List of top Questions asked in AP EAPCET

A charge \( Q \) is to be divided between two objects. The values of the charges on the objects so that the electrostatic force between them will be maximum is:

In a space having electric field \( \vec{E} = A(x\hat{i} + y\hat{j}) \), the potential at point (10 m, 20 m) is zero. Then the potential at the origin is: (Given \( A = 10 \, \text{Vm}^{-2} \))

A Carnot engine operating between temperatures \( T_H = 600\,K \) and \( T_C = 300\,K \), absorbs \( Q_H = 800 \, \text{J} \) of heat from the source. The mechanical work done per cycle is:

An object of mass 3 kg is tied by a string of negligible mass to a ceiling and held such that the string is taut. The object is released suddenly such that the string remains taut. It’s acceleration when released is (acceleration due to gravity = 10 m s$^{-2}$)

A force $\vec{F} = 4\hat{i} - 15\hat{j}$ N acts on a body resulting in a displacement of $6\hat{i}$. If the body had kinetic energy of 7 joules at the beginning of the displacement, the kinetic energy at the end of the displacement is

Consider a force $F = K x^3$ which acts on a particle at rest. The work done by the force for the displacement of 2 m is ($K = 2 \, \text{N m}^{-3}$)

The centre of mass of a homogeneous semi-circular plate of radius $r$ is located at A as shown in the figure. The distance $OA$ is

A sphere rolls down from the top of an inclined plane which makes an angle $30^\circ$ with the edge of a horizontal roof of a house. If the highest and lowest points of the inclined plane are 8.75 m and 3.75 m respectively from the ground then the horizontal distance from the lower edge of the roof at which the sphere hits the ground is (acceleration due to gravity = 10 m s$^{-2}$)

At the moment $t = 0$, a time dependent force $F = at$ (where $a$ is constant equal to 1 N s$^{-1}$) is applied on a body of mass 1 kg resting on a smooth horizontal plane as shown in the figure. If the direction of this force makes an angle $45^\circ$ with the horizontal, then the velocity of the body at the moment it leaves the plane is (acceleration due to gravity = 10 m s$^{-2}$)

The dot product of unit vectors $\hat{n}_1$ and $\hat{n}_2$ that are parallel to $5\hat{i} + 12\hat{j}$ and $3\hat{i} + 4\hat{j}$ respectively is

The derivative of the function $f(x) = \sin(x^2)$ at $x = \sqrt{\pi}$ is

The integrating factor of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is a solution of the differential equation

By multiplying with $e^{\int P dx}$ on both sides of the equation $\frac{dy}{dx} + P(x)y = Q(x)$, the left side of the equation takes the form $\frac{d}{dx} (y f(x))$, then $f(x) =$

Let \( f(x) = \int \frac{e^{3x}}{4 + 8e^{2x} + e^{4x}} \, dx \), and \( g(x) = \int \frac{2\, dx}{e^{3x} + 8e^x + 4e^{-x}} \), then \( f(x) - g(x) = \) ?

If \( a, b, c, d \ne 0 \) and \( \int \frac{at + b}{ct + d} dt = t \), then find \( \int f(x) dx \) where \( f(x) = \frac{ax + b}{cx + d} \)

Evaluate: \( \int_{0}^{4} \frac{x + 2}{\sqrt{4x - x^2}}\, dx \)

If the line \( ax + by + c = 0 \) is a normal to the curve \( xy = 1 \), then which of the following is true?

Find the area of the triangle formed by the tangent to the curve \( xy = a^2 \) at \( (x_1, y_1) \) and the coordinate axes.

Two curves \( y = a^x \) and \( y = b^x \) intersect at some angle \( \alpha \). Find \( \tan \alpha \).

Evaluate \( \int \frac{dx}{1 + a \cos x} \), given \( a>|\sec \theta| \)

In the interval \( (7, \infty) \), the function \( f(x) = |x - 5| + 2|x - 7| \) is:

If \( \frac{d^n y}{dx^n} = y_n \) and \( y = e^{\sqrt{x}} + e^{-\sqrt{x}} \), then find the value of \( 4x y_2 + 2 y_1 \).

Evaluate: \[ \lim_{n \to \infty} \sqrt{2} \left[ \frac{ \left(2 + \sqrt{2} \right)^n + \left(2 - \sqrt{2} \right)^n }{ \left(2 + \sqrt{2} \right)^n - \left(2 - \sqrt{2} \right)^n } \right] \]

A plane makes intercepts \(-2, \frac{4}{3}, \frac{-4}{5}\) on the X, Y, Z axes respectively. Find the direction cosines of a normal to the plane.

The centroid of a triangle with vertices \( (a, 1, 3), (-2, b, -5), (4, 7, c) \) lies at the origin. Find \( a^2 + b^2 + c^2 \).