List of top Questions asked in AP EAPCET- 2022

An iron sphere of diameter $ D $ and mass $ M $ is immersed in hot water, causing its temperature to rise by $ \Delta T $. If $ \alpha $ is the coefficient of linear expansion, the change in surface area of the sphere is:
If two vectors $ \vec{A} $ and $ \vec{B} $ are mutually perpendicular, then the component of $ \vec{A} \cdot \vec{B} $ along the direction of $ \vec{A} + \vec{B} $ is
A body is travelling with $ 10\, \text{m/s} $ on a rough horizontal surface. Its velocity after $ 2\, \text{s} $ is $ 4\, \text{m/s} $. The coefficient of kinetic friction between the block and the plane is
(acceleration due to gravity = $ 10\, \text{m/s}^2 $)

A small disc of mass $ m $ slides down with initial velocity zero from the top (A) of a smooth hill of height $ H $, having a horizontal portion (BC) as shown in the figure. If the height of the horizontal portion of the hill is $ h $, then the maximum horizontal distance covered by the disc from the point D is

A block of mass $ 50\, \text{kg} $ is pulled at a constant speed of $ 4\, \text{m/s} $ across a horizontal floor by an applied force of $ 500\, \text{N} $ directed $ 30^\circ $ above the horizontal. The rate at which the force does work on the block in watts is:
A particle is executing simple harmonic motion with instantaneous displacement: $$ x = A \sin^2\left( \omega t - \frac{\pi}{4} \right) $$ The time period of oscillation of the particle is:
Solve the differential equation: $$ \frac{dy}{dx} = \cos^2(3x + y) $$
The general solution of the differential equation: $$ \cos^2 x \cdot \frac{dy}{dx} + y = \tan x $$ is $$ y = \tan x - 1 + C e^{-\tan x} $$ If this solution satisfies $ y\left(\frac{\pi}{4}\right) = 1 $, then find $ C $.
Assertion (A): The order of the differential equation of a family of circles with constant radius is two.
Reason (R): An algebraic equation involving two arbitrary constants corresponds to the general solution of a second order differential equation.
The energy $ E $ of a system is a function of time $ t $ and is given by: $$ E(t) = \alpha t - \beta t^3 $$ where $ \alpha $ and $ \beta $ are constants. Find the dimensions of $ \alpha $ and $ \beta $.
A student is at a distance of 16 m from a bus. The bus starts accelerating from rest with $ a = 9 \, \text{m/s}^2 $. The minimum velocity with which the student should run towards the bus to catch it is $ \alpha \sqrt{2} \, \text{m/s} $. Find $ \alpha $.
The component of a vector $ \vec{P} = 3\hat{i} + 4\hat{j} $ along the direction $ \hat{i} + 2\hat{j} $ is
Evaluate: $$ \int \frac{e^{\tan^{-1}x}}{1+x^2} \left[\left(\sec^{-1}(\sqrt{1+x^2})\right)^2 + \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)\right] dx $$
Evaluate: $$ \int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} = ? $$
Let $$ I_n = \int (\cos^n x + \sin^n x) \, dx,\quad \text{and} \quad \frac{n - 1}{n} I_{n - 2} = \frac{\sin x \cos x}{n} f(x) $$ Then, what is $ f(x) $?
Let $ T>0 $ be a fixed number. If $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function such that $ f(x + T) = f(x) $, then: $$ \text{If } I = \int_0^T f(x)\, dx,\quad \text{then } \int_0^{5T} f(2x)\, dx = ? $$
If $$ \int_1^3 x^n \sqrt{x^2 - 1} \, dx = 6 $$ Then find the value of $ n $.
If $ [\cdot] $ denotes the greatest integer function, then evaluate: $$ \int_{-1}^1 \left( x \cdot [1 + \sin \pi x] + 1 \right) dx $$
Given: $$ \lim_{n \to \infty} \left[ \frac{n}{(n+1)\sqrt{2n+1}} + \frac{n}{(n+2)\sqrt{2(n+2)}} + \frac{n}{(n+3)\sqrt{3(2n+3)}} + \cdots \text{(n terms)} \right] = \int_0^1 f(x)\, dx $$ Then find $ f(x) $.
If $ x = f(\theta) $, $ y = g(\theta) $, then find: $$ \frac{d^2 y}{dx^2} $$
If the normal drawn at a point $ P $ on the curve $ 3y = 6x - 5x^3 $ passes through the origin $(0, 0)$, then the positive integral value of the abscissa of point $ P $ is:
The line joining the points $ (0, 3) $ and $ (5, -2) $ is a tangent to the curve: $$ y = \frac{C}{x + 1} $$ Then find $ C $.
The function $ y = x^3 - ax^2 + 48x + 7 $ is increasing for all real values of $ x $. Find the interval in which $ a $ lies:
If $ a, b>0 $, then the minimum value of: $$ y = \frac{b^2}{a - x} + \frac{a^2}{x},\quad \text{for } 0<x<a $$
The point on the curve $ y = x^2 + 4x + 3 $ that is closest to the line $ y = 3x + 2 $ is: