List of top Mathematics Questions

If $ \tan\left( \frac{\pi}{4} + \alpha \right) = \tan^3\left( \frac{\pi}{4} + \beta \right) $, then compute: $$ \tan(\alpha + \beta) \cot(\alpha - \beta) = ? $$
If $ A + B + C + D = 2\pi $, then $$ \sin A + \sin B + \sin C + \sin D = ? $$
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 = ? $
In triangle $ ABC $, if $ C = 120^\circ $, $ c = \sqrt{19} $, and $ b = 3 $, then $ a = ? $
Evaluate: $$ \sin \frac{\pi}{12} \cdot \sin \frac{2\pi}{12} \cdot \sin \frac{3\pi}{12} \cdot \sin \frac{4\pi}{12} \cdot \sin \frac{5\pi}{12} \cdot \sin \frac{6\pi}{12} $$
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
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
If $ x^2 - 4x + 5 + a>0 $ for all $ x \in \mathbb{R} $ whenever $ a \in (\alpha, \beta) $, then $ 4\beta + \alpha = $
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
Two values of $ (-8 - 8\sqrt{3}i)^{1/4} $ are
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: \[ \left| \int_{-\pi/4}^{\pi/3} \tan\left(x - \frac{\pi}{6}\right) dx \right| \]
Evaluate the integral: \[ \int \frac{(3x - 2)\tan\left(\sqrt{9x^2 - 12x + 1}\right)}{\sqrt{9x^2 - 12x + 1}} \, dx =\ ?\]
Evaluate the integral: \[ \int \frac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} \, dx =\ ? \]
Evaluate the integral: \[ \int \frac{1}{9\cos^2 x - 24 \sin x \cos x + 16 \sin^2 x} \, dx = \]
If the tangent drawn at the point \((\alpha, \beta)\) on the curve \[ x^{2/3} + y^{2/3} = 4 \] is parallel to the line \[ \sqrt{3}x + y = 1, \] then \( \alpha^2 + \beta^2 =\)
\[ \text{If the function } f(x) = \begin{cases} 1 + \cos x, & x \leq 0 \\ a - x, & 0 < x \leq 2 \\ x^2 - b^2, & x > 2 \end{cases} \text{ is continuous everywhere, then } a^2 + b^2 =\ ? \]
\[ \text{If } \lim_{x \to 0} \frac{\cos 2x - \cos 4x}{1 - \cos 2x} = k, \text{ then evaluate } \lim_{x \to k} \frac{x^k - 27}{x^{k+1} - 81} \]
A plane \( \pi \) is passing through the points \( A(1, -2, 3) \) and \( B(6, 4, 5) \). If the plane \( \pi \) is perpendicular to the plane \( 3x - y + z = 2 \), then the perpendicular distance from \( (0, 0, 0) \) to the plane \( \pi \) is
If \( A(0,0,0),\ B(3,4,0),\ C(0,12,5) \) are the vertices of a triangle ABC, then the x-coordinate of its incenter is:
The distance between the tangents of the hyperbola \( 2x^2 - 3y^2 = 6 \) which are perpendicular to the line \( x - 2y + 5 = 0 \) is:
If a tangent to the hyperbola \( xy = -1 \) is also a tangent to the parabola \( y^2 = 8x \), then the equation of that tangent is:
Let \( \theta \) be the angle between the circles \( S = x^2 + y^2 + 2x - 2y + c = 0 \) and \( S' = x^2 + y^2 - 6x - 8y + 9 = 0 \). If \( c \) is an integer and \( \cos\theta = \dfrac{5}{16} \), then the radius of the circle \( S = 0 \) is: