List of top Mathematics Questions

Let \( A (2, 3, 5) \) and \( C(-3, 4, -2) \) be opposite vertices of a parallelogram \( ABCD \). If the diagonal \( \overrightarrow{BD} = \hat{i} + 2 \hat{j} + 3 \hat{k} \), then the area of the parallelogram is equal to

Let \( \alpha, \beta \in \mathbb{N} \) be roots of the equation \( x^2 - 70x + \lambda = 0 \), where \( \frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N} \). If \( \lambda \) assumes the minimum possible value, then \[ \frac{\left( \sqrt{\alpha - 1} + \sqrt{\beta - 1} \right)(\lambda + 35)}{|\alpha - \beta|} \] is equal to____.

The number of ways to distribute the 21 identical apples to three children’s so that each child gets at least 2 apples.

A parabola has vertex \((2, 3)\), equation of directrix is \(2x – y = 1\) and equation of ellipse is \(\frac {x^2}{a^2} +\frac {y^2}{b^2} =1\), \(e=\frac {1}{\sqrt 2}\) and ellipse passing through focur of parabola then square of length of latus rectum of ellipse is

Let \(y = y(x)\) be the solution of the differential equation \(\sec(x) \frac{dy}{dx} + [2(1 - x) \tan(x) + x(2 - x)] = 0\) such that \(y(0) = 2\). Then \(y(2)\) is equal to:

Let \( f : \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = \frac{1}{2} \). If the \( \lim_{x \to 0} \frac{\int_{0}^{x} f(t) \, dt}{e^{x^2} - 1} = \alpha \), then \( 8\alpha^2 \) is equal to:

Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \rightarrow P(A) \) such that \( a \in f(a), \, \forall a \in A \) is \( m^n \), \( m \) and \( n \in \mathbb{N} \) and \( m \) is least, then \( m + n \) is equal to \(\_\_\_\_\_\_\_\_\_\).

Let \( y = y(x) \) be the solution of the differential equation \( (1 - x^2) \, dy = \left[ xy + \left( x^3 + 2 \right) \sqrt{3 \left( 1 - x^2 \right)} \right] dx \), \( -1 < x < 1, y(0) = 0 \). If \( y\left( \frac{1}{2} \right) = \frac{m}{n} \), \( m \) and \( n \) are coprime numbers, then \( m + n \) is equal to \(\_\_\_\_\_\_\_\_\_\).

The number of solution of equation \(e^{sinx} - 2e ^{-sinx} = 2\) is

The value of \(\frac {120}{\pi^3}|∫_0^\pi\frac {x^2sinx.cosx}{(sinx)^4+(cosx)^4}dx|\) is

If 2^nd, 8^th, 44^th terms of A.P. are 1^st, 2^nd and 3^rd terms respectively of G.P. and first term of A.P. is 1 then the sum of first 20 terms of A.P. is

Let \( g : \mathbb{R} \rightarrow \mathbb{R} \) be a non-constant twice differentiable function such that \( g'\left(\frac{1}{2}\right) = g'\left(\frac{3}{2}\right) \). If a real-valued function \( f \) is defined as \[ f(x) = \frac{1}{2} \left[ g(x) + g(2 - x) \right], \] then

A line passing through the point \( A(9, 0) \) makes an angle of \( 30^\circ \) with the positive direction of the x-axis. If this line is rotated about \( A \) through an angle of \( 15^\circ \) in the clockwise direction, then its equation in the new position is:

If y = f(x) is solution of differential equation \((x^2 - 1) dy = ((x^3 + 1) + \sqrt{(1 - x^2)}dx \) and \(y(0) = 2\) then find \(y(\frac{1}2)\)

In a class there are 40 students. 16 passed in Chemistry, 20 passed in Physics, 25 passed in Math. 15 students passed in both Math and Physics.15 students passed in both Math and Chemistry and 10 students passed in both Physics and Chemistry. Find the maximum number of students that passed in all the subjects.

If the circles \( (x+1)^2 + (y+2)^2 = r^2 \) and \( x^2 + y^2 - 4x - 4y + 4 = 0 \) intersect at exactly two distinct points, then

The minimum value of \(| z+ \frac{3+4i}{2}|, |z| \leq 1\) is,

If the hyperbola \(x² - y² cosec²θ = 5\) and ellipse \(x²cosec²θ + y² = 5\) has eccentricity \(e_H\) and \(e_E\) respectively and \(e_H = \sqrt 7e_E\), then \(θ\) is equal to

Number of integral terms in the expansion of \( \left( \sqrt{7} z + \frac{1}{6 \sqrt{z}} \right)^{824} \) is equal to ______.

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

Let \( \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \) be two vectors such that \( |\vec{a}| = 1 \), \( \vec{a} \times \vec{b} = 2 \), and \( |\vec{b}| = 4 \). If \( \vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b} \), then the angle between \( \vec{b} \) and \( \vec{c} \) is equal to:

Let vertex \(A(2, 3, 1), B(3, 2, -1), C(-2, 1, 3)\). If \(AD\) is angle bisector of angle \(A\), then projection of \(\overrightarrow{AD}\) on \(\overrightarrow{AC}\) is equal to:

If the foot is perpendicular from (1, 2, 3) to the line \(\frac{x+1}{2} = \frac{y-2}{5} = \frac{z-1}{1}\) is \(( a, \beta, \gamma)\), then find \(a + \beta + \gamma\)

Five people are distributed in four identical rooms. A room can also contain zero people. Find the number of ways to distribute them.