List of top Questions asked in KEAM

The area bounded by the parabola $y = x^2 + 2$ and the lines $y = x$, $x = 1$ and $x = 2$ (in square units) is:

Three coins are tossed simultaneously. Then the probability that exactly two tails appear is:

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:

The value of \[ \sin^6 15^\circ + \cos^6 15^\circ \text{ is equal to:} \]

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

The solution of the inequality $ |3x - 4| \leq 5 $ is

\[ \int_0^{\frac{\pi}{4}} (\tan^3 x + \tan^5 x) \, dx \]

The common point of the two straight lines \[ \overrightarrow{r_1} = (i - 2j + 3k) + s(2i + j + k) \quad {and} \quad \overrightarrow{r_2} = (-i + 2j + 7k) + t(i + j + k), \quad t, s \in \mathbb{R} \] is:

The point of intersection of the lines $\frac{x-3}{2} = \frac{y-2}{2} = \frac{z-6}{1}$ and $\frac{x-2}{3} = \frac{y-4}{2} = \frac{z-1}{3}$ is:

If $ z $ is a complex number of unit modulus, then \[ \left| \frac{1+z}{1+ \overline{z}} \right| \] equals:

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:

The sum up to $ n $ terms of the series $ \frac{1}{\sqrt{1} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{11}} + \cdots $ is:

Let $ \overrightarrow{a} = 2i + 3j - 4k, \quad \overrightarrow{b} = i + j - k, \quad \overrightarrow{c} = -i + 2j + 3k, \quad \overrightarrow{d} = i + j + k. $ Then \[ (\overrightarrow{a} \times \overrightarrow{b}) \cdot (\overrightarrow{c} \times \overrightarrow{d}) = \]

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

If $ a = \tan^{-1}\left(\frac{4}{3}\right) $ and $ b = \tan^{-1}\left(\frac{1}{3}\right) $, where $ 0<a, b<\frac{\pi}{2} $, then $ a - b $ is:

If $ \overrightarrow{a} $ and $ \overrightarrow{b} $ are two nonzero vectors and if $ |\overrightarrow{a} \times \overrightarrow{b}| = |\overrightarrow{a} \cdot \overrightarrow{b}| $, then the angle between $ \overrightarrow{a} $ and $ \overrightarrow{b} $ is equal to

If $\sec \theta + \tan \theta = 2 + \sqrt{3}$, then $\sec \theta - \tan \theta$ is:

The value of $ x $ that satisfies the equation:

\[ \begin{vmatrix} x & 1 & 1 \\ 2 & 2 & 0 \\ 1 & 0 & -2 \end{vmatrix} = 6 \]

Let $ f(x) = \log_e(x) $ and let $ g(x) = \frac{x - 2}{x^2 + 1} $. Then the domain of the composite function $ f \circ g $ is:

Let $ f(x) = \frac{x^2 + 40}{7x} $, $ x \neq 0 $, $ x \in [4,5] $. The value of $ c $ in $ [4,5] $ at which $ f'(c) = -\frac{1}{7} $ is equal to:

The shortest distance between the parallel straight lines \[ \overrightarrow{r_1} = \hat{k} + s(\hat{i} + \hat{j}), \quad t, s \in \mathbb{R} \quad {and} \quad \overrightarrow{r_2} = \hat{j} + t(\hat{i} + \hat{j}), \] 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 $ x \neq 0, y \neq 0 $, then the value of \[ \cot^{-1}\left(\frac{x}{y}\right) + \cot^{-1}\left(\frac{y}{x}\right) \] is: