List of top Mathematics Questions asked in KEAM

\[ \int \left( \frac{\log_e t}{1+t} + \frac{\log_e t}{t(1+t)} \right) dt \]

The coefficient of \( x^{14}y \) in the expansion of \( (x^2 + \sqrt{y})^9 \) is:

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,

The value of \[ \frac{\cos^{-1}(0) + \sin^{-1}\left( \frac{\sqrt{3}}{2} \right) + \cos^{-1}\left( \frac{1}{2} \right)}{\sin^{-1}(1) + \cos^{-1}\left( \frac{\sqrt{3}}{2} \right) + \sin^{-1}\left( \frac{1}{\sqrt{2}} \right)} \] is equal to:
Variance of 6, 7, 8, 9 is

The value of \[ \left(\frac{10i}{(2-i)(3-i)}\right)^{2024} \] is equal to:

The value of \( x \) that satisfies the equation:

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

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 centre of the circle \( (x-3)(x+1)+(y-1)(y+3)=0 \) is:

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:

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:

If \( f(x) = \frac{|x|}{1 + |x|}, \, x \in \mathbb{R} \), then \( f'(-2) \) is equal to:
The value of \[ \lim_{x \to 1} \frac{\frac{1}{2x + 1} - \frac{1}{3}}{x - 1} \] is equal to:
If \[ \begin{vmatrix} x & 2 & -1 \\ 1 & x & 5 \\ 3 & 2 & x \end{vmatrix} = 0, \quad \text{then the real value of} \quad x \quad \text{is:} \]
Given that: \[ \frac{1}{\tan A - \tan B} = \]
If \[ \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2}, \] then \( x + y + z = \):
If \( z \) is a complex number of unit modulus, then \[ \left| \frac{1+z}{1+ \overline{z}} \right| \] equals:

For \(1 \leq x<\infty\), let \(f(x) = \sin^{-1}\left(\frac{1}{x}\right) + \cos^{-1}\left(\frac{1}{x}\right)\). Then \(f'(x) =\)

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

A particle is moving along the curve \( y = 8x + \cos y \), where \( 0 \leq y \leq \pi \). If at a point the ordinate is changing 4 times as fast as the abscissa, then the coordinates of the point are:

\[ \cos^{-1} \left( \cos \left( \frac{-7\pi}{9} \right) \right) = \]
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

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:

If \( \overrightarrow{a} = \alpha \hat{i} + \beta \hat{j} \) and \( \overrightarrow{b} = \alpha \hat{i} - \beta \hat{j} \) are perpendicular, where \( \alpha \neq \beta \), then \( \alpha + \beta \) 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: