List of top Mathematics Questions

If $ \theta = \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{7} \right) + \tan^{-1} \left( \frac{1}{13} \right) + \tan^{-1} \left( \frac{1}{21} \right) + \tan^{-1} \left( \frac{1}{31} \right) $, then $ \tan \theta = ? $

If $ \tanh^{-1} x = \coth^{-1} y = \log \sqrt{5} $, then find $ \tan^{-1}(xy) = ? $

In triangle $ ABC $, $ 2A + C = 300^\circ $. If the circumradius is 8 times the inradius, then $ \sin\frac{C}{2} = ? $

In triangle $ ABC $, if $ C = 120^\circ $, $ c = \sqrt{19} $, and $ b = 3 $, then $ a = ? $

Evaluate: $$ \sin \frac{\pi}{12} \cdot \sin \frac{2\pi}{12} \cdot \sin \frac{3\pi}{12} \cdot \sin \frac{4\pi}{12} \cdot \sin \frac{5\pi}{12} \cdot \sin \frac{6\pi}{12} $$

In the binomial expansion of $ (p - q)^{14} $, if the sum of 7^th and 8^th terms is zero, then $$ \frac{p + q}{p - q} = ? $$

5 men and 4 women are seated in a row. If the number of arrangements in which one particular man and one particular woman are together is $ \alpha $, and the number of arrangements in which they are not together is $ \beta $, then $ \frac{\alpha}{\beta} = $

If $ \frac{x^4}{(x-1)(x-2)} = \frac{A}{x - 1} + \frac{B}{x - 2} $, then $$ f(-2) + A + B = ? $$

The number of positive integers less than 10000 which contain the digit 5 at least once is

If a team of 4 persons is to be selected out of 4 married couples to play mixed doubles tennis game, then the number of ways of forming a team in which no married couple appears is

If a complex number $ z = x + iy $ represents a point $ P $ on the Argand plane and $$ \text{Arg} \left( \frac{z - 3 + 2i}{z + 2 - 3i} \right) = \frac{\pi}{4} $$ then the locus of $ P $ is

If $ x^2 - 4x + 5 + a>0 $ for all $ x \in \mathbb{R} $ whenever $ a \in (\alpha, \beta) $, then $ 4\beta + \alpha = $

By taking $ \sqrt{a \pm ib} = x + iy, x>0 $, if we get $$ \frac{\sqrt{21} + 12\sqrt{2}i}{\sqrt{21} - 12\sqrt{2}i} = a + ib, $$ then $ \frac{b}{a} = $ ?

The polynomial equation of degree 5 whose roots are the roots of the equation $$ x^5 - 3x^4 + 11x^2 - 12x + 4 = 0 $$ each increased by 2 is

Let $ f(x) = x^2 + 2bx + 2c^2 $ and $ g(x) = -x^2 - 2cx + b^2 $, $ x \in \mathbb{R} $. If $ b $ and $ c $ are non-zero real numbers such that $ \min f(x)>\max g(x) $, then $$ \left| \frac{c}{b} \right| $$ lies in the interval

Two values of $ (-8 - 8\sqrt{3}i)^{1/4} $ are

If $ \alpha, \beta, \gamma $ are the roots of the equation $ x^3 - 12x^2 + kx - 18 = 0 $ and one of them is thrice the sum of the other two, then $$ \alpha^2 + \beta^2 + \gamma^2 - k = ? $$

If the inverse of $$ \begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix} $$ is $$ \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix} $$ then the value of $$ \begin{vmatrix} x & x+1 & x+2 \\ x+1 & x+2 & x+3 \\ x+2 & x+3 & x+4 \end{vmatrix} $$

If $ A = \begin{bmatrix} -1 & x & -3 \\ 2 & 4 & z \\ y & 5 & -6 \end{bmatrix} $ is symmetric and $ B = \begin{bmatrix} 0 & 2 & q \\ p & 0 & 4 \\ -3 & r & s \end{bmatrix} $ is skew-symmetric, then find $ |A| + |B| - |AB| $

If the system of equations $ 2x + 3y - 3z = 3,\ x + 2y + \alpha z = 1,\ 2x - y + z = \beta $ has infinitely many solutions, then $ \frac{\alpha}{\beta} = \frac{\beta}{\alpha} $

The general solution of the differential equation \[ y + \cos x \left( \frac{dy}{dx} \right) - \cos^2 x = 0 \] is

If the degree of the differential equation corresponding to the family of curves
\[ y = ax + \frac{1}{a} \quad (\text{where } a \neq 0 \text{ is an arbitrary constant}) \] is \(r\) and its order is \(m\), then the solution of
\[ \frac{dy}{dx} - \frac{y}{2x}, \quad y(1) = \sqrt{r + m} \] is

Evaluate the integral: \[ \left| \int_{-\pi/4}^{\pi/3} \tan\left(x - \frac{\pi}{6}\right) dx \right| \]