List of top Mathematics Questions

Let $l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :
If $(27)^{999}$ is divided by $7$, then the remainder is :
The area (in s units) of the region $\{ (x , y) : x \geq 0 , x + y \leq 3, x^2 \leq 4 y$ and $y \leq 1 + \sqrt{x} \}$ is
If two different numbers are taken from the set $\{0,1,2,3, \ldots \ldots, 10\}$then the probability that their sum as well as absolute difference are both multiple of $4$, is :
Let $z \in C$, the set of complex numbers. Then the equation, $2 | z + 3i| - | z - i| = 0 $ represents :
The value of $\cos^{-1} \left(\cot\left(\frac{\pi}{2}\right)\right) + \cos^{-1} \left(\sin\left(\frac{2\pi}{3}\right)\right) $ is
If vector $\vec{r}$ with d.c.s. $l, m, n$ is equally inclined to the co-ordinate axes, then the total number of such vectors is
The equation of the plane through $(-1, 1 , 2 ) $ whose normal makes equal acute angles with co-ordinate axes is

Let $x_1$ and $x_2$ be the roots of the equation $ax^2 + bx + c = 0$ ($ac \neq 0$). Find the value of $\frac{1}{x_1} + \frac{1}{x_2}$.

The number of real roots of $$ (6 - x)^4 + (8 - x)^4 = 16 $$ 
If the numbers $a_1, a_2, \ldots, a_n$ are different from zero and form an arithmetic progression, then $$ \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \dots + \frac{1}{a_n} $$ is equal to
If \( |a| < 1 \) and \( |b| < 1 \), then the sum of the series $$ a(a + b) + a^2(a^2 + b^2) + a^3(a^3 + b^3) + \cdots $$ 
If \[ a = \gamma \hat{i} + \delta \hat{j} + 2\zeta \hat{k}, \, b = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, \, r \times a = b \times a \, r \times b = a \times b \] then a unit vector in the direction of \( r \) is

Calculate

\[ \begin{vmatrix} x & y & x + y \\ y & x + y & x \\ x + y & x & y \end{vmatrix} \] 

The value of \[ \lim_{x \to 0} \int_0^x \sec^2 t \, dt \] is:
For all \( n \in \mathbb{N}, 72n - 48n - 1 \) is divisible by
The sum \[ \sum_{k=0}^5 \left( 5C_k \right)^2 \] is equal to

If \[ f(x) = \begin{cases} \frac{1 - \sin x}{(n - 2x)^2} & \text{if} \quad x \neq \frac{\pi}{2} \log (\sin x) \cdot \log \left( 1 + \frac{\pi}{4x + x^2} \right) & \text{if} \quad x = \frac{\pi}{2} \end{cases} \] is continuous at \( x = \frac{\pi}{2} \), then \( k \) is equal to

Given that \( \alpha_1, \alpha_2, \alpha_3 \) are the roots of \( 3x^3 - x^2 - 10x + 8 = 0 \), then the value of \( \alpha_1^2 + \alpha_2^2 + \alpha_3^2 \) is
The integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x + \cos x}}{\sin x + \cos x} \, dx \] is equal to
The value of the integral \[ \int_0^0 9 [x - 2\lfloor x \rfloor] \, dx \] where \( [.] \) denotes the greatest integer function, is

The inverse of matrix \[ \begin{pmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\3 & -3 & 4 \end{pmatrix} \] is

From the bottom of a pole of height h, the angle of elevation of the top of a tower is \( \alpha \). The pole subtends an angle \( \beta \) at the top of the tower. The height of the tower is

Choose the most appropriate option. 
If \( A \) is a square matrix such that \( A^2 = A \) and \( B = I \), then \( AB + BA + I - (I - A)^2 \) is equal to:

Let \( f(x) = (x^3 + 2)^{30} \). If \( f^{(n)}(x) \) is a polynomial of degree \( 20 \), where \( f^{(n)}(x) \) denotes the \( n + h \)-order derivative of \( f(x) \), then the value of \( n \) is: