List of top Mathematics Questions

If the elements of matrix \( A \) are the reciprocals of the elements of the matrix \( \begin{pmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{pmatrix} \), where \( \omega \) is a complex cube root of unity, then
Evaluate \( \int \frac{x^2}{(x+1)^2(x+2)^2} \, dx \)
Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \)
If \( P(A) = \frac{2}{5}, \, P(B) = \frac{1}{4}, \, P(A \cup B) = \frac{1}{2}, \) then \( P(A' \cup B') = \)
The centre and radius of a circle \[ x = 4a \left( \frac{1 - t^2}{1 + t^2} \right), \quad y = \frac{8at}{1 + t^2} \] are respectively
If \[ y = \tan^{-1}\left( \frac{\sqrt{1 + \cos \frac{x}{2}}}{\sqrt{1 - \cos \frac{x}{2}}} \right) \] then \( \frac{dy}{dx} \) is
If the planes \[ 2x - 5y + z = 8 \quad \text{and} \quad 2 \lambda x - 15y + 2z + 6 = 0 \] are parallel to each other, then the value of \( \lambda \) is
The p.d.f. of a continuous random variable \( X \) is given by \[ f(x) = \begin{cases} \frac{1}{2} & \text{if } 0<x<2, \\ 0 & \text{otherwise}, \end{cases} \] and if \[ a = P(X \leq \frac{1}{2}), \quad b = P(X>\frac{1}{2}), \] then the relation between \( a \) and \( b \) is
The p.d.f. of a continuous random variable \( X \) is given by \[ f(x) = \begin{cases} \frac{1}{2} & \text{if } 0<x<2, \\ 0 & \text{otherwise}, \end{cases} \] and if \[ a = P(X \leq \frac{1}{2}), \quad b = P(X>\frac{1}{2}), \] then the relation between \( a \) and \( b \) is
If \( m_1 \) and \( m_2 \) are slopes of the lines represented by \( \left( \sec^2 \theta - \sin^2 \theta \right) x - 2 \tan \theta \, xy + \sin^2 \theta \, y^2 = 0 \), then \( |m_1 - m_2| = \)
The maximum value of \[ Z = 3x + 5y, \] subject to \[ x + 4y \leq 24, \quad y \leq 4, \quad x \geq 0, \quad y \geq 0 \] is
If \[ y = \tan^{-1}\left( \frac{\sqrt{1 + \cos \frac{x}{2}}}{\sqrt{1 - \cos \frac{x}{2}}} \right) \] then \( \frac{dy}{dx} \) is
If the line \( \vec{r} = (i - 2j + 3k) + \lambda(2i + j + 2k) \) is parallel to the plane \( \vec{r} = (3i - 2j - mk) = 5 \), then the value of \( m \) is
A die is thrown 100 times, then the standard deviation of getting an even number is
If \( \frac{1-\tan\theta}{1+\tan\theta} = \frac{1}{\sqrt{3}}, \) where \( \theta \in \left(0,\frac{\pi}{2}\right) \), then \( \theta = \)
Bacteria increases at the rate proportional to the number of bacteria present. If the original number \( N \) doubles in \( 4 \) hours, then the number of bacteria will be \( 4N \) in
The length of the perpendicular from the point \( P(a,b) \) to the line \( \dfrac{x}{a} + \dfrac{y}{b} = 1 \) is
If \( f(x) = \begin{cases \dfrac{81^x - 9^x}{k^x - 1}, & x \neq 0
2, & x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of \( k \) is}
If \( \dfrac{d^2y}{dx^2} = \sin x + e^x \), \( y(0) = 3 \) and \( \left.\dfrac{dy}{dx}\right|_{x=0} = 4 \), then the equation of the curve is
The general solution of \( \tan 3x = 1 \) is
Evaluate \( \displaystyle \int_{0}^{1} \tan^{-1}\!\left(\frac{2x}{1-x^2}\right)\,dx \)
If \( A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 0 & 3 & -5 \end{pmatrix} \) where \( A_{ij} \) is the cofactor of the element \( a_{ij} \) of matrix \( A \), then \( a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23} = \)
The direction cosines of a line which lies in the \( ZOX \) plane and makes an angle of \( 30^\circ \) with the \( Z \)-axis are
If \( \cosec\theta + \cot\theta = 5 \), then \( \sin\theta = \)
The eccentricity of the hyperbola \( 16x^2 - 3y^2 - 32x - 12y - 44 = 0 \) is