List of top Mathematics Questions asked in KEAM

If \( f(x) = \begin{cases} x^2 & \text{for } x < 0 \\ 5x - 3 & \text{for } 0 \leq x \leq 2 \\ x^2 + 1 & \text{for } x > 2 \end{cases} \), then the positive value of \( x \) for which \( f(x) = 2 \) is:
Let a relation \( R \) on the set of natural numbers be defined by \( (x, y) \in R \) if and only if \( x^2 - 4xy + 3y^2 = 0 \) for all \( x, y \in \mathbb{N} \). Then the relation is:
If A and B are two sets, such that A has 20 elements, \( A \cup B \) has 32 elements, and \( A \cap B \) has 10 elements, the number of elements in the set B is:
If the straight lines \( \frac{x - 3}{2} = \frac{y - 4}{3} = \frac{z - 6}{-1} \) and \( \frac{x - 2}{a} = \frac{y + 3}{b} = \frac{z + 4}{-1} \) are parallel, then \( a^2 + b^2 \) is:
The lines \( \frac{x + 3}{-2} = \frac{y}{1} = \frac{z - 4}{3} \) and \( \frac{x - 1}{\mu} = \frac{y - 1}{\mu + 1} = \frac{z}{\mu + 2} \) are perpendicular to each other. Then the value of \( \mu \) is:
The equation of the curve passing through \( (1, 0) \) and which has slope \( \left( 1 + \frac{y}{x} \right) \) at \( (x, y) \), is:
The general solution of the differential equation \( (x + y)^2 \frac{dy}{dx} = 1 \) is:
The solution of the differential equation \( x + y \frac{dy}{dx} = 0 \), given that at \( x = 0 \), \( y = 5 \), is:
The order and degree of the following differential equation: \( \frac{d^2 y}{dx^2} - 2x = \sqrt{y} + \frac{dy}{dx} \), respectively, are:
The area of the region bounded by the curve \( y = 3x^2 \) and the x-axis, between \( x = -1 \) and \( x = 1 \), is:
The value of \( \int_{-1}^{1} x^2 \sin x \, dx \) is equal to:
The integral \( \int_{-2}^4 x^2 |x| \, dx \) is equal to:
The integral \( \int_5^{10} \left\lfloor x \right\rfloor dx \) is equal to (where \( \left\lfloor x \right\rfloor \) denotes the greatest integer function):
The value of \( \int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\tan x + \sin x}{1 + \cos^2 x} \, dx \) is equal to:
The value of \( \int_0^1 x(1 - x)^{10} \, dx \) is equal to:
The integral \( \int e^x \sec x (1 + \tan x) \, dx \) is equal to:
The integral \( \int xe^x \, dx \) is equal to:
The integral \( \int \sqrt{1 + \sin 2x} \, dx \) is equal to:

A train starts from X towards Y at 3 pm (time \( t = 0 \)) with velocity \( v(t) = 10t + 25 \) km per hour, where \( t \) is measured in hours.
Then the distance covered by the train at 5 pm (in km) is:

The integral \( \int \frac{1 + x^2 + x^4}{(1 - x^3)(1 + x^3)} \, dx \) is equal to:
If \( f(x) = x|x| \), then \( f'(-1) + f'(1) \) is equal to:
If a continuous function \( f \) is defined as \[ f(x) = \left\{ \begin{array}{ll} ax + 1, & x < 2 \\ x^2 + 7, & x \geq 2 \end{array} \right. \] then the value of \( a \) is:
The derivative of \( t^2 + t \) with respect to \( t-1 \) at \( t = -2 \), is equal to:
The function \( f(x) = (x - 4)^2 (1 + x)^3 \) attains a local extremum at the point:
The limit \( \lim_{x \to 0} \frac{3 \sin^2 2x}{x^2} \) is equal to: