List of top Mathematics Questions asked in KEAM

Let \( \vec{a} = \hat{i} - \hat{j} \), \( \vec{b} = \hat{j} - \hat{k} \), and \( \vec{c} = \hat{k} - \hat{i} \), then the value of \( \vec{b} \cdot (\vec{a} + \vec{c}) \) is:
Let \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) be vectors such that \( \mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0} \). If \( |\mathbf{u}| = 3 \), \( |\mathbf{v}| = 4 \), and \( |\mathbf{w}| = 5 \), then \( \mathbf{u} \cdot \mathbf{v} + \mathbf{w} \cdot \mathbf{u} \) is:
If a vector makes angles \( \frac{\pi}{3}, \frac{\pi}{4} \) and \( \gamma \) with \( \hat{i}, \hat{j} \), and \( \hat{k} \), respectively, where \( \gamma \in \left( \frac{\pi}{2}, \pi \right) \), then the angle \( \gamma \) is:
Evaluate \( \cos \left( \cot^{-1} \left( \frac{7}{24} \right) \right) \):
If \( 3 \sin \theta + 5 \cos \theta = 5 \), then the value of \( 5 \sin \theta - 3 \cos \theta \) is:
The value of \( \tan \left[ \tan^{-1} \left( \frac{3}{4} \right) + \tan^{-1} \left( \frac{2}{3} \right) \right] \) is:
If \( \alpha = \tan^2 x + \cot^2 x \), where \( x \in \left( 0, \frac{\pi}{2} \right) \), then \( \alpha \) lies in the interval:
The domain of the function \( f(x) = \frac{\sin^{-1} \left( x-3 \right)}{\sqrt{9 - x^2}} \) is:
If \( \sin \theta = \frac{b}{a} \), then \( \frac{\sqrt{a+b}} {\sqrt{a-b}} + \frac{\sqrt{a-b}} {\sqrt{a+b}} \) is equal to:
If \( \frac{\sec^2 15^\circ - 1}{\sec^2 15^\circ} \) equals:
Let \( A = (a_{ij}) \) be a square matrix of order 3 and let \( M_{ij} \) be the minors of \( a_{ij} \). If \( M_{11} = -40, M_{12} = -10, M_{13} = 35 \), and \( a_{11} = 1, a_{12} = 3, a_{13} = -2 \), then the value of \( |A| \) is equal to:
If the determinant of the matrix \( \begin{bmatrix} |x| & 1 & 2 \\ 4 & 1 & x \\ 1 & -1 & 3 \end{bmatrix} \) equals -10, then the values of \( x \) are:
If the points \( (2, -3), (\lambda, -1) \) and \( (0, 4) \) are collinear, then the value of \( \lambda \) is equal to:
If \( B = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix} \) is the adjoint of a \( 3 \times 3 \) matrix \( A \) and \( |A| = 4 \), then the value of \( \alpha \) is:

If 

then the value of \( x - y \) is:

The constant term in the expansion of \( \left( x^3 + \frac{1}{x^2} \right)^{10} \) is:
If \( _nP_r = 480 \) and \( _nC_r = 20 \), then the value of \( r \) is equal to:
The coefficient of \( x^3 \) in the binomial expansion of \( \left( \frac{1}{\sqrt{x}} - x \right)^6 \) is:
Evaluate \( \binom{10}{1} + \binom{10}{2} + \dots + \binom{10}{10} \):
The numbers \( a, b, c, d \) are in G.P. with common ratio \( r \). If \( \frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + d^3} \) are also in G.P., then the common ratio is:
The second term of a G.P. is 4, then the product of the first three terms is:
If \( z_1 \) and \( z_2 \) are two complex numbers with \( |z_1| = 1 \), then \( \left| \frac{z_1 - z_2}{1 - z_1 \overline{z_2}} \right| \) is equal to:
If \( \left( \frac{1 - i}{1 + i} \right)^{10} = a + ib \), then the values of \( a \) and \( b \) are, respectively:
Let \( z \) be a complex number satisfying \( |z + 16| = 4|z + 1| \). Then:
Let \( X \) and \( Y \) be subsets of \( \mathbb{R} \). If \( f : X \rightarrow Y \) given by \( f(x) = -8(x + 5)^2 \) is one-to-one, then the codomain \( Y \) is: