List of top Questions asked in AP EAPCET

If \( f(x) \) is defined as follows:

\[ f(x) = \begin{cases} \frac{x - \lfloor x \rfloor}{x - 2} & \text{if } x = 2 \\ \frac{|x - \lfloor x \rfloor|}{a^2 + (x - \lfloor x \rfloor)^2} & \text{if } 1 < x < 2 \\ 2a - b & \text{if } x = 1 \end{cases} \]

Then the limit \( \displaystyle \lim_{x \to 0} \frac{\sin(ax) + x \tan(bx)}{x^2} \) is:

Let \( [x] \) represent the greatest integer not more than \( x \). The discontinuous points of the function \[ f(x) = \frac{5 + [x]}{\sqrt{11 + [x]} - 6x + 2 + [x]} \] lie in the interval:

If \( x^x y^y = e^e \), then \( \left( \frac{d^2 y}{dx^2} \right)_{(e, e)} = \):

If \( f(x) = |x - 5| + |x + 5| + |x - 4| + |x + 4| \), then \[ \frac{f'(1) - f'(-6)}{f'(1) + f'(-6)} \]

If \( f(x + ay) + g(x - ay) = 0 \), then \( \frac{dy}{dx} = \):

If the angle between the curves \( y = e^{(x+4)} \) and \( x^2 y = 1 \) at the point \( (1, 1) \) is \( \theta \), then \[ \sin \theta + \cos \theta = \dots \]

If a line is moving between the coordinate axes such that the sum of the intercepts made by it on the coordinate axes is always 12, then the equation of that line which forms a triangle of maximum area with the coordinate axes is:

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 \( \int x^3 (\log x)^2 dx \):

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:

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:

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:

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:

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:

The temperature of the sink of a Carnot's engine is 300 K and the efficiency of the engine is 0.25. If the temperature of the source of the engine is increased by 100 K, the efficiency of the engine increases by

The specific heat capacity of a monatomic gas at constant volume is \( x % \) of its specific heat capacity at constant pressure. Then \( x \) =