List of top Mathematics Questions

The maximum value of \[ E = 16\sin\frac{x}{2}\cos^3\frac{x}{2} \] where \(x\in[0,\pi]\), is

If function \(y(x)\) satisfies the differential equation \[ \frac{dy}{dx}+\left[\frac{6x^2+e^{-2x}(3x^2+2x^3+4)}{(x^3+2)(2+e^{-2x})}\right]y = e^{-2x}+2 \] such that \(y(0)=\frac{3}{2}\) and \[ y(1)=\alpha(e^{-2}+2) \] then \(\alpha\) is equal to

If the quadratic equation \[ (\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0 \quad (\lambda \ne -2) \] has two positive roots then the number of possible integral values of \(\lambda\) is

From point \(B(4,8)\) on the parabola \(y^2 = 16x\), two perpendicular chords \(BA\) and \(BC\) are drawn. Given that the locus of the centroid of triangle \(BAC\) is another parabola with length of the latus rectum equal to \(\ell\), then \(3\ell\) is equal to

Area bounded between the curves \[ x = -2y^2 \] \[ x = 1 - 4y^2 \] is

If the system of equations \[ x + y + z = 5 \] \[ x + 2y + 3z = 9 \] \[ x + 3y + \lambda z = \mu \] has infinitely many solutions, then value of \( \lambda + \mu \) is

If \( \sum_{i=1}^{10} (x_i + 2)^2 = 180 \) and \( \sumᵢ=1¹0 (xᵢ - 1)² = 90 , then the Standard Deviation is equal to

If the matrix \[ \begin{bmatrix} 1 & 3 & 1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix} \] is singular. Given a function \( f(x) = \int_{0}^{x} (t^2 + 2t + 3)\, dt \), \( \forall x \in [1, \alpha] \). If \( m \) and \( n \) are the maximum and minimum values of the function \( f(x) \), then the value of \( 3(m - n) \) is:

If \(\sqrt{3}i, 1\) are the roots of \(x^{3} + ax^{2} + bx + c = 0\) where \(a, b, c \in \mathbb{R}\). Then \(\int_{-1}^{1} (x^{3} + ax^{2} + bx + c)dx\) is

The value of \( x \) which satisfies the equation \( \sin^{-1}\left(\frac{2}{3}\sqrt{1 - x^2}\right) = \cot^{-1}\left(2\sqrt{x}\right) \), is

Let \( f : \mathbb{R} \to \mathbb{R} \), \( f(x) = \frac{2x^2 - 3x + 2}{3x^2 + x + 3} \), then \( f(x) \) is

Let \( R \) be a relation such that \( R = \{ (x, y) \in \mathbb{N} \times \mathbb{N} \mid x + y \le 7 \} \). Minimum number of elements to be added in \( R \) so that it becomes transitive is:

If \( (1 - x)^{10} = \sum_{r=0}^{10} a_r \, x^r (1 - x)^{30 - 2r} \), then find \( \frac{9a_9}{a_{10}} \).

Given that quadratic equation \( (k^2 - 15k + 27) x^2 + 9(k - 1)x + 18 = 0 \) has one root twice of other. Then find the length of the latus rectum of the parabola \( y^2 = 6kx \):

Given vectors \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{b} = \hat{j} - \hat{k} \). Another vector \( \vec{c} \) satisfy equations \( \vec{a} \cdot \vec{c} = 3 \) and \( \vec{a} \times \vec{c} = \vec{b} \), then find \( \vec{a} \cdot (\vec{c} - 2\vec{b}) \):

Consider \( e_1 \) and \( e_2 \) be roots of the equation \( x^2 - a x + 2 = 0 \). Set of exhaustive values of 'a' for which \( e_1 \) and \( e_2 \) are eccentricities of hyperbolas then \( a \in [\alpha, \beta) \) and set of values of 'a' for which \( e_1 \) and \( e_2 \) are eccentricity of the parabola and ellipse is \( (\gamma, \infty) \) then \( (\alpha^2 + \beta^2 + \gamma^2) \) equal:

Solution of differential equation \( \frac{dy}{dx} + \frac{y(x - \sqrt{x^2 - 1})}{x^2 - x\sqrt{x^2 - 1}} = \frac{x}{x^2 - x\sqrt{x^2 - 1}} \) satisfies the condition \( y(1) = 1 \), then find \( [y(\sqrt{5})] \). (Here [·] denotes greatest integer function):

Find the area bounded by \( 0 \le y \le 6 - x \), \( y^2 + 3 \le 4x \) and \( x>0 \):

If domain of \( f(x) = \sin^{-1} \left( \frac{x + |x|}{3} \right) \) is \( [\alpha, \beta) \), then \( (\alpha^2 + \beta^2) \) is:

If the ratio of y-ordinates of points on the parabola \( y^2 = 12x \) is 1 : 2 and the length of the chord joining these points is \( 3\sqrt{13} \), then find the angle subtended by the chord at the focus of the parabola:

Given two lines \( L_1: \frac{x-1}{3} = \frac{y-2}{2} = \frac{z+1}{1} \) and \( L_2: \frac{x+2}{1} = \frac{y-1}{1} = \frac{z}{1} \). Third line \( L_3 \) is perpendicular to both lines \( L_1 \) & \( L_2 \). Find acute angle between lines \( L_1 \) & \( L_2 \).

Given point \( P(6, 4\sqrt{5}) \) satisfies the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] where eccentricity is a root of the equation \( 9e^2 - 21e + 10 = 0 \). Then find the length of latus rectum of the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{2(1+b^2)} = 1. \]

If there are (n + 1) coins of which n coins are fair and one is double headed. A coin is randomly selected and tossed, if probability that head occurs is \( \frac{9}{16} \) then n is:

If \( x_1, x_2, x_3, \dots, x_{25} \) 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 observation, is:

Let the set of all values of \( K \in \mathbb{R} \) such that the equation, \( z(\bar{z} + 2 + i) + K(2 + 3i) = 0 \), \( z \in \mathbb{C} \) has at least one solution, be the interval \( [\alpha, \beta] \). Then \( 9(\alpha + \beta) = \)