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List of top Mathematics Questions

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:

Let \[ A = \begin{pmatrix} 3 & -2 & 1 \\ -1 & 3 & -1 \end{pmatrix} \] and \[ B = \begin{pmatrix} 1 \\ \alpha \\ -1 \end{pmatrix}. \] If \[ AB = \begin{pmatrix} -2 \\ 6 \end{pmatrix}, \] then the value of $ \alpha $ is equal to:

Let $ f(x) = \begin{cases} x^2 - \alpha, & \text{if } x < 1 \\ \beta x - 3, & \text{if } x \geq 1 \end{cases} $. If $ f $ is continuous at $ x = 1 $, then the value of $ \alpha + \beta $ is:

Let $ \overrightarrow{AB} = i + 2j - 2k $ and $ \overrightarrow{AC} = i - j + k. $ Then the area of $ \triangle ABC $ is:

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:

A particle is moving along the curve $ y = 8x + \cos y $, where $ 0 \leq y \leq \pi $. If at a point the ordinate is changing 4 times as fast as the abscissa, then the coordinates of the point are:

Let $ A $ and $ B $ be two sets each containing more than one element. If $ n(A \times B) = 155 $, then $ n(A) $ is equal to:

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

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

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

The value of the limit $\lim_{x \to 0} \frac{(2 + \cos 3x) \sin^2 x}{x \tan(2x)}$ is equal to:

If \[ \frac{1}{1 - \tan x} = \frac{3 + \sqrt{3}}{2}, \quad 0 \leq x \leq \frac{\pi}{2}, \] then the value of $ x $ is equal to:

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:

The value of \[ \frac{\cos^{-1}(0) + \sin^{-1}\left( \frac{\sqrt{3}}{2} \right) + \cos^{-1}\left( \frac{1}{2} \right)}{\sin^{-1}(1) + \cos^{-1}\left( \frac{\sqrt{3}}{2} \right) + \sin^{-1}\left( \frac{1}{\sqrt{2}} \right)} \] is equal to:

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:

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:

Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}} \, dx \]

The critical points of the function $ f(x) = (x-3)^3(x+2)^2 $ are:

Let $f(x) = a^{3x}$ and $a^5 = 8$. Then the value of $f(5)$ is equal to:

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:

The value of $ x $ that satisfies the equation:

\[ \begin{vmatrix} x & 1 & 1 \\ 2 & 2 & 0 \\ 1 & 0 & -2 \end{vmatrix} = 6 \]

If a line makes angles $\alpha$, $\beta$, and $\gamma$ with the positive directions of the x, y, and z-axis respectively, then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ equals:

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

The area bounded by the parabola $y = x^2 + 2$ and the lines $y = x$, $x = 1$ and $x = 2$ (in square units) is:

If $ A = \begin{bmatrix} 0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x \end{bmatrix} $ is a singular matrix, then $ x $ is equal to