List of top Mathematics Questions asked in KEAM

The integral

\[ \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx \] is:
Let \( \left\lfloor x \right\rfloor \) be the greatest integer less than or equal to \( x \). Then \[ \lim_{x \to 0^-} \frac{x \left( \left\lfloor x \right\rfloor + |x| \right)}{|x|} \] is equal to:

The limit: \[ \lim_{x \to 0} \frac{\sin \left( \pi \sin^2 x \right)}{x^2} \] is equal to:

The area bounded by the curves \( y = x^2 \) and \( y = 2x \) in the first quadrant, is equal to:
The solution of \( (y \cos y + \sin y) \, dy = (2x \log x + x) \, dx \) is:
The value of \( \int_{-4}^{-2} \left[ (x+3)^3 + 2 + (x+3)\cos(x+3) \right] \, dx \) is equal to:
If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]
The value of \[ \sin^6 15^\circ + \cos^6 15^\circ \text{ is equal to:} \]
Let \[ A = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}. \] If \( XA = B \), then \( X \) is:
If \[ \begin{vmatrix} x & 2 & -1 \\ 1 & x & 5 \\ 3 & 2 & x \end{vmatrix} = 0, \quad \text{then the real value of} \quad x \quad \text{is:} \]
If \[ \theta \in \left( 0, \frac{\pi}{3} \right) \quad \text{and} \quad \begin{vmatrix} 0 & -\sin^2 \theta & -2 - 4 \cos 6\theta \\ 0 & \cos^2 \theta & -2 - 4 \cos 6\theta \\ 1 & \sin \theta & \cos 2\theta \end{vmatrix} = 0, \] then \( \theta \) is equal to:
If

\[ A = \begin{pmatrix} x & 2 \\ 2 & x \end{pmatrix} \quad \text{and} \quad \det(A^2) = 25, \] then \( x \) is equal to:
Let \( A \) be a symmetric matrix and \( B \) be a skew symmetric matrix. If

\[ A + B = \begin{pmatrix} 1 & 3 \\ -2 & 5 \end{pmatrix}, \] then \( A - B \) is equal to:
The coefficient of \( x^{17} \) in

\[ (1 - x)^{13} (1 + x + x^2)^{12} \] is:
Number of integers greater than 7000 can be formed using the digits 2, 4, 5, 7, 8 is (Repetition of digits is not allowed):
Evaluate the sum \[ \sum_{n=1}^{24} \left( i^n + i^{n+1} \right) \] is:
The sum up to \( n \) terms of the series \( \frac{1}{\sqrt{1} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{11}} + \cdots \) is:
The coefficient of \( x^3 \) in the expansion of \[ \frac{1}{(1 + 2x)^{-10}} \] is:

Let \( a, b, c \) be positive numbers such that \( abc = 1 \). Then the minimum value of \( a + b + c \) is:

Let \[ f(x) = |\sin x| + |\cos x|, \quad x \in \mathbb{R}. \] The period of \( f(x) \) is:
If

\[ R = \{(x, y) : x, y \in \mathbb{Z}, \, x^2 + 3y^2 \leq 7 \} \] is a relation on the set of integers \( \mathbb{Z} \), then the range of the relation \( R \) is:
Let \( \mathbb{N} \) be the set of all natural numbers. Let \( R \) be a relation defined on \( \mathbb{N} \) given by \( aRb \) if and only if \( a + 2b = 11 \). Then the relation \( R \) is:
If \( a^2 + b^2 = 1 \), then:

\[ \frac{1 + (a - ib)}{1 + (a + ib)} \] is equal to:
Let \( z \) be a non-zero complex number such that \[ z = \frac{16}{\bar{z}}. \] Then the locus of \( z \) is:
Real part of \[ \left( \frac{1+i}{1-i} \right) \left( \frac{2+i}{2-i} \right) \] is: