List of top Mathematics Questions

Let the mean and the variance of seven observations \(2,4,\alpha,8,\beta,12,14\), \( \alpha < \beta \), be \(8\) and \(16\) respectively. Then the quadratic equation whose roots are \(3\alpha+2\) and \(2\beta+1\) is :

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

The sum \( 1 + \frac{1}{2}(1^2+2^2) + \frac{1}{3}(1^2+2^2+3^2) + \ldots \) upto \(10\) terms is equal to:

Let \( S = \{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0 \}. \) Then \( \displaystyle \sum_{z \in S} z^8 \) is equal to:

Consider the quadratic equation \( (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0, \; n \in \mathbb{R}. \) Let \( \alpha \) be the minimum value of the product of its roots and \( \beta \) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \( \alpha \) and the common ratio is \( \dfrac{\alpha}{\beta} \), is:

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3} \). Then \( f \) is:

Let $y = y(x)$ be the solution of the differential equation $x \sin \left( \frac{y}{x} \right) dy = \left( y \sin \left( \frac{y}{x} \right) - x \right) dx, y(1) = \frac{\pi}{2}$ and let $\alpha = \cos \left( \frac{y(e^{12})}{e^{12}} \right)$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \le 6$, is _________.

If $\frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to _________.

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left( \frac{1}{x^3} - x^4 \right)^n, x \neq 0,$ is zero, then the value of $n$ is _________.

Two players A and B play a series of games of badminton. The player who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways in which player A wins the series is _________.

Let $A = \{1, 2, 3, 4, 5, 6\}$. The number of one-one functions $f: A \to A$ such that $f(1) \ge 3, f(3) \le 4$ and $f(2) + f(3) = 5$, is _________.

The value of the integral $\int_{\pi/6}^{\pi/3} \left( \frac{4 - \csc^2 x}{\cos^4 x} \right) dx$ is:

Let $f : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f \left( \frac{x+y}{3} \right) = \frac{f(x)+f(y)}{3}$ for all $x, y \in \mathbb{R}$, and $f'(0) = 3$. Then the minimum value of the function $g(x) = 3 + e^x f(x)$, is:

The value of the integral $\int_0^\infty \frac{\log_e (x)}{x^2 + 4} dx$ is:

The product of all possible values of $\alpha$, for which $\lim_{x \to 0} \frac{1-\cos(\alpha x)\cos((\alpha+1)x)\cos((\alpha+2)x)}{\sin^2((\alpha+1)x)} = 2$, is:

The square of the distance of the point of intersection of the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})$, $a \neq 0$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})$ from the origin is:

Let $\vec{a} = \sqrt{7}\hat{i}+\hat{j}-\hat{k}$ and $\vec{b} = \hat{j} + 2\hat{k}$. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0}$ and $\vec{r} \cdot \vec{a} = 0$, then $|3\vec{r}|^2$ is equal to:

The sum of all the integral values of p such that the equation $3\sin^2x + 12\cos x - 3 = p, x \in \mathbb{R}$, has at least one solution, is:

In an equilateral triangle PQR, let the vertex P be at (3, 5) and the side QR be along the line x + y = 4. If the orthocentre of the triangle PQR is ($\alpha, \beta$), then 9($\alpha + \beta$) is equal to:

Let P be a moving point on the circle $x^2 + y^2-6x-8y + 21 = 0$. Then, the maximum distance of P from the vertex of the parabola $x^2 + 6x + y + 13 = 0$ is equal to:

Let a focus of the ellipse E: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be S(4, 0) and its eccentricity be $\frac{4}{5}$. If the point P(3, $\alpha$) lies on E and O is the origin, then the area of $\Delta$POS is equal to:

The mean deviation about the mean for the data

56 is equal to:

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

Let tan A, tan B, where A, B $\in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, be the roots of the quadratic equation $x^2 - 2x - 5 = 0$. Then $20 \sin^2\left(\frac{A+B}{2}\right)$ is equal to: