List of top Mathematics Questions

The set \( \{ x \in \mathbb{R} : 16(2^x)>16^{x-1} \} \) is:
If \( \omega, \omega^2 \) are the cube roots of unity, \( k \) is a positive integer, and \[ (1 - \omega^2)^{3k} + (1 - \omega^2 + 0)^{3k+1} + (1 + \omega - 0)^{3k+1}, \] then find \( k \).
If \( \alpha, \beta \) are the two real roots of the 4th roots of unity, and \( \gamma, \delta \) are the other two roots of it, then the sum of the eccentricities of the conics \( |z - \alpha| + |z - \beta| = 4 \) and \( |z - \gamma| + |z - \delta| = 6 \) is:
If \( z = (1 - i^3)(x + i) \) is a purely imaginary number for \( x = x_1 \), and if \( z \) is a purely real number for \( x = x_2 \), then find \( x_1, x_2 \).
If \[ \begin{vmatrix} x & 4 & -1 \\ 2 & 1 & 0 \\ 0 & 2 & 4 \end{vmatrix} = 0, \text{ then find the value of } x. \]
For real numbers \( a \) and \( b \), if \[ 4a + i(3a - b) = b - 6i \] and \[ z = a + \frac{b}{4}i, \] then find \[ \frac{|z|}{a}. \]
Let \( \vec{a}, \vec{b}, \vec{c} \) be any three non-coplanar vectors. If \( m, n \) are scalars such that \( \vec{a} + \vec{b} = m\vec{c} \) and \( \vec{b} + \vec{c} = n\vec{a} \), then \( 3\vec{a} + 2\vec{b} + 2\vec{c} = \):
If \( 2i\hat{i} - j\hat{j} - 3j\hat{k} \) and \( -3i\hat{i} + 4j\hat{j} - 4k\hat{k} \) are the position vectors of three points A, B, and C respectively, then \( \Delta ABC \) is:
In \( \Delta ABC \), \( A, B, C \) are in arithmetic progression and \( a : c = 1 : 2 \). If \( b = 4\sqrt{3} \) cm, then the area of \( \Delta ABC \) (in sq. cm) is:
If \( \frac{17x - 2}{12x^2 - x - 20} = \frac{A}{ax + 5} + \frac{B}{3x + b} \), then \( a \cdot A + b \cdot B \) is:
If \( \alpha, \beta \) are acute angles such that \( \sin \beta = 2 \sin \alpha \) and \( 3 \cos \beta = 2 \cos \alpha \), then \( \sec (\alpha + \beta) \) is:
In \( \Delta ABC \), \( \left( \tan \frac{A}{2} + \tan \frac{B}{2} \right) \tan \frac{C}{2} = \):
In \( \Delta ABC \), if \( \angle C = 90^\circ \), then \( \left( \frac{I_1 - I_3}{I_1} \right) \left( \frac{I_2 - I_3}{I_2} \right) = \):
If \( \sinh x = \frac{5}{12} \), then \( \cosh \frac{x}{2} \) is:
Match the ranges of the functions given in List - A with those of the items given in List - B.
Match the ranges of the functions given
If \( \frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = \):
In \( \Delta ABC \), if \( \sin 2A + \sin 2B + \sin 2C = \frac{\cos A + \cos B + \cos C - 1}{\cos A + \cos B + \cos C - 1} \), then:
Find the value of \( \cos 12^\circ \cdot \cos 24^\circ \cdot \cos 36^\circ \cdot \cos 48^\circ \cdot \cos 72^\circ \cdot \cos 84^\circ \):
The number of all 8-digit odd numbers is:
If \( f(n) = n!(31-n)! \), where \( n \in \{ 0, 1, 2, \dots, 31 \} \), then the minimum value of \( f(n) \) is:
The number of all four-digit numbers which begin with 4 and end with either zero or five is:
The degree of the polynomial \( \left( x + \sqrt{x^4 - 1} \right)^9 + \left( x - \sqrt{x^4 - 1} \right)^9 \) is:
If the numerically greatest term in the expansion of \( (2 - 3x)^9 \) when \( x = 1 \) is \( P_1^q P_2^r P_3^s P_4^t \) (where \( P_1 < P_2 < P_3 < P_4 \) are the first four prime numbers), then \( \alpha + \beta + \gamma + \delta = \):
If the sum of the cubes of the roots of the equation \( x^3 - ax^2 + bx - c = 0 \) is zero, then \( a^3 + 3c = \):
If \( \alpha, \beta, \gamma \) are the roots of \( x^3 + 2x + 5 = 0 \), then \( \sum \frac{\beta + \gamma}{\alpha^2} \) is: