List of top Mathematics Questions

Evaluate the sum \[ \sum_{n=1}^{24} \left( i^n + i^{n+1} \right) \] is:
Simplify the following expression: \[ \frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x} \] The simplified form is:
Simplify the following expression: \[ \cos 18^\circ \cos 42^\circ \cos 78^\circ. \] The simplified form is:
The value of \[ \sin^6 15^\circ + \cos^6 15^\circ \text{ is equal to:} \]
Let \( \vec{a} \) and \( \vec{b} \) be two unit vectors. Let \( \theta \) be the angle between \( \vec{a} \) and \( \vec{b} \). If \( \theta \neq 0 \) or \( \pi \), then

\[ \left| \vec{a} - (\vec{a} \cdot \vec{b}) \vec{b} \right|^2 \] is equal to:
Find the value of \[ \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right). \] The answer is:

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 \( 0 \leq x \leq 5 \), then the greatest value of \( \alpha \) and the least value of \( \beta \) satisfying the inequalities \( \alpha \leq 3x + 5 \leq \beta \) are, respectively,

The sum up to \( n \) terms of the series \( \frac{1}{\sqrt{1} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{11}} + \cdots \) is:

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:

The common point of the two straight lines \[ \overrightarrow{r_1} = (i - 2j + 3k) + s(2i + j + k) \quad {and} \quad \overrightarrow{r_2} = (-i + 2j + 7k) + t(i + j + k), \quad t, s \in \mathbb{R} \] is:

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 centre of the circle \( (x-3)(x+1)+(y-1)(y+3)=0 \) 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 bag contains 10 green balls and 5 red balls. If two balls are selected randomly, then the probability that both are green balls, is:
\[ \cos^{-1} \left( \cos \left( \frac{-7\pi}{9} \right) \right) = \]
The shortest distance between the parallel straight lines \[ \overrightarrow{r_1} = \hat{k} + s(\hat{i} + \hat{j}), \quad t, s \in \mathbb{R} \quad {and} \quad \overrightarrow{r_2} = \hat{j} + t(\hat{i} + \hat{j}), \] is:
Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}} \, dx \]
The integrating factor of the differential equation \[ x \frac{dy}{dx} + 2y = x e^x \] is:
The angle between the two straight lines \[ \overrightarrow{r_1} = (4i - k) + t(2i + j - 2k), \quad t \in \mathbb{R}, \quad {and} \quad \overrightarrow{r_2} = (i - j + 2k) + s(2i - 2j + k), \quad s \in \mathbb{R} \] is:
If \( \left( 1 + \cos \frac{\pi}{8} \right) \left( 1 + \cos \frac{7\pi}{8} \right) \) =
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 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: