List of top Questions asked in AP EAPCET

The derivative of the function $f(x) = \sin(x^2)$ at $x = \sqrt{\pi}$ is

The integrating factor of the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is a solution of the differential equation

By multiplying with $e^{\int P dx}$ on both sides of the equation $\frac{dy}{dx} + P(x)y = Q(x)$, the left side of the equation takes the form $\frac{d}{dx} (y f(x))$, then $f(x) =$

Let \( f(x) = \int \frac{e^{3x}}{4 + 8e^{2x} + e^{4x}} \, dx \), and \( g(x) = \int \frac{2\, dx}{e^{3x} + 8e^x + 4e^{-x}} \), then \( f(x) - g(x) = \) ?

If \( a, b, c, d \ne 0 \) and \( \int \frac{at + b}{ct + d} dt = t \), then find \( \int f(x) dx \) where \( f(x) = \frac{ax + b}{cx + d} \)

Evaluate: \( \int_{0}^{4} \frac{x + 2}{\sqrt{4x - x^2}}\, dx \)

If the line \( ax + by + c = 0 \) is a normal to the curve \( xy = 1 \), then which of the following is true?

Find the area of the triangle formed by the tangent to the curve \( xy = a^2 \) at \( (x_1, y_1) \) and the coordinate axes.

Two curves \( y = a^x \) and \( y = b^x \) intersect at some angle \( \alpha \). Find \( \tan \alpha \).

Evaluate \( \int \frac{dx}{1 + a \cos x} \), given \( a>|\sec \theta| \)

In the interval \( (7, \infty) \), the function \( f(x) = |x - 5| + 2|x - 7| \) is:

If \( \frac{d^n y}{dx^n} = y_n \) and \( y = e^{\sqrt{x}} + e^{-\sqrt{x}} \), then find the value of \( 4x y_2 + 2 y_1 \).

Evaluate: \[ \lim_{n \to \infty} \sqrt{2} \left[ \frac{ \left(2 + \sqrt{2} \right)^n + \left(2 - \sqrt{2} \right)^n }{ \left(2 + \sqrt{2} \right)^n - \left(2 - \sqrt{2} \right)^n } \right] \]

A plane makes intercepts \(-2, \frac{4}{3}, \frac{-4}{5}\) on the X, Y, Z axes respectively. Find the direction cosines of a normal to the plane.

The centroid of a triangle with vertices \( (a, 1, 3), (-2, b, -5), (4, 7, c) \) lies at the origin. Find \( a^2 + b^2 + c^2 \).

Find the locus of the incenter of the triangle formed by: \[ xy - 4x - 4y + 16 = 0,\quad x + y = 5 \]

Find the set of all points at a distance of at least 2 units from the point \( (-3, 0) \).

Given points \( A(-3, 3), B(1, 1), C(1, -1), D(-2, -2) \), find the angle between diagonals \( AC \) and \( BD \).

A circle touches the y-axis at a distance 4 units from the origin and cuts off an intercept 6 from the x-axis. Find its equation.

A circle touches both coordinate axes and its center lies on the line \( x - 2y - 3 = 0 \). What is its equation?

A circle of diameter \( R \) touches the parabola \( x^2 + y^2 - 4y = 0 \) and passes through the point \( (4, 5) \). Which of the following is correct?

From a pack of 52 cards, 3 cards are drawn at random. What is the probability that one is an ace, one is a queen, and one is a jack?

A discrete random variable \( X \sim B(n, p) \) (i.e., binomial distribution). Given: \[ P(X = 2) = P(X = 3) \] Then what is the mean of \( X \)?

Two distinct points are each at a unit distance from the line \( 4x + 3y - 10 = 0 \) and lie on the line \( x + y = 4 \). If they are separated by distance \( d \), then:

Let \( P(x, y) \) lie on one of the lines: \[ \sqrt{3}x - y + 2 = 0 \quad \text{or} \quad \sqrt{3}x + y - 2 = 0 \] and suppose it is at a distance 5 units from their point of intersection. What is the distance from \( (0, 0) \) to the foot of the perpendicular from \( P \) onto the \( y \)-axis?