List of top Mathematics Questions

Let \( A \) be a symmetric matrix and \( B \) be a skew symmetric matrix. If

\[ A + B = \begin{pmatrix} 1 & 3 \\ -2 & 5 \end{pmatrix}, \] then \( A - B \) is equal to:
The length of the latus rectum of the parabola \( y^2 = x \) is:

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:

The angle between the lines \[ \frac{x-1}{6} = \frac{y-5}{8} = \frac{z-3}{10} \quad {and} \quad \frac{x+1}{2} = \frac{2y+3}{2} = \frac{z+3}{2} \] is:
Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}} \, dx \]
Simplify the following expression: \[ (\sec A - \cos A)(\tan A - \cot A) \] The simplified form is:
The integrating factor of the differential equation \[ x \frac{dy}{dx} + 2y = x e^x \] is:
Evaluate the integral:

\[ \int \frac{4x \cos \left( \sqrt{4x^2 + 7} \right)}{\sqrt{4x^2 + 7}} \, dx \]

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

Let \( f(x) = 2 - 7 \sin{\left( \frac{2x}{7} \right)} \). Then the maximum value of \( f(x) \) is:

The value of \( x \) that satisfies the equation:

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

If \( |x| \leq 1 \), then \( \sin \left( 2 \sin^{-1} x + \cos^{-1} x \right) \) is equal to:

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

If \( 0 \leq \alpha \leq \frac{\pi}{2} \) and \(\sin \left(\alpha - \frac{\pi}{12}\right) = \frac{1}{2}\), then \(\alpha\) is equal to:

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:
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:

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:

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

The point of intersection of the lines \(\frac{x-3}{2} = \frac{y-2}{2} = \frac{z-6}{1}\) and \(\frac{x-2}{3} = \frac{y-4}{2} = \frac{z-1}{3}\) 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:
The value of \[ \lim_{x \to 1} \frac{\frac{1}{2x + 1} - \frac{1}{3}}{x - 1} \] is equal to:
Let \( \overrightarrow{AB} = i + 2j - 2k \) and \( \overrightarrow{AC} = i - j + k. \) Then the area of \( \triangle ABC \) is:
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: