List of top Mathematics Questions asked in AP EAPCET

The substitution \( x = vy \) converts which one of the following differential equations to an equation solvable by the variable separable method?

The area (in square units) bounded by the curves \( x^2 = 9y \), \( (x - 6)^2 = 9y \), and the X-axis is:

Evaluate the integral: \[ \int \frac{e^x \left( 2 + \sin(2x) \right)}{1 + \cos(2x)} \, dx \]

If \( I = \int_{-a}^{a} \left( x^4 - 2x^2 \right) \, dx \), then \( I \) is minimum at \( a = \):

If \( f(x) = \int \frac{dx}{x^2 + 2} \) and \( f(\sqrt{2}) = 0 \), then \( f(0) = \)

If \( [x] \) is the greatest integer not exceeding \( x \), then \[ \int_{-0.5}^{1.5} x^2 [x] \, dx = \]

Evaluate the integral: \[ \int_0^{50\pi} \sqrt{1 - \cos 2x} \, dx \]

If \[ \int \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} \, dx = Ax + B \log \left( 9e^{2x} - 4 \right) + C, \text{ then } (A, B) = \]

If \( f(x) \) is a differentiable function, \( f'(x) \geq 5 \) for \( x \in [2,6] \), \( f(2) = 4 \) and \( f(3) = 15 \), then a possible value of \( f(6) \) is:

If the tangent drawn to the curve \( y = x^3 - ax^2 + x + 1 \) at each point \( x \in \mathbb{R} \), is inclined at an acute angle with the positive direction of \( X \)-axis, then the set of all possible values of \( a \) is:

If \( y = \log x \), then the value of \( x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y \) at the point \( \left( \sqrt{e}, \sqrt{e} \right) \) is:

Let the equation of the tangent at a point \( P \) on the parabola \( x^2 - 4x - 4y + 16 = 0 \) be \( 2x - y - 5 = 0 \). If the equation of the normal drawn at \( P \) to this parabola is \( ax + y + c = 0 \), then find \( ac \):

Let \( f(x) \) be a real valued function. If \( f'(x) \) is a constant for all \( x \in \mathbb{R}, f(0) = 1 \) and \( f'(0) = 2 \), then

If \( f : \mathbb{R} \to \mathbb{R} \) is defined by: \[ f(x) = \begin{cases} \sin x - \sin \left( \frac{x}{2} \right), & x<0 \\ \frac{x}{\sqrt{x^2 + \sqrt{x^2}}}, & x>0 \end{cases} \] If \( f \) is continuous on \( \mathbb{R} \), then \( f(0) = \)

If \( \theta \) is the acute angle between the tangents drawn from the point \( (2, 3) \) to the hyperbola \( 5x^2 - 6y^2 - 30 = 0 \), then \( \tan \theta \) is:

The difference between the focal distances of any point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text{ is } 6. \text{ If } (\sqrt{13}, k) \text{ is an end point of a latus rectum of this hyperbola, then } k = \]

Let the circle \( S: x^2 + y^2 + 2gx + 2fy + c = 0 \) cut the circles \( x^2 + y^2 - 2x + 2y - 2 = 0 \) and \( x^2 + y^2 + 4x - 6y + 9 = 0 \) orthogonally. If the centre of the circle \( S = 0 \) lies on the line \( 2x + 3y - 2 = 0 \), then \( 2g + f = \)

Let \( S \) be the circumcircle of the triangle formed by the line \( x - 2y - 4 = 0 \) with the coordinate axes. If \( P(-2, -4) \) is a point in the plane of the circle \( S \) and \( Q \) is a point on \( S \) such that the distance between \( P \) and \( Q \) is the least, then \( PQ = \)

The equation of the pair of tangents drawn from the point \( (1, 1) \) to the circle \( x^2 + y^2 + 2x + 2y + 1 = 0 \) is:

If the line passing through the points \( (5,1,a) \) and \( (3,b,1) \) crosses the YZ plane at the point \( \left( 0, \frac{17}{2}, \frac{-13}{2} \right) \), then \( a + b = \):

If the chord of contact of the point \( P(h, k) \) with respect to the circle \( x^2 + y^2 - 4x - 4y + 8 = 0 \) meets the circle in two distinct points and it also makes an angle \( 45^\circ \) with the positive X-axis in the positive direction, then \( (h, k) \) cannot be:

The distance between two parallel planes \( ax + by + cz + d_1 = 0 \) and \( ax + by + cz + d_2 = 0 \) is given by \( \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \). If the plane \( 2x - y + 2z + 3 = 0 \) has the distances \( \frac{1}{3} \) units from the planes \( 4x - 2y + 4z + \lambda = 0 \) and \( 2x - y + 2z + \mu = 0 \) respectively, then the maximum value of \( \lambda + \mu \) is:

If the coordinates of the point of contact of the circles \( x^2 + y^2 - 4x + 8y + 4 = 0 \) and \( x^2 + y^2 + 2x = 0 \) is \( (a, b) \), then \( a + 2b \) is: