List of top Mathematics Questions asked in AP EAMCET

Define the functions \( f, g \) and \( h \) from \( \mathbb{R} \) to \( \mathbb{R} \) such that: \[ f(x) = x^2 - 1, \quad g(x) = \sqrt{x^2 + 1} \]

Consider the following statements:

If a real-valued function \[ f(x) = \begin{cases} \frac{2x^2 + k(2x) + 9}{3x^2 - 7x - 6}, & \text{for } x \neq 3, \\ l, & \text{for } x = 3\\ \end{cases} \] is continuous at \( x = 3 \) and \( l \) is a finite value, then \( l - k = \):

Find the general solution of the differential equation: \[ (xy + y^2)dx - (x^2 - 2xy)dy = 0. \] is

Given that: \[ If M = \int_{0}^{\infty} \frac{\log t}{1 + t^3} \, dt, and \quad N = \int_{-\infty}^{\infty} \frac{e^{2t} t}{1 + e^{3t}} \, dt. \] Then, the relation between \( M \) and \( N \) is:

Evaluate the integral: \[ \int_{-2}^{2} (4 - x^2)^{\frac{5}{2}} \, dx. \]

Evaluate the following limit: \[ \lim_{x \to \infty} \left[ \left(1 + \frac{1}{n^3} \right)^{\frac{1}{n^3}} \left(1 + \frac{8}{n^3} \right)^{\frac{8}{n^3}} \left(1 + \frac{27}{n^3} \right)^{\frac{9}{n^3}} \dots (2n)^{\frac{1}{n}} \right]. \]

Evaluate the integral: \[ I = \int_{-5\pi}^{5\pi} \left(1 - \cos 2x \right)^{\frac{5}{2}} dx. \]

Evaluate the integral: \[ \int e^{4x^2 + 8x -4} (x+1) \cos(3x^2 + 6x -4) \, dx.= \]

Evaluate the integral: \[ \int \frac{1}{(1 + x^2) \sqrt{x^2 + 2}} \, dx. \]

Evaluate the integral: \[ \int \left[ (\log_2 x)^2 + 2 \log_2 x \right] dx. \]

If \[ \int \log \left( 6\sin^2x + 17\sin x + 12 \right)^{\cos x} \, dx = f(x) + c \] then, \[ f\left(\frac{\pi}{2}\right) = \]

Evaluate the integral: \[ \int \sin^4 x \cos^4 x \, dx. \]

If \[ y = 1 + x + x^2 + x^3 + \dots \quad \text{and} \quad |x|<1, \text{ then } y'' = \]

The semi-vertical angle of a right circular cone is \( 45^\circ \). If the radius of the base of the cone is measured as 14 cm with an error of \( \left(\frac{\sqrt{2}-1}{11} \right) \) cm, then the approximate error in measuring its total surface area is (in sq. cm).

The interval containing all the real values of \( x \) such that the real valued function \[ f(x) = \sqrt{x} + \frac{1}{\sqrt{x}} \] is strictly increasing is:

If the curves \[ 2x^2 + ky^2 = 30 \quad \text{and} \quad 3y^2 = 28x \] cut each other orthogonally, then \( k = \)

If \[ f(x) = 5 \cos^3 x - 3 \sin^3 x \quad \text{and} \quad g(x) = 4 \sin^3 x + \cos^2 x, \] then the derivative of \( f(x) \) with respect to \( g(x) \) is:

If \[ y = \tan^{-1} \frac{x}{1+2x^2} + \tan^{-1} \frac{x}{1+6x^2} + \tan^{-1} \frac{x}{1+12x^2}, \] then \( \left(\frac{dy}{dx}\right)_{x=\frac{1}{2}} \) =

Evaluate the limit: \[ \lim\limits_{x \to \infty} \frac{[2x - 3]}{x}. \]

Evaluate the limit: \[ \lim\limits_{x \to 0} \frac{\cos 2x - \cos 3x}{\cos 4x - \cos 5x}.= \]

The foot of the perpendicular drawn from a point \( A(1,1,1) \) onto a plane \( \pi \) is \( P(-3,3,5) \). If the equation of the plane parallel to the plane \( \pi \) and passing through the midpoint of \( AP \) is \[ ax - y + cz + d = 0, \] then \( a + c - d \) is: