List of top Mathematics Questions asked in AP EAPCET

If the tangent drawn to the curve \( y = x^3 \) at point \( (\alpha, \beta) \) cuts again the curve at another point \( (\alpha_1, \beta_1) \), then \( \frac{\beta_1}{\beta} = \):
Evaluate the integral \[ \int \frac{dx}{(2ax + x^2)^{3/2}} \]
If a normal drawn to the ellipse \[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \] touches the hyperbola \[ \frac{x^2}{4} - \frac{y^2}{3} = 1, \] then the square of the slope of that normal is:
If \( f(x) = 3x + \frac{12}{x} \) is continuous on \( \mathbb{R} - \{0\} \) and \( M \) is its maximum value, then \( \displaystyle \lim_{x \to M} f(x) = \):
If \[ \lim_{x \to 2} \frac{1 + \sqrt{1 + 4\log_2 x}}{2 + (2x + \sin^2 x + 2\cos x)(2x - 4)} = m, \] then \( m(m - 1) = \):
If \[ \frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right) = \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2}, \] then the value of \( A + B = \):
A function \( f(x) \) is defined as: \[ f(x) = \begin{cases} ax^2 + bx + c, & x \leq -1
2x^2 + 4x + 1, & -1<x<1
cx^2 + bx + a, & x \geq 1 \end{cases} \] If \( f(x) \) is continuous on \( \mathbb{R} \) and \( \lim_{x \to -\frac{3}{2}} f(x) = 14 \), then \( \lim_{x \to 2} f(x) = \):
If \[ S = 2x^2 + 2y^2 - 8x + 8y - 7 = 0 \] is the circle passing through the points of intersection of the circles \[ x^2 + y^2 - kx - ky + 1 = 0 \quad \text{and} \quad x^2 + y^2 - kx + ky - 2 = 0, \] then the length of the tangent drawn from the point \( (k, k) \) to the circle \( S \) is:
The point of intersection of the tangents drawn at the points where the line \[ 2x - y + 3 = 0 \] meets the circle \[ x^2 + y^2 - 4x - 6y + 4 = 0 \] is:
Let the equation \[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \] represent a point circle (not at the origin). Then which one of the following conditions must hold?
Let \( \ell \) be the directrix of the parabola \( 9y^2 + 12y + 9x - 14 = 0 \), and \( \ell_1 \) be the line passing through the vertex of this parabola and the origin. If \( (h, k) \) is the point of intersection of \( \ell \) and \( \ell_1 \), then \( h + k = \):
Let the point \( L \) lying in the first quadrant be one end of a latus rectum of the ellipse \[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \] Let \( P \) and \( Q \) be the points where the normal drawn at \( L \) meets the major and minor axes. Then the distance between \( P \) and \( Q \) is:
Let X-axis be the transverse axis and Y-axis be the conjugate axis of a hyperbola \( H \). Let the eccentricity of \( H \) be the reciprocal of the eccentricity of the ellipse \[ \frac{x^2}{4} + \frac{y^2}{2} = 1 \] If \( (5, 4) \) lies on \( H \), then the length of the transverse axis is:
The distance of the origin from the external centre of similitude for the circles \[ x^2 + y^2 - 8x - 10y - 8 = 0 \quad \text{and} \quad x^2 + y^2 + 2x - 2y - 2 = 0 \] is:
The equation of the line common to the pair of lines \[ (p^2 - q^2)x^2 + (q^2 - r^2)xy + (r^2 - p^2)y^2 = 0 \] and \[ (l - m)x^2 + (m - n)xy + (n - l)y^2 = 0 \] is:
If \( A(2, 3, 5), B(\alpha, 3, 3), C(7, 5, \beta) \) are the vertices of a triangle. If the median through \( A \) is equally inclined with the coordinate axes, then \( \frac{\beta}{\alpha} = \):
Let the locus of the point of intersection of the perpendicular tangents drawn to the circle \[ x^2 + y^2 + 6x - 4y - 12 = 0 \] be the circle \( S \). Then the equation of the tangent drawn to \( S \) which is perpendicular to the line \( 6x - 4y + k = 0 \) is:
The lines \( L_1: y - x = 0 \) and \( L_2: 2x + y = 0 \) intersect the line \( L_3: y + 2 = 0 \) at points \( P \) and \( Q \) respectively. The bisector of the acute angle between \( L_1 \) and \( L_2 \) intersects \( L_3 \) at \( R \). Statement-1: \( PR : RQ = 2\sqrt{2} : \sqrt{5} \) Statement-2: In any triangle, the bisector of an angle divides the triangle into two similar triangles.
In \( \triangle ABC \), if the midpoints of the sides \( AB, BC, CA \) are respectively \( (l, 0, 0), (0, m, 0), (0, 0, n) \), then: \[ \frac{AB^2 + BC^2 + CA^2}{l^2 + m^2 + n^2} = ? \]
The equation of the plane passing through the point \( (1, 2, 2) \) and perpendicular to the planes \[ x - y + 2z = 3 \quad \text{and} \quad 2x - 2y + z + 12 = 0 \] is:
If \( t \in \mathbb{R} \setminus \{-1\} \), then the locus of the point \[ \left( \frac{3at}{1 + t^3}, \frac{3at^2}{1 + t^3} \right) \] is:
The centre of a square of side 4 units length is \( (3, 7) \), and one of the diagonals is parallel to the line \( y = x \). If \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \) are the vertices of this square, then \[ \frac{y_1 y_2 y_3 y_4}{x_1 x_2 x_3 x_4} = ? \]
The area (in square units) of the triangle formed by the lines \( x = 0 \), \( y = 0 \), and \( 3x + 4y = 12 \) is:
If \( X \) is a random variable such that \[ P(X = -2) = P(X = -1) = P(X = 2) = P(X = 1) = \frac{1}{6}, \quad P(X = 0) = \frac{1}{3}, \] then the mean of \( X \) is:
Let origin be the centroid of an equilateral triangle ABC and one of its sides is along the straight line \( x + y = 3 \). If \( R \) and \( r \) are its circumradius and inradius respectively, then \( R + r = \):