List of top Questions asked in AP EAPCET

Evaluate the integral: \[ \int \frac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} \, dx =\ ? \]
Evaluate the integral: \[ \int \frac{1}{9\cos^2 x - 24 \sin x \cos x + 16 \sin^2 x} \, dx = \]
The interval in which the curve represented by \( f(x) = 2x + \log\left(\frac{x}{2 + x}\right) \) is increasing is
If the extreme value of the function \( f(x) = \frac{4}{\sin x} + \frac{1}{1 - \sin x} \) in \(\left[0, \frac{\pi}{2}\right]\) is \(m\) and it exists at \(x = k\), then \(\cos k =\)}
The displacement \(S\) of a particle measured from a fixed point \(O\) on a line is given by \[ S = t^3 - 16t^2 + 64t - 16. \] Then the time at which the displacement of the particle is maximum is
If the tangent drawn at the point \((\alpha, \beta)\) on the curve \[ x^{2/3} + y^{2/3} = 4 \] is parallel to the line \[ \sqrt{3}x + y = 1, \] then \( \alpha^2 + \beta^2 =\)
\[ \text{If } y = |\cos x - \sin x| + |\tan x - \cot x|, \text{ then } \left( \frac{dy}{dx} \right)_{x = \frac{\pi}{3}} + \left( \frac{dy}{dx} \right)_{x = \frac{\pi}{6}} = \]
\[ \text{Assertion (A): If } y = f(x) = (|x| - |x - 1|)^2, \text{ then } \left.\frac{dy}{dx}\right|_{x = 1} = 1 \] \[ \text{Reason (R): If } \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \text{ exists, then it is called the derivative of } f(x) \text{ at } x = a. \] Then:
\[ \text{If } x = 2 \cos^3 \theta \text{ and } y = 3 \sin^2 \theta, \text{ then } \frac{dy}{dx} =\ ? \]
\[ \text{If the function } f(x) = \begin{cases} 1 + \cos x, & x \leq 0 \\ a - x, & 0 < x \leq 2 \\ x^2 - b^2, & x > 2 \end{cases} \text{ is continuous everywhere, then } a^2 + b^2 =\ ? \]
\[ \text{If } \lim_{x \to 0} \frac{\cos 2x - \cos 4x}{1 - \cos 2x} = k, \text{ then evaluate } \lim_{x \to k} \frac{x^k - 27}{x^{k+1} - 81} \]
\[ \lim_{y \to 0} \frac{\sqrt{1 + \sqrt{1 + y^4}} - \sqrt{2}}{y^4} = \ ? \]
A plane \( \pi \) is passing through the points \( A(1, -2, 3) \) and \( B(6, 4, 5) \). If the plane \( \pi \) is perpendicular to the plane \( 3x - y + z = 2 \), then the perpendicular distance from \( (0, 0, 0) \) to the plane \( \pi \) is
If \( A = (0, 4, -3),\ B = (5, 0, 12),\ C = (7, 24, 0) \), then \( \angle BAC = \)
If \( A(0,0,0),\ B(3,4,0),\ C(0,12,5) \) are the vertices of a triangle ABC, then the x-coordinate of its incenter is:
The distance between the tangents of the hyperbola \( 2x^2 - 3y^2 = 6 \) which are perpendicular to the line \( x - 2y + 5 = 0 \) is:
If a tangent to the hyperbola \( xy = -1 \) is also a tangent to the parabola \( y^2 = 8x \), then the equation of that tangent is:
The angle between the tangents drawn from a point \( (-3, 2) \) to the ellipse \( 4x^2 + 9y^2 - 36 = 0 \) is:
PQ is a focal chord of the parabola \( y^2 = 4x \) with focus S. If \( P = (4,4) \), then SQ = ?
Let \( \theta \) be the angle between the circles \( S = x^2 + y^2 + 2x - 2y + c = 0 \) and \( S' = x^2 + y^2 - 6x - 8y + 9 = 0 \). If \( c \) is an integer and \( \cos\theta = \dfrac{5}{16} \), then the radius of the circle \( S = 0 \) is:
A circle \( S = x^2 + y^2 - 16 = 0 \) intersects another circle \( S' = 0 \) of radius 5 units such that their common chord is of maximum length. If the slope of that chord is \( \dfrac{3}{4} \), then the centre of such a circle \( S' = 0 \) is:
Length of the common chord of two circles of same radius is \( 2\sqrt{17} \). If one of the two circles is \( x^2 + y^2 + 6x + 4y - 12 = 0 \), then the acute angle between the two circles is:
If the circle passing through the points \( (3,5), (5,5), (3,-3) \) cuts the circle \( x^2 + y^2 + 2x + 2fy = 0 \) orthogonally, then the value of \( f \) is:
If \( Q \) is the inverse point of \( P(-1, 1) \) with respect to the circle \( x^2 + y^2 - 2x + 2y = 0 \), then the line containing \( Q \) is:
If the angle between the lines joining the origin to the points of intersection of \( x + 2y + \lambda = 0 \) and \( 2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0 \) is \( \dfrac{\pi}{2} \), then a value of \( \lambda \) is: