List of top Questions asked in AP EAPCET

Evaluate the integral \( \int \frac{x^2 - 2}{x^3 \sqrt{x^2 - 1}} \, dx \)

Evaluate the integral: \[ \int_{-\pi}^{\pi} \frac{2x(1 + \sin x)}{1 + \cos^2 x}\, dx \]

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

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 \( a>|\sec \theta| \)

If the slope of the line through \( (0, 0) \) which is tangent to the curve \( y = x^2 + x + 16 \) is \( m \), then find the value of \( m - 4 \).

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.

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

Evaluate the limit: \[ \lim_{x \to \infty} \frac{A + e^{mx}}{x + Ae^{mx}} \]

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

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.

A ray makes angles \( \alpha, 2\alpha, 3\alpha \) with the coordinate axes OX, OY, OZ respectively. Then possible values of \( \alpha \) are:

Find the distance between the directrices of the ellipse: \[ \frac{x^2}{36} + \frac{y^2}{20} = 1 \]

Find the distance between the point \( (6, 4\sqrt{3}) \) and the focus of the parabola \( y^2 = 8x \)

For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the locus of the point of intersection of the tangents at the endpoints of normal chords is:

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?

A circle passes through \( (0, a) \) and \( (b, h) \), and its center is at \( (c, 0) \). Find \( c \).

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

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.

The pairs of straight lines \( 2x^2 + axy + 3y^2 = 0 \) and \( 2x^2 + bxy - 3y^2 = 0 \) share a common line, and the other lines are perpendicular. Then values of \( a \) and \( b \) are?

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 \( (-3, 0) \).