List of top Mathematics Questions

The sum of all possible values of \( \theta \in [0, 2\pi] \), for which the system of equations:
\( x \cos 3\theta - 8y - 12z = 0 \)
\( x \cos 2\theta + 3y + 3z = 0 \)
\( x \sin \theta + y + 3z = 0 \)
has a non-trivial solution, is equal to:

\[ \int_{-1}^{1} \frac{x^3 + |x| + 1}{x^2 + |x| + 1} \, dx \]

The area (in square units) of the region \(\{(x,y): x^2 - 8x \leq y \leq -x\}\) is:

If the percentage change in the radius of a sphere is \(2\%\), then find the percentage change in the volume of the sphere.

If \(\left(2\alpha+1,\;\alpha^2-3\alpha,\;\frac{\alpha-1}{2}\right)\) is the image of \((\alpha,2\alpha,1)\) in the line \[ \frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}, \] then the value of \(\alpha\) is:

P is a point on \[ \frac{x^2}{9}+\frac{y^2}{4}=1 \] as \(P(3\cos\alpha,2\sin\alpha)\). Q is a point on \[ x^2+y^2-14x+14y+82=0 \] R is a point on line \[ x+y=5 \] If the centroid of triangle \(PQR\) is \[ \left(\cos\alpha+2,\;\frac{2\sin\alpha}{3}+3\right) \] find the sum of possible ordinates of \(R\).

The value of \( \sum_{n=1}^{10} \frac{528}{n(n+1)(n+2)} \) is equal to:

If \( \lim_{x \to 0} \frac{1 - \cos(\alpha x)\cos((\alpha + 1)x)\cos((\alpha + 2)x)}{\sin^2((\alpha + 1)x)} = 2 \), then the product of all possible values of \( \alpha \) is:

If \( f(x) \) satisfies the relation \( f\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y)}{3} \) and \( f(0) = 3 \), then the minimum value of \( g(x) = 3 + e^x f(x) \) is:

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 \( \frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4} \):

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:

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:

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

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

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:

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 \( \alpha, \beta \) 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:

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 \( S_1: x^2 + y^2 - 6x - 8y + 21 = 0 \) 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:

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: