List of top Mathematics Questions asked in KEAM

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

The integral \(\int e^x \sqrt{e^x} \, dx\) equals:

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

Find the value of \[ \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right). \] The answer 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
The expression
\[ \frac{1 + \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)}{1 + \cos\left(\frac{\pi}{5}\right) - i \sin\left(\frac{\pi}{5}\right)} \] 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 integrating factor of the differential equation \[ x \frac{dy}{dx} + 2y = x e^x \] is:
The solution of the inequality \( |3x - 4| \leq 5 \) is
The value of \[ \sin^6 15^\circ + \cos^6 15^\circ \text{ is equal to:} \]
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:
The value of \[ \lim_{x \to 1} \frac{\frac{1}{2x + 1} - \frac{1}{3}}{x - 1} \] is equal to:

The minimum value of the function \( f(x) = x^4 - 4x - 5 \), where \( x \in \mathbb{R} \), is:

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

The line \(y = 5x + 7\) is perpendicular to the line joining the points \((2, 12)\) and \((12, k)\). Then the value of \(k\) is equal to:

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
\[ P(A \mid B) = ? \]
Simplify the following expression: \[ (\sec A - \cos A)(\tan A - \cot A) \] The simplified form is:

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:

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

The angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\). If \(\|\vec{a}\| = 5\) and \(\|\vec{b}\| = 10\), then \(\|\vec{a} + \vec{b}\|\) is equal to:

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 \( a_1 = 3 \) and \( a_n = n \cdot a_{n-1} \), for \( n \geq 2 \), then \( a_6 \) is equal to
If \( 2z = 7 + i\sqrt{3} \), then the value of \( z^2 - 7z + 4 \) is:
The period of \( 2 \sin 4x \cos 4x \) is: