List of top Mathematics Questions

The volume integral of the function \( f(r, \theta, \phi) = r^2 \cos\theta \) over the region \( 0 \leq r \leq 2, 0 \leq \theta \leq \frac{\pi}{3} \) and \( 0 \leq \phi \leq 2\pi \) is:
Consider a system of \( N \) particles obeying classical statistics, each of which can have an energy 0 or \( E \). The system is in thermal contact with a reservoir maintained at a temperature \( T \). Let \( k \) denote the Boltzmann constant. Which one of the following statements regarding the total energy \( U \) and the heat capacity \( C \) of the system is correct?
Let \[ f(x, y) = \frac{xz y}{x^2 + y^2 + z^2}, \quad (x, y) \neq (0, 0). \] Then \[ \frac{\partial f}{\partial x} \text{ and } f \text{ are bounded and unbounded.} \]
Let \( 0<a_1<b_1 \), For \( n \geq 1 \), define \[ a_{n+1} = \sqrt{a_n b_n} \quad \text{and} \quad b_{n+1} = \frac{a_n + b_n}{2}. \] Then which one of the following is NOT TRUE?
For \( a>0, b>0 \), let \[ \mathbf{F} = \frac{xj - yk}{b x^2 + a y^2}. \] Let \[ C = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 = a^2 + b^2\}. \] Then the line integral \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \]
The area of the surface \[ z = \frac{xy}{3} \] intercepted by the cylinder \[ x^2 + y^2 \leq 16 \] lies in the interval \[ \left( 20\pi, 22\pi \right) \]
The flux of the vector field \[ \mathbf{F} = \left( \frac{2\pi x + 2x^2 y^2}{\pi} \right) \hat{i} + \left( \frac{2\pi x y - 4y}{\pi} \right) \hat{j} \] along the outward normal, across the ellipse \( x^2 + 16y^2 = 4 \) is equal to
If \( \lim_{t \to \infty} \int_0^t e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \), then \[ \lim_{t \to \infty} \int_0^t x^2 e^{-x^2} dx = \]
Find the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{n} \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot \frac{k}{n} \right) = \]
Evaluate the integral: \[ \int_0^1 \int_x^1 \sin(y^2) \, dy \, dx \]
The number of generators of the additive group \( \mathbb{Z}_{36} \) is equal to

There are two integers 34041 and 32506, when divided by a three-digit integer $n$, leave the same remainder. What is the value of $n$?

What is the remainder when $1!+2!+3!+\cdots+100!$ is divided by $7$? 

If $(67^{67}+67)$ is divided by $68$, the remainder is: 

Find the value of $1(1!)+2(2!)+3(3!)+\cdots+20(20!)$. 

Find the remainder when  \[6^{\underbrace{66\cdots6}_{100 \text{ times}}}\] is divided by 10.
 

Find the value of  $\log_{20} 100 + \log_{20} 1000 + \log_{20} 10000 \quad \bigl[\textit{Assume that } \log 2 = 0.3\bigr].$ 
 

Find the remainder when the $41$-digit number $1234\ldots$ is divided by $8$. 

In a survey of $500$ TV viewers: $285$ watch football (F), $195$ hockey (H), $115$ basketball (B); $45$ watch F&B, $70$ watch F&H, $50$ watch H&B, and $50$ watch none. How many watch exactly one of the three games? 

Nine squares are chosen at random on a chessboard. What is the probability that they form a square of size $3\times 3$? 

$A,B,C,D$ are four towns, any three of which are non-colinear. In how many ways can we construct three roads (each road joins a pair of towns) so that the roads do not form a triangle?

In the figure, $\triangle APB$ is formed by three tangents to a circle with centre $O$. If $\angle APB=40^\circ$, then the measure of $\angle BOA$ is 

If $1+\sin^2(2A)=3\sin A\cos A$, then what are the possible values of $\tan A$?

If \[ 2\sin\alpha + 15\cos^{2}\alpha = 7, \quad 0^\circ < \alpha < 90^\circ, \] find \(\cot\alpha\). 

In the given figure, $ABCD$ is a rectangle. $P$ and $Q$ are the midpoints of sides $CD$ and $BC$ respectively. Then the ratio of area of shaded portion to the area of unshaded portion is: