List of top Mathematics Questions

The equation of the pair of asymptotes of the hyperbola $x^2 - 2y^2 - 8x + 8y + 4 = 0$ is
Evaluate the integral: \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1 + \sqrt{\cot x}} \, dx = \]
The positive value of \( x \) satisfying the equation \[ \int_0^x (1 - t) \, dt = \frac{1}{2} \] is:
If matrix \( D_1 = \text{diag}(a, b, c) \), matrix \( D_2 = \text{diag}(3, 3, 3) \) and \( A \) is a skew-symmetric matrix of order 3, then
\[ \text{Tr}(D_1 D_2 A + D_2 D_1 + D_1 A + D_2 A) - \text{Tr}(D_1 + D_2) = ? \]
If the roots of the equation \( 3x^2 + 4kx + 3 = 0 \) are non-real, then \( k \) lies in the interval:
If \(u = \sin\left(\frac{x}{y}\right),\ x = e^t,\ y = t^2\), then \[ t^6 \left(\frac{du}{dt}\right)^2 \div e^{2t}(t - 2)^2 = \]
If the cofactors of the elements 3, 7, and 6 of the matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6 \end{bmatrix} \) are \( a, b, c \) respectively, then evaluate: \[ \left[ a \ b \ c \right] \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} + \left[ a \ b \ c \right] \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix} \]
Tangent \( L_1 = 3x - 4y - 8 = 0 \) and the chord \( L_2 = x + y - 1 = 0 \) are at a distance of 2 and \( \sqrt{2} \) units respectively from the centre of a circle \( S \). \((h, k)\) is the centre of \( S \) such that \( h^2 + k^2 = 13 \). If the midpoint of the chord \( L_2 = 0 \) is \( (\alpha, \beta) \) and the radius of the circle is \( r \), then \( \alpha + \beta + r = \):
The area (in square units) bounded by \[ x = 4, \quad y = -4, \quad \text{and} \quad y = x \quad \text{is:} \]
If \( f'(x) = a \cos x + b \sin x \), \( f'(0) = 4 \), \( f(0) = 3 \), and \( f\left( \frac{\pi}{2} \right) = 5 \), then \( f(x) = \)
Let the circle \( S \) be concentric with the circle \( x^2 + y^2 - 2x + ky + 4 = 0 \). If one of the diameters of \( S \) lies along the line \( 3x - 2y + 4 = 0 \) and the length of the diameter is 6, then the radius of the circle \( S \) is
If A(2,3) and B(2,-3) are two points, then the equation of the locus of a point $P$ such that $PA + PB = 8$ is
The line \( x + y = k \) meets the curve \( x^2 + y^2 - 2x - 4y + 2 = 0 \) at two points \( A \) and \( B \). If \( O \) is the origin and \( \angle AOB = 90^\circ \), then the value of \( k \) (\( k>1 \)) is
If the equation \( x^4 + ax^3 + bx^2 + cx + d = 0 \) has three equal roots, then the root is:
Evaluate \( \cot 16^\circ \cot 44^\circ + \cot 44^\circ \cot 76^\circ - \cot 76^\circ \cot 16^\circ \):
If \( \cosh x = \frac{5}{4} \), then \( \tanh 3x = \)
Let \( A = \begin{pmatrix} 0 & 3 & 5 & -7 \\ 8 & 0 & -1 & 0 \\ 6 & -1 & 0 & 0 \end{pmatrix} \) and \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \). If \( D = [\alpha \, \beta \, \gamma]^T \) is the solution of \( X^T B^T = A^T X \), then \( D^T A = \)
If \( f(x) \) is a function such that \( f'(x) = \sqrt{f^2(x) - 1} \) and \( f(0) = 1 \), then \( f(1) = \):
A pair of dice is thrown. Then the probability that either of the dice shows 2 when their sum is 6 is:
If the mean and varian of a binomial variable $X$ are 2 and 1 respectively, then $P(X>1) = $
If \[ \frac{d}{dx}\left( \frac{x^2}{(x+2)(2x+3)} \right) = \frac{-A}{(x+2)^2} + \frac{B}{(2x+3)^2}, \] then the value of \( A + B = \):
If \( 2i\hat{i} - j\hat{j} - 3j\hat{k} \) and \( -3i\hat{i} + 4j\hat{j} - 4k\hat{k} \) are the position vectors of three points A, B, and C respectively, then \( \Delta ABC \) is:
If $\vec{a} = 2\hat{i} - t\hat{j} - 2\hat{k}$ and $\vec{b} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ are two vectors such that $\vec{a} \cdot \vec{b}$ is minimum, then $t=$
The probability that a non-leap year contains 53 Sundays is
If p and q are the x and y intercepts respectively of the line passing through the points \( (a \cos \alpha, b \sin \alpha) \) and \( (a \cos \beta, b \sin \beta) \), then: