List of top Questions asked in AP EAPCET

If the points of contact of the tangents drawn from \( (0,0) \) to the curve \( y = x^2 + 3x + 4 \) are \( (\alpha, \beta) \) and \( (\gamma, \delta) \), then \( \beta + \delta = \):

Evaluate the integral: \[ \int \frac{dx}{4 + 5 \cos x} \]

If \( f(x) = \int_0^x \frac{5t^8 + 7t^6}{(t^2 + 2t + 1)^2} dt \) and \( f(0) = 0 \), then the value of \( f(1) \) is:

Evaluate the integral: \[ \int_0^{\frac{\pi}{4}} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \]

Evaluate the integral \( \int x^3 (\log x)^2 dx \):

If \[ \int \frac{1}{x(\log x)^2 + 4\log x - 1} \, dx = A \log [ \log x + B ] + K \] where \( K \) is the constant of integration, then \[ \int \frac{1}{x(\log x)^2 + 4\log x - 1} \, dx = A \log [ \log x + C ] + K. \]

If \( f(x) = \int_0^x \left[ (a+1)(t+1)^2 - (a-1)(t^2 + t + 1) \right] dt \), then a possible positive value of \( a \), for which \( f'(x) = 0 \) has equal roots, is:

Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{\sin \left( \frac{\pi}{4} + x \right) + \sin \left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} \, dx \]

Evaluate the integral: \[ \int_0^1 \left( \sqrt{10} \right)^{2x} \, dx \]

The area (in sq. units) of the region bounded by the curves \( y = 4 \cos x \) and \( y = -|\cos x| \) from \( x = -\frac{\pi}{2} \) to \( x = \frac{\pi}{2} \) is:

If \[ x^a \frac{dy}{dx} = y^\beta \left( \gamma \log x + \delta y + 1 \right) \] is a homogeneous differential equation, then

The general solution of the differential equation \( \tan x \tan y \, dx + \cos^2 x \csc^2 y \, dy = 0 \) is:

If \( \alpha \) and \( \beta \) are respectively the order and degree of the differential equation \[ y = e^{\frac{d^2y}{dx^2}}, \] then the value of \( \alpha + \alpha^\beta + \alpha^{2\beta} + \dots + \alpha^{2023\beta} \) is:

Among the following, the unit of permeability is NOT represented by

A truck moving with a constant velocity \( 12 \, \text{ms}^{-1} \) crosses a car moving from rest with uniform acceleration \( 2 \, \text{ms}^{-2} \). The distance the car has to travel from the starting point to cross the truck again is:

If \( \mathbf{F} = (4\hat{i} - 10\hat{j}) \, \text{N} \) and \( \mathbf{r} = (-5\hat{i} - 3\hat{j}) \, \text{m} \), then \( \mathbf{r} \times \mathbf{F} \) is:

Two forces whose magnitudes are in the ratio 5:3 are acting at a point at an angle of \( 60^\circ \) simultaneously. If the resultant of the two forces is 35 N, then the magnitudes of two forces respectively are:

A body of mass 2 kg is moving with a constant acceleration of \( \left( 2\hat{i} + 3\hat{j} - \hat{k} \right) \, \text{ms}^{-2} \). If the displacement made by the body is \( \left( 3\hat{i} - \hat{j} + 2\hat{k} \right) \, \text{m} \), then the work done is:

In the case of non-conservative forces, the following statement is correct:

Two particles of masses 5 g and 3 g are separated by a distance of 40 cm. The centre of mass of the system of these two particles lies:

As shown in the figure, a block of weight 20 N is connected to the top of a smooth inclined plane by a massless spring of constant \( 8\pi \, \text{Nm}^{-1} \). If the block is pulled slightly from its mean position and released, the period of oscillations is:

The orbital velocity of a body near the surface of a planet 'A' is equal to escape velocity of a body from the planet 'B'. If the masses of planets A and B are same, the ratio of their radii is:

A spherical ball of volume 2000 cm\(^3\) is subjected to a hydraulic pressure of 15 atm. If the change in volume is \( 5 \times 10^{-2} \) cm\(^3\), the bulk modulus of the material of the spherical ball is:

The gauge pressure at a depth of 50 m in a sea is

The emissivity of a perfect black body is increased to 16 times by increasing its temperature. If the initial temperature is \( T \), then final temperature of that black body is: