = \log_e(\sec e^{x^2})$ then $\frac{dy}{dx} =$

If $f(x) = 2x^3 - 15x^2 + 36x + 1$, $x \in [1, 5]$, then the absolute minimum value of $f(x)$ is:

If $y = 3^x + e^x + x^x + x^3$, then the value of $\frac{dy}{dx}$ at $x = 3$ is

If $\int_0^1 \frac{e^x}{1 + x} dx = m$, then the value of $\int_0^1 \frac{e^x}{(1 + x)^2} dx$ is:

If $f(x) = x^3 e^{-x}$, then the value of $f''(1)$ is equal to

If $\int_0^a 3x^2dx = 8$, then the value of $a$ is:

If $y^{1/m} + y^{-1/m} = 2x$, then the value of $(x^2 - 1)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ 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

If $x = t^{1/2}$, $y = t^{3/2}$, then $\frac{dy}{dx}$ =

If $f(x) = x^2 - 4x + 13, x \in \mathbb{R}$, then which of the following are correct? (A) $x = 2$ is a stationary point of $f(x)$. (B) $f(x)$ is increasing function on $(2, \infty)$ (C) $f(x)$ have maxima at $x = 2$ (D) $f(2) = 9$ Choose the correct answer from the options given below:

If $y = \sqrt{2024x + 2025}$, then which of the following is correct?

If $x^2 - y^2 = 1$, then which of the following is correct? (A) $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = x^2$ (B) $(x^2 - 1)\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (C) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (D) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = 1$ Choose the correct answer from the options given below:

If $e^y = \log x$, then which of the following is true?

If $f(x) = |x| + |x - 5|$, then which of the following statements are TRUE? (A) f is a continuous function every where (B) f is a continuous function except $x = 5$ and $x = 0$ (C) f is a continuous function except $x = 0$ but not differentiable at $x = 5$ (D) f is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$ Choose the correct answer from the options given below:

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 $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$, where $|x| < 1, |y| < 1$ and $x ≠ y$, then

If $\int \frac{x^4}{x-2}dx = px + qx^2 + rx^3 + sx^4 + t\log |x - 2| + C$, where C is an arbitrary constant and p, q, r, s, t are real numbers, then the correct arrangement of p, q, r, s, t is:

If $\int \sqrt\frac{1-x}{{1+x}} dx = a\sqrt{1-x^2} + \beta \sin^{-1}x + C$, Where C is an arbitrary constant, then which of the following are TRUE? (A) $\alpha = 1$ (B) $\alpha = -1$ (C) $\beta = 1$ (D) $\beta = -1$ Choose the correct answer from the options given below:

If $\int e^x\left(\frac{x-1}{(x+1)^3}\right)dx = \frac{Ae^x}{(x+1)^B} + C$, where C is constant of integration then which of the following are correct? (A) $A = -1$ (B) $A = 1$ (C) $B = 3$ (D) $B = 2$ Choose the correct answer from the options given below:

If $\int \frac{(1 + x \log x)}{xe^{-x}} dx = e^x f(x) + C$, where C is constant of integration, then $f(x)$ is

If $\int \frac{dx}{(x-1)^3/^4. (x+2)^5/^4} = a[1 - g(x)]^b + c$, where $c$ is a constant of integration, then which of the following are true? (A) $a = \frac{2}{3}$ (B) $\beta = \frac{3}{4}$ (C) $3\alpha + 4\beta = 5$ (D) $g(x) = \frac{3}{(x+2)}$ Choose the **correct** answer from the options given below:

If $\int \frac{2x - 5}{(2x - 3)^3} e^{2x} dx = \frac{\lambda e^{2x}}{(2x - 3)^2} + C$, where $C$ is an arbitrary constant then the value of $\lambda$ is

If $I = \int \frac{x}{x - \sqrt{x^2 - 4}} dx = \alpha x^3 + \beta(x^2 - 4)^{\frac{3}{2}} + \gamma$, where $\gamma$ is constant of integration, then

In the following differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ order and degree is:

In which of the following interval the function $f(x) = x^x, x > 0$ is strictly increasing?

In which of the following interval, the function $f(x) = \frac{x}{\log x}$ is decreasing?

In which of the following intervals, the function $f(x) = -x^2 - 2x + 15$ is decreasing?

Interval in which the function $f$ given by $f(x) = \tan x - 4x$, $x \in (0, \frac{\pi}{2})$ is strictly decreasing is

$\int \left(\frac{\cos x - \sin x}{1 + \sin 2x}\right) dx$ is equal to

$\int \frac{dx}{2\sin^2 x + 5\cos^2 x}$ is equal to

$\int \frac{\log_e x}{(1 + \log_e x)^2} dx$ is equal to

$\int \frac{dx}{9x^2 - 16}$ is equal to

$\int \frac{dx}{(1+5\sin^2 x)}$ is equal to

$\int_{3\pi/8}^{\pi/8} \frac{ \tan^{2025} x}{ \tan^{2025} x + \cot^{2025} x} dx$ is equal to

$\int \left(\frac{1}{\log_e x} - \frac{1}{(\log_e x)^2}\right)dx$ is equal to

$\int_0^8 (x^{\frac{2}{3}} + 1) dx$ is equal to

$\int_0^2 x(2-x)^n dx$ is equal to

$\int_{\pi/6}^{\pi/3} \frac{tan x}{tan x + cot x} dx$ is equal to

$\int \frac{(x-3)e^x}{(x-1)^3} dx$ is equal to

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{tanx}}$ is equal to

$\int \frac{1}{(x + 1)(x + 2)} dx$ is equal to

$\int_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a-x}} dx$ is equal to

$\int \frac{f'(x)}{f(x) \log_e[f(x)]} dx$ is equal to

$\int \frac{e^{7\log_e x} - e^{6\log_e x}}{e^{4\log_e x} - e^{3\log_e x}} dx$ is equal to: (Here, c is an arbitrary constant)

$\int \left(\frac{1}{log_e t} - \frac{1}{(log_e t)^2}\right) dt$ is equal to

$\int_0^1 tan^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ is equal to

$\int_{-1}^1 \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1}dx$ is equal to

$\int \sin x \sin 2x \sin 3x dx$ is equal to

$\int_0^{\pi/2} \frac{\sin^8 x}{\sin^8 x + \cos^8 x} dx$ is equal to

$\int \frac{\cos 2x - \cos 2α}{\cos x - \cos α} dx$ is equal to

$\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3 - x} + \sqrt{x}} dx$ is equal to

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx$ is equal to

$\int_{-\pi}^{\pi} \frac{e^{\sin x}}{e^{\sin x} + e^{-\sin x}}dx$ is equal to

$\int \frac{dx}{\sqrt{5 - 4x - x^2}}$ is equal to

$\int \frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}dx$ is equal to

$\int_1^4 |x - 2|dx$ is equal to

$\int \frac{1}{x(x^5-1)} dx$ is equal to

$\int e^{2x}(\sin x + \frac{1}{2}\cos x) dx$ is equal to

$\int_{-1}^{1}(x^7 + x^5 + x^3 + x + 1)dx$ is equal to

$\int \frac{dx}{e^x + e^{-x}}$ is equal to

$\int_1^{\sqrt{3}} \frac{1}{1+x^2} dx$ is equal to:

$\int_0^1 x e^x dx$ is equal to

$\int \frac{x^3 - 1}{x^2} dx$ is equal to

$\int_{-\frac{5}{2}}^{\frac{5}{2}} |x| dx$ is equal to

$\int \frac{(x^4 - x)^{1/4}}{x^5} dx$ is equal to

$\int_0^{\pi/2} \sqrt{1 - \sin 2x}\,dx$ is equal to:

$\int_0^1 \frac{dx}{\sqrt{1+x} - \sqrt{x}}$ is equal to

$\int \frac{\sqrt{16+(\log x)^2}}{x} dx$ is equal to (where C is an arbitrary constant)

$\int \sqrt{1 + \frac{x^2}{9}} dx$ is equal to (Where C is an arbitrary constant)

$\int (x^4 + x^2 + 1)d(x^2)$ is equal to: (where c is an integration constant)

$\int (e^{x\log a} + e^{a\log x}) dx$ is equal to (where $a > 1$)

$\int \frac{x}{(x-1)(x-2)} dx$ is equal to ( where $C$ is a constant of integration)

$\int\limits_{\sqrt{log_e 2}}^{\sqrt{log_e 4}} xe^{x^2} dx$ is equal to

$\int \frac{e^x(1 + x)dx}{\cos^2(e^x x)}$ is equal to

$\int e^{(x \log 5)}e^x dx$, is: Where $C$ is the constant of integration.

Let $f(x)=\begin{cases}\dfrac{|x|}{x},&x\ne0\\1,&x=0\end{cases}$ and $g(x)=\begin{cases}x\sin\left(\dfrac{1}{x}\right),&x\ne0\\0,&x=0\end{cases}$ Then at the origin, which one of the following is true?

Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE? (A) $f'(x) = 3(x - 1)(x - 3)$ (B) The critical points of the function are $x = 1$ and $x = 3$ (C) $x = 1$ is the point of local minimum (D) The local maximum value is $-4$ Choose the correct answer from the options given below:

Let $f(x) = x^2 + \frac{250}{x}$ be any function defined on $\mathbb{R} - \{0\}$, where $\mathbb{R}$ is the set of real numbers. Then which of the following are TRUE? (A) $f'(x) = 2x + \frac{250}{x^2}$ (B) $x = 5$ in the only critical point of $f(x)$ (C) minimum value of $f(x)$ is 75 (D) maximum value of $f(x)$ is 50. Choose the **correct** answer from the options given below:

Let $y(x) = a(x + 1) \log(x + 1) + bx + 5$ be the solution of the differential equation e$^\frac{dy}{dx} = x + 1_{;}y(0) = 5$, then the value of $(a + b)$ is:

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

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:

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:

Let $x = t^2, y = t^3$. Then $\frac{d^2y}{dx^2}$ is equal to

Let $y = \cos(\sin x^2)$, then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to

Let $y = \sin(\cos x^2)$, then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to

Let $e^y(x+1) = 1$. Then which of the following are TRUE? (A) $\frac{d^2y}{dx^2} = -\frac{1}{(x+1)^2}$ (B) $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$ (C) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = -1$ (D) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = 1$ (E) $\left.\frac{d^2y}{dx^2}\right|_{x=1} = \frac{1}{4}$ 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?

Let $f(x) = 4x^3 - 18x^2 + 27x - 5$, $x \in R$. Then which of the following statements are TRUE? (A) $f''(x) = 24x - 36$ (B) f has local maxima at $x = \frac{3}{2}$ but no minima (C) f has neither maxima nor minima (D) f has both maxima and minima Choose the correct answer from the options given below:

Let $f(x) = \log_e(\sin x), x \in (0, \pi)$, then which of the following statements is/are TRUE? (A) $f(x)$ is increasing on $(0, \pi/2)$ (B) $f(x)$ is decreasing on $(\pi/2, \pi)$ (C) $f(x)$ is increasing on $(0, \pi)$ (D) $f(x)$ is decreasing on $(0, \pi)$ Choose the correct answer from the options given below:

Let $e^{\alpha y} + e^{\beta y} + \gamma x^2 + \delta \log|x| + C = 0$, where $C \in \mathbb{R}$ be a particular solution of the differential equation $x(e^{2y} - 1)dy + (x^2 - 1)e^ydx = 0$ and passes through the point $(1, 1)$. The value of $(\alpha + \beta + \gamma + \delta - C)$ is

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:

Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) The minimum value of } f(x) = (2x - 1)^2 + 3 & \text{(I) } 4 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) The maximum value of } f(x) = -|x + 1| + 4 & \text{(II) } 10 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) The minimum value of } f(x) = \sin(2x) + 6 & \text{(III) } 3 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) The maximum value of } f(x) = -(x - 1)^2 + 10 & \text{(IV) } 5 \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:

Match List-I with List-II Consider the function f(x) = 2x³ - 21x² + 36x + 80, x∈[0, 6]. Then | List-I | List-II | |---|---| | (A) one of its critical points is at x = | (I) -28 | | (B) Its absolute maximum value is | (II) -42 | | (C) Its absolute minimum value is | (III) 97 | | (D) Its second derivative at x = 0 is | (IV) 6 | Choose the correct answer from the options given below:

Match List-I with List-II [.] denotes the greatest integer function. | List-I | List-II | |---|---| | (A) $\int_0^3 [x]dx$ | (I) $\frac{1}{2}$ | | (B) $\int_0^1 [2x]dx$ | (II) 1 | | (C) $\int_0^1 [3x]dx$ | (III) $\frac{3}{2}$ | | (D) $\int_0^1 [4x]dx$ | (IV) 3 | Choose the correct answer from the options given below:

Match List-I with List-II (Given that $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \frac{dx}{\sqrt{a^2 - x^2}} =$ | (I) $\log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | | (B) $\int \sqrt{a^2 - x^2} dx =$ | (II) $\sin^{-1} \frac{x}{a} + c$ | | (C) $\int \sqrt{x^2 - a^2} dx =$ | (III) $\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c$ | | (D) $\int \frac{dx}{\sqrt{x^2 - a^2}} =$ | (IV) $\frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | Choose the correct answer from the options given below:

Match **List-I** with **List-II**. Here [x] denotes the greatest integer function $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = [x] & \text{(I) is continuous everywhere but not differentiable at } x=-1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x-1| & \text{(II) is continuous everywhere except at all integral values} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = e^{|x|} & \text{(III) is continuous everywhere but not differentiable at } x=1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x+1| & \text{(IV) is continuous everywhere but not differentiable at } x=0 \\[1.2ex] \hline \end{array}$ 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:

Match List-I with List-II | List-I | List-II | |------------|-------------| | (A) Degree of this differential equation $\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$ | (I) 1 | | (B) Order of this differential equation $e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$ | (II) 4 | | (C) Degree of $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$ | (III) not defined | | (D) Order of the differential equation $2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$ | (IV) 2 | Choose the correct answer from the options given below:

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:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\int \dfrac{dx}{x^2 - 16}$ | (I) $\dfrac{1}{8} \log \left\vert \dfrac{4 + x}{4 - x} \right\vert + c$, Where C is an arbitrary constant, | | (B) $\int \dfrac{dx}{x^2 + 16}$ | (II) $\log \left\vert x + \sqrt{x^2 - 16} \right\vert + c$, Where C is an arbitrary constant, | | (C) $\int \dfrac{dx}{16 - x^2}$ | (III) $\dfrac{1}{8} \log \left\vert \dfrac{x - 4}{x + 4} \right\vert + c$, Where C is an arbitrary constant, | | (D) $\int \dfrac{dx}{\sqrt{x^2 - 16}}$ | (IV) $\dfrac{1}{4} \tan^{-1} \left( \dfrac{x}{4} \right) + c$, Where C is an arbitrary constant, | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Integral | Solution: C is an arbitrary constant | | --- | --- | | (A) $\int \frac{dx}{x^2 + 25}$ | (I) $\frac{1}{10} \log \left\vert \frac{5 + x}{5 - x} \right\vert + C$ | | (B) $\int \frac{dx}{x^2 - 25}$ | (II) $\log \vert x + \sqrt{x^2 - 25}\vert + C$ | | (C) $\int \frac{dx}{25 - x^2}$ | (III) $\frac{1}{5} \tan^{-1} \left( \frac{x}{5} \right) + C$ | | (D) $\int \frac{dx}{\sqrt{x^2 - 25}}$ | (IV) $\frac{1}{10} \log \left\vert \frac{x - 5}{x + 5} \right\vert + C$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The value of $\int_{0}^{4} \vert x\vert \, dx$ is | (I) 3 | | (B) The value of $\int_{-2}^{2} \vert x\vert \, dx$ is | (II) -1 | | (C) The value of $\int_{0}^{3} [x] \, dx$ is | (III) 8 | | (D) The value of $\int_{-1}^{1} [x] \, dx$ is | (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) $(y'')^3 + (y')^4 - 6 = (y''')^2$ | (I) Order = 1, Degree = 2 | | (B) $\sqrt{(y')^2 + 5} = y''$ | (II) Order = 2, Degree = 3 | | (C) $(y')^2 = (2 + y'')^{3/2}$ | (III) Order = 2, Degree = 2 | | (D) $y = xy' + \sqrt{a^2(y')^2 + b^2}$ | (IV) Order = 3, Degree = 2 | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Interval for increasing/decreasing of function f(x) | | --- | --- | | (A) $f(x) = x\vert x\vert $ | (I) Decreases on $(0, \infty)$ | | (B) $f(x) = x^2 + 2x - 5$ | (II) Increases on $(3, \infty)$ | | (C) $f(x) = x^2 - 6x + 9$ | (III) Decreases on $(-\infty, -1)$ | | (D) $f(x) = -x^2$ | (IV) Increases on $(-\infty, \infty)$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Functions | Integrals | | --- | --- | | (A) $\int \frac{dx}{x^2 - 4}, x \neq \pm 2$ | (I) $\log \vert x + \sqrt{4 + x^2}\vert + C$, where $C$ is an arbitrary constant | | (B) $\int \frac{1}{\sqrt{16 - x^2}} dx; \vert x\vert < 4$ | (II) $\sin^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | | (C) $\int \frac{1}{16 + x^2} dx$ | (III) $\frac{1}{4} \log \left\vert \frac{x-2}{x+2} \right\vert + C$, where $C$ is an arbitrary constant | | (D) $\int \frac{1}{\sqrt{4 + x^2}} dx$ | (IV) $\frac{1}{4} \tan^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | Choose the correct answer from the options given below:

Match List-I with List-II: | List-I | List-II | | --- | --- | | **Differential Equations** | **Degree/Order** | | (A) Degree of the differential equation $\frac{d^3y}{dx^3} + 2 \log x.y = 0$ | (I) 3 | | (B) Order of the differential equation $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^4 + xy = 0$ | (II) 2 | | (C) Degree of the differential equation $\left(\frac{d^4y}{dx^4}\right)^2 + \left(\frac{dy}{dx}\right)^3 + x^2y = 0$ | (III) 1 | | (D) Order of the differential equation $\frac{d^3y}{dx^3} + y\left(\frac{dy}{dx}\right)^3 = 0$ | (IV) 4 |

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 | General solution | | --- | --- | | (A) $\dfrac{dy}{dx} = \dfrac{y}{x}; x \neq 0$ | (I) $y = cx; c \text{ is an arbitrary constant}$ | | (B) $x dx - y dy = 0; y \neq 0, x \neq 0$ | (II) $x^2 - y^2 = c; c \text{ is an arbitrary constant}$ | | (C) $\dfrac{(x^2 - 1)}{y^2 + 1}\dfrac{dx}{dy} = 1$ | (III) $2x + 3y = c; c \text{ is an arbitrary constant}$ | | (D) $2 dx + 3 dy = 0$ | (IV) $(x^3-y^3) = c + 3(x+y); c \text{ is an arbitrary constant}$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $f(x) = x \sin x$ | (I) is not continuous at $x = -3$ | | (B) $f(x) = \frac{\vert x\vert }{x}, x \neq 0$ and $f(x) = 1 \text{ at } x = 0$ | (II) is continuous everywhere | | (C) $f(x) = x - [x]$, $[x]$ denotes greatest integer function | (III) is not differentiable at $x = 1$ | | (D) $f(x) = e^{\vert x - 1\vert }$ | (IV) is not continuous at $x = 0$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Integral | Value | | --- | --- | | (A) $\int_{-1}^{1} (\vert x\vert + 1) dx$ | (I) 0 | | (B) $\int_{-2}^{2} \vert x + 1\vert dx$ | (II) 2 | | (C) $\int_{-1}^{1} 3\vert x^2\vert dx$ | (III) 5 | | (D) $\int_{-1}^{1} x\vert x\vert dx$ | (IV) 3 | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) Point of minima of $f(x) = \vert x+1\vert $ | (I) 1 | | (B) Minimum value of $f(x) = \vert x\vert $ | (II) -1 | | (C) Maximum value of $f(x) = 1 - x^2$ | (III) 2 | | (D) Minimum value of $f(x) = 2 + \sin^2 x$ | (IV) 0 | Choose the correct answer from the options given below:

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:

Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Property** | | (A) $f(x) = \begin{cases} \frac{x}{\vert x \vert} & : x \neq 0 \\ 0 & : x = 0 \end{cases}$ | (I) continuous but not differentiable at $x= 0$ | | (B) $f(x) = \vert x \vert$ | (II) continuous but not differentiable at $x=1$ | | (C) $f(x) = \vert x^2 - 1 \vert$ | (III) discontinuous at $x = 0$ | | (D) $f(x) = \vert x - 1 \vert$ | (IV) continuous but not differentiable at $x = 1, -1$ | Choose the **correct** answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Points of Non-Differentiability | | --- | --- | | (A) $f(x) = \vert x\vert + 1$ | (I) Not differentiable at $x = 3$ only | | (B) $f(x) = \vert x - 3\vert $ | (II) Not differentiable at $x = -3$ only | | (C) $f(x) = \vert x + 3\vert $ | (III) Not differentiable at $x = 3, -3$ only | | (D) $f(x) = \vert x^2 - 9\vert $ | (IV) Not differentiable at $x = 0$ only | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | Function | Derivative | | --- | --- | | (A) $y = \sin^{-1} x + \sin^{-1} \sqrt{1 - x^2}; \vert x\vert < 1$ | (I) $\frac{dy}{dx} = \frac{1}{2y-1}$ | | (B) $y = \sqrt{x + y}, x+y > 0 \text{ and } y \neq \frac{1}{2}$ | (II) $\frac{dy}{dx} = 10^x \log_e 10$ | | (C) $y = \log_{10} x, x > 0$ | (III) $\frac{dy}{dx} = 0$ | | (D) $y = 10^x$ | (IV) $\frac{dy}{dx} = \frac{1}{x \log_e 10}$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Degree of the differential equation $\frac{d^2y}{dx^2} = e^{dy/dx}$ is | (I) 2 | | (B) Order of the differential equation $(\frac{dy}{dx})^2 + \frac{d^3y}{dx^3} = 0$ is | (II) not defined | | (C) Degree of the differential equation $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 5x^2 = 0$ | (III) 3 | | (D) If p is the order and q is the degree of the differential equation $\frac{dy}{dx} + 3y = e^x$, then p + q is | (IV) 1 | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of differential equation $\frac{d^3y}{dx^3} = e^{\frac{dx}{dy}}$ | (I) 2 | | (B) The order of differential equation $\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0$ | (II) 4 | | (C) The sum of order and degree of differential equation $\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^5 = x$ | (III) not defined | | (D) The number of arbitrary constants in the general solution of a differential equation of order 2 | (IV) 3 | 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:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Integrating factor of $xdy - (y + x^2)dx = 0$ | (I) $x^2$ | | (B) Integrating factor of $xdy + (2y + x^2)dx = 0$ | (II) $x^3$ | | (C) Integrating factor of $(3y - x^2)dx + xdy = 0$ | (III) $x$ | | (D) Integrating factor of $(y + 3x^2)dx + xdy = 0$ | (IV) $\frac{1}{x}$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\int_{-a}^a f(x) dx = 0$ | (I) 0 | | (B) $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ | (II) 1 | | (C) $\int_{-\pi}^{\pi} \cos x dx$ | (III) $f$ is an odd function | | (D) $\int_{-1}^1 x^{101} dx + 1$ | (IV) $f(2a-x) = f(x)$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)$ | (I) 4 | | (B) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$ | (II) 1 | | (C) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0$ | (III) Not defined | | (D) The degree of differential equation $1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3$ | (IV) 3 | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$ | (I) $50\sqrt{x}$ | | (B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$ | (II) $-\frac{75}{x^2\sqrt{x}}$ | | (C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$ | (III) $\frac{-25}{x\sqrt{x}}$ | | (D) Marginal revenue if demand function $P=50\sqrt{x}$ | (IV) $75\sqrt{x}$ | Choose the **correct** answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Maximum value of $f(x) = \sin^2 x - \cos^2 x$, $\forall x \in (\pi, 2\pi)$ is | (I) 0 | | (B) Minimum value of $f(x) = \sin x \cos x$ | (II) 1 | | (C) Point of Minima of $f(x) = x^x$ $(x > 0)$ | (III) $-\frac{1}{2}$ | | (D) Maximum value of $f(x) = -x^{2026}$ | (IV) $\frac{1}{e}$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | Definite integral | Value | | (A) $\int_0^1 \frac{2x}{1 + x^2} dx$ | (I) 2 | | (B) $\int_{-1}^1 sin^3 x \cos^4 x dx$ | (II) $log_e\left(\frac{3}{2}\right)$ | | (C) $\int_0^{\pi} \sin x dx$ | (III) $log_e 2$ | | (D) $\int_2^3 \frac{2}{x^2 - 1} dx$ | (IV) 0 | Choose the correct answer from the options given below:

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:

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equation | Order and degree of differential equation | | (A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$ | (I) Order = 1, Degree = 2 | | (B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$ | (II) Order = 2, Degree = 1 | | (C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$ | (III) Order = 2, Degree = 2 | | (D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$ | (IV) Order = 2, Degree = Not defined | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equations | Order and degree | | (A) $ydx + x\log(y/x)dy - 2xdy = 0$ | (I) Order : 2, degree:1 | | (B) $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$ | (II) Order :1, degree:1 | | (C) $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$ | (III) Order : 3, degree:2 | | (D) $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$ | (IV) Order : 1, degree: Not defined | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | Differential equation | Integrating factor | | (A) $x\frac{dy}{dx} - y = 2x^2$ | (I) $e^{-y}$ | | (B) $\frac{dy}{dx} + \frac{y}{x} = 2x$ | (II) $\frac{1}{x}$ | | (C) $x\frac{dy}{dx} + 2y = x^2logx$ | (III) $x$ | | (D) $\frac{dx}{dy} - x = y$ | (IV) $x^2$ | 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:

Match List-I with List-II | List-I | List-II | |---|---| | (Differential equation) | (Order and Degree) | | (A) $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$ | (I) order = 3, degree = 1 | | (B) $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0$ | (II) order = 3, degree not defined | | (C) $2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0$ | (III) order = 2, degree = 3 | | (D) $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0$ | (IV) order = 2, degree = 1 | Choose the correct answer from the options given below:

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equation** | **Degree** | | (A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$ | (I) 3 | | (B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$ | (II) 1 | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$ | (III) not defined | | (D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$ | (IV) 2 | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equation | Integrating Factor | | (A) $y dx + (x - y^3)dy = 0$ | (I) $e^{-x}$ | | (B) $x\frac{dy}{dx} + y = x^2$ | (II) $\frac{1}{x}$ | | (C) $\frac{dy}{dx} - y = e^x$ | (III) $y$ | | (D) $x dy - y dx = x^3 dx$ | (IV) $x$ | Choose the correct answer from the options given below:

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:

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Integrating Factor** | | (A) $\frac{dy}{dx} + 2xy = 1$ | (I) $x$ | | (B) $x\frac{dy}{dx} + 2xy = 1$ | (II) $e^{2x}$ | | (C) $x\frac{dy}{dx} + y = 1$ | (III) $x^2$ | | (D) $x\frac{dy}{dx} + 2y = 2$ | (IV) $e^{x^2}$ | Choose the correct answer from the options given below:

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equations** | **Order and degree** | | (A) $\frac{dy}{dx} + e^y = 0$ | (I) order 2, degree not defined | | (B) $\frac{d^2y}{dx^2} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2}$ | (II) order 2, degree 1 | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + e^{(\frac{dy}{dx})} = 0$ | (III) order 1, degree 1 | | (D) $\frac{d^2y}{dx^2} + x\frac{dy}{dx} - 2y = logx; x > 0$ | (IV) order 2, degree 2 | Choose the **correct** answer from the options given below:

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Function** | **Increasing on the interval** | | (A) $f(x) = -x^2 - 2x + 1$ | (I) $(-\infty, -1)$ | | (B) $f(x) = x^2 + 1$ | (II) $(1, \infty)$ | | (C) $f(x) = x^2 - 2x + 3$ | (III) $(-\infty, 0)$ | | (D) $f(x) = -x^2$ | (IV) $(0, \infty)$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (Function) | (Derivative with respect to 'x') | | (A) $f(x) = x^x$ | (I) $ax^{a-1}$ | | (B) $f(x) = a^x$ | (II) 0 | | (C) $f(x) = a^a$ | (III) $a^x log_e a$ | | (D) $f(x) = x^a$ | (IV) $x^x(1 + log_e x)$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (Parametric equations) | $\left(\frac{dy}{dx}\right)$ | | (A) $x = \frac{2}{t}, y = 2t$ | (I) $4t^2$ | | (B) $x = t^3, y = 3t + 2$ | (II) $2(t+1)$ | | (C) $x = \log t, y = 2t^2$ | (III) $-t^2$ | | (D) $x = e^t, y = 2te^t$ | (IV) $t^{-2}$ | Choose the correct answer from the options given below:

Match **List-I** with **List-II** The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ for $x \in [2,5]$ has | List-I | List-II | |---|---| | (A) absolute maximum value | (I) 5 | | (B) absolute minimum value | (II) 60 | | (C) point of absolute maxima | (III) 3 | | (D) point of absolute minima | (IV) 32 | Choose the **correct** answer from the options given below:

Match List-I with List-II The function $f(x) = (x - 1)(x + 1)^2$ has | List-I | List-II | |---|---| | (A) A local maxima at $x = $ ____ | (I) $\frac{1}{3}$ | | (B) A local minima at $x = $ ____ | (II) 0 | | (C) The local minimum value of $f(x) = $ ____ | (III) -1 | | (D) The local maximum value of $f(x) = $ ____ | (IV) $-\frac{32}{27}$ | Choose the correct answer from the options given below:

Match List-I with List-II (where $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \tan x \, dx$ | (I) $\log\vert \sec x + \tan x\vert + c$ | | (B) $\int \cot x \, dx$ | (II) $\log\vert \sec x\vert + c$ | | (C) $\int \sec x \, dx$ | (III) $\log\vert \sin x\vert + c$ | | (D) $\int \cosec x \, dx$ | (IV) $\log\vert \cosec x - \cot x\vert + c$ | Choose the correct answer from the options given below:

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:

Particular solution of the differential equation $x(1 + y^2)dx - y(1 + x^2)dy = 0$, given $y = 0$ when $x = 1$, is

$\int e^{-x}(\cot x + \cosec^2 x)dx =$

$\int_0^2 (|x| + |x - 2|) dx =$

$\int \frac{e^{2x} - 1}{e^{2x} + 1} dx =$

$\int \frac{dx}{x^3\sqrt{(1 + x^4)}} =$

Shown below is the graph of parabola $y^2=x$, the area (in sq. units) of the shaded region is: 

Solution of the differential equation $\frac{dy}{dx} = \sqrt{1 + x^2 + y^2 + x^2y^2}$ is : (Here $C$ is an arbitrary constant)

Solution of the differential equation $y\log_e y dx - x dy = 0$ is (Where c is an arbitrary constant)

The absolute maximum value of the function $f(x) = 4x - \frac{1}{2}x^2$ in the interval $\left[-2, \frac{9}{2}\right]$ is

The area bounded by $y = 3x + 1$, $x = 0$, $y = 0$ and $x = a$ is 8 Sq.units. Then value of $a$ (where $a > 0$) is

The area bounded by the curve $y = 4 + 3x - x^2$ and $x$-axis is equal to

The area bounded by the curve $y = \log x, y = 0$ and $x = e$, is

The area bounded by the x-axis and the parabola $y = 3x-x^2$ is:

The area enclosed between the graph of y = x³ and the lines x = 0, y = 1, y = 8 is

The area (in sq. units) bounded by the curve $y = \cos x$ and x-axis between $x = 0$ and $x = \frac{3\pi}{2}$ is

The area (in sq. units) bounded by the parabola $y^2 = 16x$ and its latus rectum is

The area (in sq. units) bounded by the parabola $y^2 = 4ax$, its latus rectum and the $x$-axis in the first quadrant is:

The area (in sq. units) of the bigger portion of region enclosed by the curves $4x^2 + 9y^2 = 36$ and $2x + 3y = 6$ is

The area (in sq. units) of the region bounded by the parabola $y^2 = 4x$ and the line $x = 1$ is

The area (in sq. units) of the region $\{(x,y): 3x^2 \leq y \leq |x|\}$ is equal to

The area (in sq. units) of the region enclosed by the ellipse $16x^2 + 25y^2 = 400$ is

The area (in sq. units) of the region enclosed by the curve $9x^2 + 4y^2 = 36$ is

The area (in sq. units) of the region bounded by the lines $y = 2x + 3$, the x – axis and the ordinates $x = -2$ and $x = 2$ is equal to

The area (in sq. units) of the region bounded by $y = 2\sqrt{1 - x^2}$, $x \in [0, 1]$ and $x$-axis is equal to

The area (in sq. units) of the region bounded by the curve $y = x^5$, the x-axis and the ordinates $x = -1$ and $x = 1$ is equal to

The area (in sq. units) of the region bounded by the curve $y = \sin x, -2\pi \leq x \leq 2\pi$ and $x - axis$ is equal to

The area (in sq. units) of the region bounded by the curve $y = \sqrt{16-x^2}$ and x-axis is

The area (in Sq. units) of the region bounded by $y = -2$, $y = 2$, $x = y^3$ and $x = 0$ is equal to

The area (in sq. units) of the region bounded by the parabola $y^2 = 8x$ and the line $x = 2$ is

The area (in sq. units) of the region bounded by $y = -1, y = 2, x = y^3$ and $x = 0$ is equal to

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 area (in sq. units) of the region bounded by the line $y = x + 2$, $x = 0$, $x = 1$ and $y = 0$ is

The area (in sq. units) of the region bounded by the curve $y = 2x^3$, $x$ - axis and ordinates $x = -1$ and $x = 1$ is:

The area (in sq. units) of the region bounded by the curve $x^2 = 250y$, $y = 0$ and $x = 50$ is

The area (in square units) bounded by the curve $y = \cos x$ between $x = 0$ and $x = 2\pi$ in first quadrant is equal to:

The area (in square units) of the region bounded by the curve $x^2 = y$ and the straight line $y = 4$ in the first quadrant is equal to

The area (in square units) of the region enclosed between the lines $x + y = 2$, $x = 0$, $x = 3$ and $x$-axis is equal to

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 (in sq.units) of region bounded by $y^2 = 9x$, $x = 2$, $x = 4$ and the $x$-axis in the first quadrant is

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

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

The area (in sq.units) of the region enclosed by the curve $y = \cos x$, $\frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ and the x - axis is:

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 area of the region bounded by $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is $\frac{m}{n}$ sq. units, where $\gcd(m, n) = 1$, then $m - n$ is equal to:

The area of the region bounded by parabola $x^2 = 4y$, straight line $x = 2$ and $x$-axis, is

The area of the region bounded by the curve $y = x + 1$, $x = axis$ and the lines $x = 2$ and $x = 3$ is

The area of the region bounded by the parabola $y^2 = x$ and the straight line $2y = x$ is

The area of the region bounded by the curves $y = x^2 + 2$ and $x$-axis, between $x = 0$ and $x = 3$ in the first quadrant is:

The area of the region bounded by the curves $y = x$ and $y = x^3$ is:

The area of the region bounded by the parabola $y^2 = 8x$ and its latus rectum in the first quadrant, is

The area of the region bounded by the line $y = 2x$ and the x-axis between $x = -2$ and $x = 2$ is

The area of the region bounded by the curves $y = x^2 + 2$, $y = x$, $x = 0$ and $x = 2$ is

The area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant, is

The area of the region (in square units) bounded by $x=1, x=2$ and the curve $y^2 = 4x$ in the first quadrant is

The area of the region $\{(x, y): x^2 + y^2 \leq 1 \leq x + y\}$ is

The area of the smaller region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and the straight line $3x + 4y = 12$ is:

The area of the smaller region of the circle $x^2 + y^2 = 8$ cut off by the line $x = 2$ is

The area (sq.units) bounded by the curve y = sinx, π ≤ x ≤ 2π and the x-axis is

The curve $x = y^2$ and $xy = k$ cut orthogonally, then $k^2$ is equal to:

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 demand for a certain product is represented by the function $p = 150 + 10x - x^2$ (in Rs.) where $x$ is the number of units demanded and $p$ is the price per unit, then the value of marginal revenue, when 10 units are sold is

The demand for a certain product is represented by the function $p = 300 + 25x - x^2$ (in rupees), where x is the number of units demanded and p is the price per unit, then the marginal revenue when 15 units are sold, 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:

The demand function for a commodity is $p = 35 - 2x - x^2$, then the consumer's surplus at equilibrium price $p_0 = 20$ is

The demand function (in Rs.) for a product is given by $P = 20 - 0.25x$, where P is the price per unit and x is the number of units sold, then the price of one unit, when the revenue is maximized, is:

The derivative of $(\log x)^x$ with respect to $\log x$ is

The differential equation of the family of curves $y = Ae^{3x} + Be^{-3 x}$, where $a$ and $\beta$ are arbitrary constants, is

The differential equation representing the curve $y = e^{2x}(a + bx)$, where a, b are arbitrary constants is

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$, $x \neq 0$ where A and B are arbitrary constants, is given by

The edge of a cube is increasing at a rate of 7 cm/s. The rate of change of area of the cube when its side is 3 cm is:

The edge of a cube is increasing at a rate of 7cm/s. The rate of change of area of the cube when edge of the cube is 3cm is:

The equation of tangent line to $y = 2x^2 + 7$, which is parallel to the line $4x - y + 3 = 0$ is

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

The equation of the tangent to the curve $y = \frac{(x - 3)}{(x - 1)(x - 2)}$ at the point, where it cuts x-axis is:

The function $f(x) = tanx - x$

The function $f(x) = \sin 3x$, $x \in \left[0, \frac{\pi}{2}\right]$ (A) is increasing on $\left[0, \frac{\pi}{6}\right]$ (B) is decreasing on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$ (C) is increasing on $\left[0, \frac{\pi}{2}\right]$ (D) is decreasing on $\left[0, \frac{\pi}{2}\right]$ Choose the correct answer from the options given below:

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$ is

The function $f(x) = x + \frac{1}{x}$ has

The function $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 163$ has a local maxima at

The function $f(x) = x + \frac{a^2}{x}$, $a > 0$, $x \neq 0$ has a local maxima at

The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at

The function $f(x) = x^2e^{-2x}$ increases on

The function $f(x) = x^2 - 4x + 6$ is (A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$ (B) Strictly increasing on $(2, \infty)$ (C) Strictly increasing on $(-\infty, \infty)$ (D) Strictly decreasing on $(-\infty, 2)$ Choose the correct answer from the options given below:

The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is (A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$ (B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$ (C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$ (D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$ (E) strictly increasing on $(0, \pi)$ Choose the correct answer from the options given below:

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is:

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

The function $f(x) = \frac{x}{3} + \frac{3}{x}$ is increasing in the interval:

The function $f(x) = x^4 - 2x^2$ is increasing on

The function $f(x) = \frac{x}{2} + \frac{2}{x}, x \neq 0$ is increasing on (A) $(-\infty, -2)$ (B) $(-2, 2)$ (C) $(2, \infty)$ (D) $(-1, 1)$ Choose the correct answer from the options given below:

The function $f(x) = kx^3 + 6kx^2 + 18x + 17$ is increasing on $\mathbb{R}$(set of real numbers) if:

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)

The function $f(x) = x^2 - x + 1$ is

The function, $f(x) = x - \frac{1}{x}$ is

The function $f(x) = 6 - 6x - 2x^2$

The function $f(x) = x^3 + 3x^2 + 4x + 4$, $x \in \mathbb{R}$ (set of real numbers) :

The function $f(x) = [x]$, where $[x]$ denotes the greatest integer function, is continuous at $x =$ (A) 2.9 (B) 5 (C) -3 (D) 6.5 Choose the correct answer from the options given below:

The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as $f(x) = x^2 + 2x$ is

The general solution of differential equation $\frac{dy}{dx} = e^{x+y}$ is

The general solution of the differential equation $ydx - (x + 2y^2)dy = 0$

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^2e^{-y}$ is equal to:

The general solution of the differential equation $\frac{xdy}{dx} + 4y = x^3, (x \neq 0)$ is:

The general solution of the differential equation $(x^2 - yx^2)dy + (y^2 + x^2y^2)dx = 0$ is:

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

The general solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = ax + by$ is

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

The general solution of the differential equation $\frac{dy}{dx} = xy + x + y + 1$ is

The general solution of the differential equation $x\left(\frac{dy}{dx}\right) = y + x \tan\left(\frac{y}{x}\right)$ is

The general solution of the differential equation $e^x dy + (y e^x + 2x)dx = 0$ is

The general solution of the differential equation $\frac{dy}{dx} = e^{ax+by}$ is: (Here C is an arbitrary constant)

The general solution of the differential equation $\frac{dy}{dx} + y \tan x = \sec x$

The general solution of the differential equation $(1 + e^x)dy + ye^x dx = 0$, where $y > 0$, is

The greatest integer function $f(x) = [x]$ is differentiable for all values of

The greatest possible value of '$a$' such that the function $f(x) = x^2 + a x + 1$ is always decreasing in the interval [1, 2] is:

The integral I = $\int e^x\left(\frac{x - 1}{3x^2}\right) dx$ is equal to

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 integral $\int e^x\left(\frac{x-1}{2x^2}\right)dx$ is equal to

The integral $\int \frac{2dx}{e^{2x}-1}$ is equal to:

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is

The integrating factor of the differential equation $(x log_e x)\frac{dy}{dx} + y = 2log_e x$ is

The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

The interval in which the function $g(x) = x^2 e^{-x}$ is increasing is:

The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing, is

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is

The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to

The interval on which the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly decreasing, is:

The interval(s), where the function $f(x) = \begin{cases} \frac{1-e^x}{e^{2x}-1} & : x \neq 0 \\ \frac{-1}{2} & : x = 0 \end{cases}$ is increasing, is/ are:

The largest interval, in which the function $f(x) = x^3 + 2x^2 - 1$ is increasing, is:

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is

The length of a rectangle is decreasing at the rate of 4 cm/minute and the width is increasing at the rate of 3 cm/minute, then the rate of change of the perimeter is

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

The marginal cost of production of x units of a commodity is $56 + \frac{3}{2}x$. It is known that fixed costs are Rs.115. Then the total cost of producing 50 units is:-

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is

The maximum value of f(x) = $\left(\frac{1}{x}\right)^x$ is

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

The maximum value of the function $f(x) = x^2(60 - x)$ in [20, 80] is:

The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

The nearest integral value of the shaded area shown below is: 

The number of arbitrary constants in the general solution of a differential equation with degree 1 and order 3, is

The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is

The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

The number of tangents to the curve $xy - 3y + 2 = 0$ having slope 2 is:

The order of $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \left[a \frac{d^2y}{dx^2}\right]^{\frac{1}{2}}$ is

The particular solution of the differential equation $e^x\sqrt{1-y^2}dx + \frac{y}{x}dy = 0$, given that $y = 1$, when $x = 0$ is:

The particular solution of the differential equation $xdy = (2x^2 + 1)dx, x \neq 0$, given that $y = 1$ when $x = 1$ is:

The particular solution of the differential equation $\frac{dy}{dx} + \frac{3y}{x} = 0$, $y(1) = 1$ is

The particular solution of the differential equation $\log\left(\frac{dy}{dx}\right)= 3x + 4y$ satisfying $y = 0$ when $x = 0$ is:

The particular solution of the differential equation $\left[x \sin^2\left(\frac{y}{x}\right) - y\right]dx + xdy = 0$, $y = \frac{\pi}{4}$ when $x = 1$ is

The particular solution of the differential equation $\frac{dy}{dx} = 8yx$ when $y = 1$ at $x = 0$

The particular solution of the differential equation $\frac{dy}{dx} = e^{x^2/2} + xy$, when $x = 0$, $y = 1$, is

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

The point of local maxima of the function $f(x) = (x - 2)^5(x + 2)^2$ is

The point on the curve $\frac{x^2}{4} + \frac{y^2}{9} = 1$ at which the tangent to the curve is parallel to the x-axis is

The point on the curve $y^2 = 8x$ for which the abscissa and ordinate change at the same rate, 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:

The product of order and degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + x^2y\left(\frac{d^2y}{dx^2}\right)^3 = 2x^5$ is:

The radius of spherical balloon is decreasing at the rate of 0.1cm/sec, the rate at which its volume is decreasing, when its radius is 0.5cm is

The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

The rate of change of area of a circle with respect to its circumference when radius in 6 cm, is

The rate of change of the area of a circle with respect to its radius $r$, when $r = 3$cm, is:

The rate of change of volume of a sphere with respect to its surface area, when radius is 4 cm, is equal to

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6cm is:

The real valued function $f(x) = 12x^\frac{4}{3} - 6x^\frac{1}{3}, x \in [-8, 8]$ has absolute maximum value equal to

The real valued function $f(x) = x^{15} + 5x^9 + 10$ is increasing for___________.

The semi vertical angle of a right circular cone of maximum volume of a given slant height is

The sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases when the side is 10 cm, is

The slope of the normal to the curve $y = 2x^2$ at $x = 1$ is:

The solution of the differential equation $\frac{dy}{dx} - \frac{y}{x} = 2\log_e x$

The solution of the differential equation $\frac{dy}{dx} = \sqrt\frac{{y}}{x}$ is

The solution of the differential equation $(x^2 + xy)dy = (x^2 + y^2)dx$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

The solution of the differential equation $ydx + (x - y^2)dy = 0$ is

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

The solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = 5x + 2y$ is given by

The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)

The solution of the differential equation $\frac{dr}{dt} = -rt, r(0) = r_0$ is

The solution of the differential equation $(x + 1)\frac{dy}{dx} + 1 - 2e^{-y} = 0$, $y(0) = 0$ is

The solution of the differential equation $\frac{dy}{dx} = \frac{ax + c}{by + d}$ represents a circle when

The solution of the differential equation $xdy - ydx = 0$ represents

The sum of order and degree of the differential equation $y = x\frac{dy}{dx} + 2\sqrt{1 + \left(\frac{dy}{dx}\right)^2}$ is

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

The sum of the order and degree of the differential equation representing the family of curves $y = mx + m^4$, where m is arbitrary constant, is

The sum of two positive numbers is 60. If the sum of their squares in minimum, then the absolute value of the difference of their cubes is

The total cost $c(x)$ associated with the production of $x$ units of an item is given by $c(x) = 0.001x^3 + 0.06x^2 + 20x + 500$. The marginal cost when 10 units are produced is:

The total cost function is given by $c(x) = \frac{1}{3}x^3 - 5x^2 + 30x - 15$ and selling price per unit is Rs.6. The profit is maximum if the value of x is:

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$. The marginal cost when 10 items are produced is:

The two positive numbers whose sum is 16 and the sum of whose squares is minimum then the positive numbers are:

The value of derivative of the function $\cot^{-1}\{(\cos 2x)^{1/2}\}$ at $x = \frac{\pi}{6}$ is

The Value of $\int_1^3 |2x - 1|dx$ equal to

The value of $\int \frac{x^5}{\sqrt{1 + x^3}} dx$ is

The value of $\int_{-1}^{1}|x|dx$ is

The value of $\int_{-1}^{1} |x^3 - x| dx$ is

The value of $\int_0^{\pi/2} \log_e \left(\frac{5 + 2 \sin x}{5 + 2 \cos x}\right) dx$ is

The value of $\int_{0}^{\pi/2} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} dx$ is

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|)dx$, is equal to:

The value of $\int_2^4 \frac{x}{x^2 + 1} dx$ is equal to

The value of $\int \frac{(x^4 - x)^{1/4}}{x^5} dx$ is equal to (where C is an arbitrary constant)

The value of $\int_0^1 \log_e\left(\frac{1}{x} - 1\right)dx$ is:

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|) dx$ is

The value of $\int_{-\pi/2}^{\pi/2}(x^5 + x^3\cos x)dx$ is

The value of $\int \left\{ \frac{1}{\log_e x} - \frac{1}{(\log_e x)^2} \right\} dx$ is

The value of $\int_{-5}^{5} |x + 3| dx$ is

The value of $\int_0^1 [\log x - \log(1-x)] dx$ is

The value of $\int_0^1 x e^x dx$ is:

The value of k for which the function $f(x) = \begin{cases} \frac{1-\cos 8x}{16x^2}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$ is:

The value of k for which the function, defined by, $f(x) = \begin{cases} \frac{3x + 4 \tan x}{x} & : x \neq 0 \\ k & : x = 0 \end{cases}$ is continuous at $x = 0$, is

The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is:

The value of the definite integral $I = \int_0^2 x\sqrt{2-x} dx$ is:

The value of the definite integral $\int_0^1 e^x \frac{(1-x)^2}{(1+x^2)^2}dx$ is: