List of top Mathematics Questions asked in KEAM

The coefficient of \( x^{17} \) in

\[ (1 - x)^{13} (1 + x + x^2)^{12} \] is:

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

Find the value of \[ \left| \left( \frac{1+i}{\sqrt{2}} \right)^{2024} \right|. \]

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:

Let A and B be two events such that \( P(A) = 0.4 \), \( P(B) = 0.5 \) and \( P(A \cap B) = 0.1 \). Then
\[ P(A \mid B) = ? \]

If \( a^2 + b^2 = 1 \), then:

\[ \frac{1 + (a - ib)}{1 + (a + ib)} \] is equal to:

Find the value of \[ \sin \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right). \] The answer is:

Evaluate the sum \[ \sum_{n=1}^{24} \left( i^n + i^{n+1} \right) \] is:

Simplify the following expression: \[ \frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x} \] The simplified form is:

Simplify the following expression: \[ \cos 18^\circ \cos 42^\circ \cos 78^\circ. \] The simplified form is:

Find the value of \[ \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right). \] The answer is:

If \( \sec(\alpha + \beta) = \frac{\sqrt{7}}{\sqrt{3}} \), then \( \sin(\alpha + \beta) + \tan(\alpha + \beta) \) is equal to:

If

\[ \int x e^{-x} \, dx = M e^{-x} + C, \quad \text{where } C \text{ is an arbitrary constant, then } M \text{ is equal to:} \]

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:

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 \( \alpha, \beta, \gamma \) 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 \( z \) is a complex number of unit modulus, then \[ \left| \frac{1+z}{1+ \overline{z}} \right| \] equals:

When \( y = vx \), the differential equation \[ \frac{dy}{dx} = \frac{y}{x} + \frac{f\left( \frac{y}{x} \right)}{f'\left( \frac{y}{x} \right)} \] reduces to:

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