List of top Questions asked in KEAM

The value of the limit $\lim_{t \to 0} \frac{(5-t)^2 - 25}{t}$ is equal to:

The sum of the series $ \frac{1}{2^{10}} + \frac{1}{2^{11}} + \cdots + \frac{1}{2^{19}} $ is equal to:

The vectors $\vec{a} = 4\mathbf{i} - 3\mathbf{j} - \mathbf{k}$ and $\vec{b} = 3\mathbf{i} + 2\mathbf{j} + \lambda\mathbf{k}$ are perpendicular to each other. Then the value of $\lambda$ is equal to:

Let $ f(x) = \begin{cases} x^2 - \alpha, & \text{if } x < 1 \\ \beta x - 3, & \text{if } x \geq 1 \end{cases} $. If $ f $ is continuous at $ x = 1 $, then the value of $ \alpha + \beta $ is:

Let $ f(x) = 2 - 7 \sin{\left( \frac{2x}{7} \right)} $. Then the maximum value of $ f(x) $ is:

If \[ \sin^{-1} x + \cos^{-1} y = 0, \] then $ x^2 + y^2 $ is equal to:

If $ |x| \leq 1 $, then $ \sin \left( 2 \sin^{-1} x + \cos^{-1} x \right) $ is equal to:

The angle between the lines \[ \frac{x-1}{6} = \frac{y-5}{8} = \frac{z-3}{10} \quad {and} \quad \frac{x+1}{2} = \frac{2y+3}{2} = \frac{z+3}{2} \] is:

If \[ \frac{1}{1 - \tan x} = \frac{3 + \sqrt{3}}{2}, \quad 0 \leq x \leq \frac{\pi}{2}, \] then the value of $ x $ is equal to:

The volume of the parallelepiped whose coterminous vectors are given by the vectors \[ \overrightarrow{a} = i - j + k, \quad \overrightarrow{b} = 3i + j - k, \quad \overrightarrow{c} = 5i + 2j - 7k \] is (in cubic units):

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 a line makes angles $\alpha$, $\beta$, and $\gamma$ with the positive directions of the x, y, and z-axis respectively, then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ equals:

If $ a = \frac{1 + \tan \theta + \sec \theta}{2 \sec \theta} $ and $ b = \frac{\sin \theta}{1 - \sec \theta + \tan \theta} $, then $ \frac{a}{b} $ is equal to:

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 $ 0 \leq \alpha \leq \frac{\pi}{2} $ and $\sin \left(\alpha - \frac{\pi}{12}\right) = \frac{1}{2}$, then $\alpha$ is equal to:

Let $ \overrightarrow{AB} = i + 2j - 2k $ and $ \overrightarrow{AC} = i - j + k. $ Then the area of $ \triangle ABC $ is:

An assignment of probabilities for outcomes of the sample space $ S = \{1, 2, 3, 4, 5, 6\} $ is as follows:

\[ \begin{array}{c|c c c c c c} 1 & 2 & 3 & 4 & 5 & 6 \\ \hline k & 3k & 5k & 7k & 9k & 11k \end{array} \]

If this assignment is valid, then the value of $ k $ is:

Three dice are thrown simultaneously. The probability that all the three outcomes are the same number, is:

If $ \alpha, \beta, \gamma $ are the angles made by $$ \frac{x-1}{3} = \frac{y+1}{2} = -\frac{z}{1} \text{ with the coordinate axes, then } $$ $(\cos\alpha, \cos\beta, \cos\gamma) = $

If $ 0 \leq x \leq 5 $, then the greatest value of $ \alpha $ and the least value of $ \beta $ satisfying the inequalities $ \alpha \leq 3x + 5 \leq \beta $ are, respectively,

If \[ \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2}, \] then $ x + y + z = $:

The value of the limit $\lim_{x \to 0} \frac{(2 + \cos 3x) \sin^2 x}{x \tan(2x)}$ is equal to:

The foci of the ellipse $\frac{x^2}{49} + \frac{y^2}{24} = 1$ are:

Let $ \vec{a} $ and $ \vec{b} $ be two unit vectors. Let $ \theta $ be the angle between $ \vec{a} $ and $ \vec{b} $. If $ \theta \neq 0 $ or $ \pi $, then

\[ \left| \vec{a} - (\vec{a} \cdot \vec{b}) \vec{b} \right|^2 \] is equal to:

The second term of a G.P. is $ \frac{1}{2} $. If the product of first five terms is 32, then the common ratio of the G.P. is: