List of top Mathematics Questions

If \[ (1+x)^n=C_0+C_1x+C_2x^2+\cdots+C_nx^n \] where \[ C_i={}^{n}C_i, \] then \[ (C_0+C_1)(C_1+C_2)\cdots(C_{n-1}+C_n) \] equals

If \(A=[-1,1)\) and \(B=(0,\infty)\), then the complement of \(A\cup B\) is

If a set \(A\) contains \(3\) elements and another set \(B\) contains \(4\) elements, then the number of functions from \(A\) to \(B\), which are not injective, is

If a set \(A\) contains \(5\) elements, then the number of reflexive relations on \(A\) is

The direction cosines of the vector \[ \vec a=a_1\hat i+a_2\hat j+a_3\hat k \] are

The differential equation of all circles in a plane of radius \(r\) is \[ \text{where } y_1=\frac{dy}{dx}, \qquad y_2=\frac{d^2y}{dx^2} \]

The number of arbitrary constants in the general solution and in the particular solution of a differential equation of fourth order are respectively

The sum to \(n\) terms of the series \[ 1+3+7+15+\cdots \] is

In an election, the number of candidates is one more than the number of seats. If a voter can cast his vote in 30 ways, find the number of candidates (when a voter can cast his vote for one or more seats).

20 delegates from 20 countries sit in a circle such that two particular delegates never sit together. In how many ways can they be seated?

Two motorcyclists \(A\) and \(B\) leave a place at 12 noon. \(A\) travels north at \(60\) km/hr and \(B\) travels east at \(80\) km/hr. At 2 PM, they are separating at the rate

In a class XII of a school, \(40\%\) of the students study Mathematics, \(30\%\) study Physics and \(20\%\) study Chemistry. \(20\%\) of the class study both Mathematics and Physics, \(10\%\) study both Mathematics and Chemistry and \(10\%\) study both Physics and Chemistry. \(5\%\) of the class study all the three subjects. If a student is selected at random from the class, find the probability that he studies neither Mathematics nor Physics nor Chemistry.

If \(f\) is a subset of \(Z\times Z\), then which of the following is a function from \(Z\) to \(Z\)?

The mean of 5 observations is \(4.4\) and their variance is \(8.24\). If three of the observations are \(1, 2\) and \(6\), then the other two observations are

The limit \[ \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left[ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} \right] \]

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by \[ f(x) = \begin{cases} x \left( \frac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}} \right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \] Then

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = x^5 + x^3$ and let $g(x) = f^{-1}(x)$ be the inverse of $f$. If $g''(-2) = \frac{a}{b}$ where $a$ and $b$ are positive coprime integers, then the value of $a$ is

Let $S_1$ and $S_2$ be two circles drawn inside a unit square ABCD, touching each other externally. Further, the circle $S_1$ touches the sides AD and DC; and the circle $S_2$ touches the sides AB and BC. If the area of $S_2$ is twice the area of $S_1$, then the radius of $S_1$ is

The number of solutions of the equation $|\sin(\pi x)| = \frac{1}{50}(x^2 + 1)$ in $\mathbb{R}$ is

The definite integral \[ \int_{\pi/2}^{\pi} \frac{\sin x - x \cos x}{x(x + \sin x)} \, dx \] equals

Let $S$ be the set of real values of $k$ for which the system of equations \[ kx + y + z = k \] \[ x + ky + z = k \] \[ x + y + kz = k \] has no solution. Then $|S|$ is

In a building, an elevator starts from the ground floor (that is, 0th floor) with 10 passengers and stops at every floor until the 15th floor which is the topmost floor. No new passengers enter the elevator on any floor from the 1st through the 15th floor. If all the passengers got off the elevator then the probability that at most one passenger got off at each floor is

The probability that the sum of two integers $m$ and $n$, where $m, n \in \{1, 2, \dots, 50\}$, chosen randomly and independently, being divisible by 3 is

Let a variable line $L$ meet $x$-axis and $y$-axis at points A and B, respectively. Suppose the distance of the line $L$ from the origin is 3 units. Then the equation of the locus of the point C that divides the line segment AB internally in the ratio $2 : 1$ is

Amar, Akbar, and Anthony write a test independently. The probability that Amar passes the test and Akbar fails the test is $\frac{3}{20}$. The probability that Akbar passes the test and Anthony fails the test is $\frac{1}{4}$. The probability that Amar and Anthony both pass the test is $\frac{2}{5}$. Then the probability that at least one of Amar, Akbar and Anthony fails the test is