List of top Mathematics Questions

Find \( I = \int_{-\pi/4}^{\pi/4} \frac{32 \cos^4 \theta}{1 + e^{\sin \theta}} \, d\theta \)

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

Let \( A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix} \). If \( B = [b_{ij}]_{3 \times 3} \) and \( B = A^{99} - I \), then find \( \frac{b_{31} - b_{21}}{b_{32}} \).

If $\vec{a}_R = \tan \theta_R \hat{i} + \hat{j}$ and $\vec{b}_R = \hat{i} - \cot \theta_R \hat{k}$ where $\theta_R = \frac{2^{R-1} \pi}{2^N + 1}$, then the value of $\frac{\sum_{R=1}^N |\vec{a}_R|^2}{\sum_{R=1}^N |\vec{b}_R|^2}$ is

Let a hyperbola be \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) and ellipse be \(\frac{x^2}{9} + \frac{y^2}{8} = 1\). If length of latus rectum of hyperbola is equal to minor axis of ellipse and eccentricity of hyperbola is equal to semi-major axis of ellipse, then \(2ae\) is equal to (where 'e' is the eccentricity of hyperbola)

If \(\alpha\) and \(\beta\) are the roots of the equation \(z^2 - \sqrt{6}i z - 3 = 0\), then find \(\alpha^8 + \beta^8\).

The line \(x - y = 4\) is a chord of the circle \((x - 4)^2 + (y + 3)^2 = 9\), which cuts the circle at points Q & R. If \(P(\alpha, \beta)\) lies on the circle such that \(PQ = PR\), then find \((6\alpha + 8\beta)^2\).

Let \(f(x+y) = f(x) f(y)\), \(f(0) \neq 0\). If \(x^2 g(x) = \int_0^x (t^2 f(t) + t g(t)) \, dt\), then \(g(2)\) is equal to

Consider the observations: 2, 4, \(\alpha\), \(\beta\), 6, 12, 14. If their mean is 8 and variance = 16, then the quadratic equation whose roots are \(3\alpha + 2\) and \(2\beta + 1\), is

A bag contains 6 Red and 6 black balls. 6 pair of balls are selected one by one without replacement then the probability that each of the 6 pairs contains 1 red and 1 black ball.

The shortest distance between the lines \(\frac{x-3}{-1} = \frac{y-2}{4} = \frac{z-1}{2}\) and \(\frac{x-1}{2} = \frac{y-1}{1} = \frac{z-2}{5}\) is:

Let \(\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}\), \(\vec{b} = 6\hat{i} + 3\hat{j} + 3\hat{k}\). If \(2\vec{a} + 3\vec{b}\) and \(\vec{a} - \vec{b}\) are two adjacent sides of a triangle then square of area of the triangle is

If system of equations \(x \cos \theta - 8y - 12z = 0\), \(x \cos 2\theta + y + 3z = 0\), \(x + y + 3z = 0\) has non-trivial solution, then find sum of values of \(\theta\) (where \(\theta \in [0, 2\pi]\)).

Let \(x = 9\) be a directrix of an ellipse centred at \((0, 0)\) and having eccentricity \(\frac{1}{3}\). If focus at \((\alpha, 0)\) (\(\alpha<0\)), then locus of the mid-point of the chord passing through the focus \((\alpha, 0)\) is

An ellipse has directrix \(x = 9\) & eccentricity \(= \frac{1}{3}\). If one of its focus is \((\alpha,0)\), \(\alpha<0\), then locus of the mid-point of the chord passing through \(P(\alpha,0)\) is

A lift of a 10 floor building contains 9 persons and group of 4 and 5 leave the lift on different floor and there is no stoppage of lift at 1st and 2nd floor, then find number of ways this can be done.

A 3rd order square matrix M satisfies \( M \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 2 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 2 & 1 \end{pmatrix} \) and \( M \begin{pmatrix} 0 & -1 \\ 1 & 2 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix} \). If \( M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \\ 7 \end{pmatrix} \), then \( x + y + z \) is:

Let \( A = \{1, 4, 7\} \), \( B = \{2, 3, 8\} \). Let \( R \) be a relation defined as \[ \{((a_1, b_1), (a_2, b_2)) \in (A \times B) \times (A \times B) : (a_2 + b_1) \text{ divides } (a_1 + b_2)\} \], then find the number of such relations.

Let \( f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix} \). If \( \sum_{n=1}^{k} f(n) = 98 \), then find \( k \).

Let \( f(x) + 3f\left(\frac{\pi}{2} - x\right) = \sin x \) & maximum value of f is \( \alpha \). If area bounded between \( g(x) = x^2 \) & \( h(x) = \beta x^3 \) (\( \beta >" 0 \)) is \( \alpha^2 \), then \( 30\beta^3 \) is equal to:

If distance of point (a, 2, 5) from image of point (1, 2, 7) in the line \( \frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2} \) is 4, then sum of all possible values of a is:

Let \( A = \{2, 3\} \) and \( B = \{5, 6\} \), then the number of relations from \( A \times B \) to \( A \times B \) are:

Let foci of a hyperbola be (3, 5) and (3, -4). If eccentricity ‘e’ of the hyperbola satisfies the equation \( 3e^2 - 11e + 6 = 0 \), then the length of the latus rectum of the hyperbola is:

If the sum of the first 10 terms of the series \( \frac{1}{1 + 4 \cdot 1^4} + \frac{2}{1 + 4 \cdot 2^4} + \frac{3}{1 + 4 \cdot 3^4} + \dots \) is \( \frac{m}{n} \) (where m, n are coprime), then (m + n) is:

The coefficient of \( x^2 \) in the binomial expansion of \( (2x^2 + \frac{1}{x})^{10 \) is: