List of top Questions asked in AP EAPCET- 2023

The lines \( p(x^2 + 1) + x - y + q = 0 \) and \( (p^2 + 1)x^2 + (p^2 + 1)y + 2q = 0 \) are perpendicular to a line L. Then the equation of the line L is:

If \( 2x^2 - 3xy + y^2 = 0 \) represents two sides of a triangle and \( x + y - 1 = 0 \) is its third side, then the distance between the orthocenter and the circumcenter of that triangle is:

If \( S \) is the set of all real values of \( a \) such that a plane passing through the points \( (-a^2, 1, 1), (1, -a^2, 1), (1, 1, -a^2) \) also passes through the point \( (-1, -1, 1) \), then \( S = \dots \)

The distance between the centres of similitude of the circles \( x^2 + y^2 + 6x - 8y + 16 = 0 \) and \( x^2 + y^2 - 2x - 2y + 1 = 0 \) is:

Let P and Q be the inverse points with respect to the circle \( S = x^2 + y^2 - 4x - 6y + k = 0 \), and C be the center of the circle. If \( CP.CQ = 4 \), and \( P = (1, 2) \), then \( Q = (a, b) \) and \( 2a = \dots \)

The perpendicular distance from the origin to the focal chord drawn through the point \( (4, 5) \) to the parabola \( y^2 - 4y - 3x + 7 = 0 \) is:

If \( S = \frac{x^2}{k - 7} - \frac{y^2}{11 - k} = 0, k \in \mathbb{R}, k \neq 7,11 \), then which one of the following statements is incorrect?

The function \( f(x) = \frac{\sqrt{3x^2 - 5x - 2}}{2x^2 - 7x + 5} \) has discontinuous points at \( x = \dots \)

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: