List of top Mathematics Questions asked in KEAM

Find the integrating factor of the differential equation: \[ (3 \sin x \cos x) \, dy = (1 + 3y \sin^2 x) \, dx, \quad {where} \quad 0<x<\frac{\pi}{2} \] is
Evaluate the integral: \[ \int_{0}^{3} |x - 2| \,dx \] is equal to
Evaluate the integral: \[ \int \frac{6x^3 + 9x^2}{x^4 + 3x^3 - 9x^2} dx. \]
Evaluate the integral: \[ \int \frac{\sin \theta \sin 2\theta}{1 - \cos 2\theta} d\theta. \]
The general solution of the differential equation \[ 2y \tan x + \frac{dy}{dx} = 5 \sin x \] is:
The function \( f(x) = 6x^4 - 3x^2 - 5 \) is increasing in the set:
If \( f(x) = x^2 + 2x f'(1) + f''(2) \) for all \( x \), then \( f(0) \) is equal to:
Let \( \alpha \) and \( \beta \) be real numbers such that \( f(x) \) is defined as: \[ f(x) = \begin{cases} 2x^2 + 4x + \alpha, & \text{if } x < 1 \\ \beta x^2 + 5, & \text{if } x \geq 1 \end{cases} \] and is differentiable at \( x = 1 \). Then \( \alpha + \beta \) is equal to:
If \[ y = \log_e \left( \frac{1 + 2x^2}{1 - 3x^2} \right), \] then \( \frac{dy}{dx} \) is:
Evaluate the limit: \[ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) \] is equal to
For the curve \[ y = \alpha x^2 + \cos y + \beta, \] the value of \( \frac{dy}{dx} \) at \( (1,0) \) is 2. Then the value of \( \alpha \beta \) is equal to
If \[ \lim_{x \to 9} f(x) = 6 \quad {and} \quad \lim_{x \to 9} g(x) = 3, \] then \[ \lim_{x \to 9} \frac{f(x) - 2g(x)}{g(x)} \] is equal to
If \[ x^4 + 2\sqrt{y} + 1 = 3, \] then \( \frac{dy}{dx} \) at \( (1,0) \) is equal to
Evaluate the limit: \[ \lim\limits_{x \to 6} \frac{\sqrt{x^2 + 13} - 7}{x^2 - 36}. \]
Evaluate the integral \[ \int x^5 e^{x^3} \,dx. \]
The value of \( \int_{-2}^{2} x |x| \,dx \) is:
Let \( f(x) = ax^3 + bx^2 + cx + d \). If \( f \) has a local maximum value 21 at \( x = -1 \) and a local minimum value 7 at \( x = 1 \), then \( f(0) \) is equal to:
Let \( f(x) = \log_e \left( \frac{x^2 + 30}{11x} \right) \), for \( x \in [5,6] \). Then the point \( c \in (5,6) \) at which \( f'(c) = 0 \) is:
The range of the function \( f(x) = 7\cos(10x + 4\pi) \) is
Let \[ g(x) = 4x + 3 \quad {and} \quad f(g(x)) = x^2 + 9. \] Then the value of \( f(7) \) is equal to

The area bounded by the parabola \( y = x^2 + 4 \) and the straight line passing through the points \( (-1,2) \quad {and} \quad (1,6) \) is (in square units)

Evaluate the integral \[ \int \frac{e^x}{2^x} dx. \]

Ice is coated uniformly around a sphere of radius 15 cm. If ice is melting at the rate of \( 80 { cm}^3/{min} \) when the thickness is 5 cm, then the rate of change of thickness of ice is

If \[ f(x) = \sqrt[3]{x^2} + \sqrt{x}, \] then the value of \( f'(64) \) is equal to
If the function \[ f(x) = \begin{cases} x^2, & \text{for } x < 4 \\ 5x - k, & \text{for } x \geq 4 \end{cases} \] is continuous at \( x = 4 \), then the value of \( k \) is equal to