List of top Questions asked in AP EAPCET

Given that \( \frac{d}{dx} \left[ \int_0^{\phi(x)} f(t) dt \right] = \phi'(x) f(\phi(x)) \). If \( \int_0^{x^3} f(t) dt = x^2 \sin 2\pi x \), then the value of \( f(8) \) is
\( \int_0^{1/2} \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} dx = \)
\( \int 3^{-\log_3 x^2} dx = \)
\( \lim_{n \to \infty} \frac{1}{n} \left( \frac{1}{e^{1/n}} + \frac{1}{e^{2/n}} + \frac{1}{e^{3/n}} + \dots + \frac{1}{e^{n/n}} \right) = \)
If \( f(x) = px^3 + qx^2 + rx + t \) attains local minimum and local maximum values at \( x = -2 \) and \( x = 2 \) respectively and \( p \) is a root of \( 9x^2 - 1 = 0 \), then \( p + q + r = \)
If the tangent drawn at \( A(2, 1) \) to the curve \( x = 1 + \frac{1}{y^2} \) meets the curve again at \( B \), then
The points on the curve \( y^2 = x + \sin x \) at which the normal is parallel to the Y-axis lie on
Given that the solid obtained by rotating a rectangle about one of its sides is a cylinder. If the perimeter of a rectangle is 48 cm and the volume of the cylinder formed by rotating it is maximum, then the dimensions of that rectangle is
\( \int \frac{x^2 - 1}{x^3 \sqrt{2x^4 - 2x^2 + 1}} dx = \)
Let \( f(x) = \max\{\cos x, \sin x, 0\} \). If the number of points at which \( f(x) \) is not differentiable in \( (0, 2024\pi) \) is \( 1012k \), then \( k = \)
If \( \theta \) is the angle between the asymptotes of the hyperbola \( \frac{x^2}{9} - \frac{(y - 2)^2}{16} = 1 \) and \( \cos \theta = \frac{a^2}{b^2} \), then \( a^2 = \)
If \( y = x \log \left( \frac{1}{ax} \right) \), then \( x(1 + x) \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - y = \)
Let \( f(x) = |x - 3| + |x + 5| \) and \( A = \left\{ a \in \mathbb{R} / \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \text{ exists} \right\} \). Then the number of real numbers which are in \( [-8, 3] \cup [5, \infty) \) but not in \( A \) is
Let \( P(h, k) \) be the point of contact of the tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) which is parallel to the line \( \sqrt{3}x - y + 1 = 0 \). If \( P \) lies in the fourth quadrant then \( 3h^2 - 2k = \)
\( \lim_{x \to 0} \left( \frac{(1 + y)^{1/x} - 1}{y} \right) = \)
If a parabola having its axis parallel to X-axis passes through the points \( (0, -1), (6, 1) \) and \( (-2, -3) \), then the point at which this parabola cuts the X-axis is
If \( Q \) is the inverse point of the point \( P(2, 3) \) with respect to the circle \( x^2 + y^2 - 2x - 2y + 1 = 0 \), then the circle with \( PQ \) as diameter is
If the circles \( x^2 + y^2 - 2x + 4y + c = 0 \) and \( x^2 + y^2 + 2x - 4y + c = 0 \) have four common tangents, then
If the length of the chord \( 2x + 3y + k = 0 \) of the circle \( x^2 + y^2 - 6x - 8y + 9 = 0 \) is \( 2\sqrt{5} \), then one of the values of \( k \) is
The locus of the poles of the tangents to the circle \( x^2 + y^2 - 2x + 2y - 2 = 0 \) with respect to the circle \( x^2 + y^2 = 4 \) is
Let \( S(1, 0) \) and \( S'(0, 1) \) be the foci of an ellipse such that \( SP + S'P = 2 \) for any point \( P \) on the ellipse. If \( A(x_1, y_1) \) and \( A'(x_2, y_2) \) are the end points of the major axis of this ellipse, then \( x_1 + x_2 + y_1 + y_2 = \)
Let the circle \( S \) be concentric with the circle \( x^2 + y^2 - 2x + ky + 4 = 0 \). If one of the diameters of \( S \) lies along the line \( 3x - 2y + 4 = 0 \) and the length of the diameter is 6, then the radius of the circle \( S \) is
A point on the straight line \( 3x + 5y = 15 \) which is equidistant from the coordinate axes will lie in
The coordinates of the point \( (3, -5) \) in the new system, when the origin is shifted to the point \( (-1, -1) \) by the translation of axes, is
The line \( x + y = k \) meets the curve \( x^2 + y^2 - 2x - 4y + 2 = 0 \) at two points \( A \) and \( B \). If \( O \) is the origin and \( \angle AOB = 90^\circ \), then the value of \( k \) (\( k>1 \)) is