List of top Mathematics Questions

In the expansion of \( \left( \frac{1}{x^3} - x^4 \right)^n \), if the sum of the coefficients of \( x^7 \) and \( x^{14} \) is zero, then find \( n \):

Find the square of the distance of the point \( (5, 6, 7) \) from the line \( \frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4} \):

The value of \( \int_{0}^{\pi/3} \frac{4 - \cos x \sec^3 x}{\cos^3 x} \, dx \) (or equivalent form) is:

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

If \( 3\sin t - 12\cos t - 3 = p \), then the sum of all integral values of 'p' such that the equation has at least one real root, is:

The value of \( \int_{0}^{\infty} \frac{\ln x}{x^2 + 4} \, dx \) is equal to:

Let \( p(x, y) \) be a variable point on the circle \( x^2 + y^2 - 6x - 8y + 21 = 0 \). Then the maximum possible distance from the vertex of \( y^2 + 6y + x + 13 = 0 \) is:

If \( \tan A \) and \( \tan B \) are roots of the equation \( x^2 - 2x - 5 = 0 \), then the value of \( 10\left(\sin^2\left(\frac{A+B}{2}\right)\right) \) is:

On a postcard one of the two words either KANPUR or ANANTPUR is written. If only two consecutive letters AN are visible on the postcard, then the probability that the written word is ANANTPUR, is:

If \( \vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0} \), \( \vec{a} = \sqrt{7}\hat{i} + \hat{j} + \hat{k} \), \( \vec{b} = \hat{j} - 2\hat{k} \) and \( \vec{r} \cdot \vec{a} = 0 \), then the value of \( |3\vec{r}|^2 \) is:

If the system of equations (in variables \(x, y, z\)): \(x - 2y + tz = 0\), \(3x + 5y + t^2 z = 0\), and \(6x + ty + f(t)z = 0\) has infinitely many solutions (where \(f(t)\) represents a real function), then:

If \( S_1: x^2 + y^2 - 6x - 8y + 21 = 0 \) and \( S_2: x^2 + y^2 + 6x + 8y + \lambda = 0 \), then the distance of the centre of \( S_2 \) to the farthest point on \( S_1 \) is:

Consider an equilateral \( \Delta PQR \), where \( P(3, 5) \) and the side \( QR \) lies on the line \( x + y = 4 \). If the orthocentre of \( \Delta PQR \) is \( (\alpha, \beta) \), then \( 9(\alpha + \beta) \) is equal to:

If \( \alpha, \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \) and \( |\alpha - \beta| = \sqrt{11} \), \( \alpha + \beta = 3i \), then the value of \( (\alpha^3 - \beta^3) \) is:

A and B play a tennis match which will not result in a draw. The player who wins 5 rounds first will be the winner of the match. The number of ways such that A can win the match is:

If \( \alpha = \frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) \), then the value of \( \tan(\alpha) \) is:

Let \( S_n \) be the sum of the first \( n \) terms of an A.P. If \( S_n = 3n^2 + 5n \), then the sum of the squares of the first 10 terms of the given A.P. is:

The area enclosed between the region given by \( xy \le 27 \) and \( 1 \le y \le x^2 \) is:

The value of \( \lim_{x \to 0} \frac{x^2 \sin^2 x}{x^2 - \sin^2 x} \) is equal to:

The value of \( 1^3 - 2^3 + 3^3 - 4^3 + \dots + 15^3 \) is equal to:

If \( x_1, x_2, \dots, x_{25} \) be 25 observations such that \( \sum_{i=1}^{25} (x_i + 5)^2 = 2500 \) and \( \sum_{i=1}^{25} (x_i - 5)^2 = 1000 \). Then, the ratio of Mean and Standard deviation of the given observations is:

In the expansion of \( (1 + \alpha x)^{26} \) and \( (1 - \alpha x)^{28} \), the coefficient of the middle term is the same, then the value of \( \alpha \) is:

\( \int_{-\pi/4}^{\pi/4} \frac{32 \cos^4 \theta}{1 + e^{\sin \theta}} \, d\theta \) is equal to:

If $\int_{-2}^2 ([\sin x] + |x \sin x|) dx = 2 \sin 2 - 4 \cos 2 - \beta$, then the value of $|\beta|$ where $[\cdot]$ is GIF is

If coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}$ is $\binom{100}{3} \left( \frac{101}{4} - 43n \right)$, then the value of $(k^3 + 43n)$ is