List of top Mathematics Questions asked in KEAM

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

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:

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):

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

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:

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

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

For a hyperbola, the vertices are at \( (6, 0) \) and \( (-6, 0) \). If the foci are at \( (2\sqrt{10}, 0) \) and \( -2\sqrt{10}, 0) \), then the equation of the hyperbola is:

11\( (10P_7) \) =

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 unit vectors and if \( \frac{\pi}{4} \) is the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \), then \[ (\overrightarrow{a} + (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{b}) \cdot (\overrightarrow{a} - (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{b}) \] 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 \( z \) is a complex number of unit modulus, then \[ \left| \frac{1+z}{1+ \overline{z}} \right| \] equals:

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 \[ \theta \in \left( 0, \frac{\pi}{3} \right) \quad \text{and} \quad \begin{vmatrix} 0 & -\sin^2 \theta & -2 - 4 \cos 6\theta \\ 0 & \cos^2 \theta & -2 - 4 \cos 6\theta \\ 1 & \sin \theta & \cos 2\theta \end{vmatrix} = 0, \] then \( \theta \) is equal to:

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

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

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

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

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