List of top Mathematics Questions

The number of ways of forming the ordered pairs (p, q) such that p>q by choosing p and q from the first 50 natural numbers is:
If all possible 4-digit numbers are formed by choosing 4 different digits from the given digits $ 1, 2, 3, 5, 8 $, then the sum of all such 4-digit numbers is:
The cubic equation whose roots are the squares of the roots of the equation \( x^3 - 2x^2 + 3x - 4 = 0 \) is
If the roots of the equation \(x^2 + 2ax + b = 0\) are real, distinct and differ utmost by \(2m\), then b lies in the interval
Let \((a-3)x^2 + 12x + (a+6) > 0, \forall x \in \mathbb{R} \text{ and } a \in (t, \infty)\). If \(\alpha\) is the least positive integral value of \(a\), then the roots of \((\alpha-3)x^2 + 12x + (\alpha+2) = 0\) are:
The sum of the squares of the imaginary roots of the equation $ z^8 - 20z^4 + 64 = 0 $ is:
If the least positive integer \( n \) satisfying the equation \(\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n = -1\) is \( p \) and the least positive integer \( m \) satisfying the equation \(\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^m = \text{cis}\left(\frac{2\pi}{3}\right)\) is \( q \), then \(\sqrt{p^2 + q^2}\) is equal to:
If \( z \) is a complex number such that \( \frac{z-1}{z-i} \) is purely imaginary and the locus of \( z \) represents a circle with center \( (\alpha, \beta) \) and radius \( r \), then the value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \) is:
The rank of the matrix \( \begin{bmatrix} 2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4 \end{bmatrix} \) is
If \( \alpha \) is a real root of the equation \( x^3 + 6x^2 + 5x - 42 = 0 \), then the determinant of the matrix \[ \begin{bmatrix} \alpha - 1 & \alpha + 1 & \alpha + 2 \\ \alpha - 2 & \alpha + 3 & \alpha - 3 \\ \alpha + 4 & \alpha - 4 & \alpha + 5 \end{bmatrix} \] is Options:
If \( P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix} \) is the adjoint of a matrix \( A \) and \( \det(A) = 4 \), then the value of \( \alpha \) is:
If \( 11^{12} - 11^2 = k(5 \times 10^9 + 6 \times 10^9 + 33 \times 10^8 + 110 \times 10^7 + \ldots + 33) \), then find the value of \( k \).
If \( f(x) = (x+1)^2 - 1, x \ge -1 \), then \( \{x \mid f(x) = f^{-1}(x)\} \) is:
Let \( [t] \) denote the greatest integer function and \( [t - m] = [t] - m \) when \( m \in \mathbb{Z} \). If \( k = 2[2x - 1] - 1 \) and \( 3[2x - 2] + 1 = 2[2x - 1] - 1 \), then the range of \( f(x) = [k + 5x] \) is:
If $ \alpha $, $ \beta $, and $ \gamma $ are the angles made by a vector with the $ x $-, $ y $-, and $ z $-axes respectively, then find the value of $ \sin^2\alpha + \sin^2\beta $.
Evaluate: \( \int_0^{400\pi} \sqrt{1 - \cos 2x} \, dx \)
If \( \int \frac{dx}{2\cos x + 3\sin x + 4} = \frac{2}{\sqrt{3}}f(x) + c \), then \( f\left(\frac{2\pi}{3}\right) = \)
Evaluate \( \int \frac{\sec^2 x}{\sin^7 x} \, dx - \int \frac{7}{\sin^7 x} \, dx \):
If \(y = \sqrt{\cosh x + \sqrt{\cosh x}}\), then \(\frac{dy}{dx} =\)
If $ A(0, 1, 2) $, $ B(2, -1, 3) $, and $ C(1, -3, 1) $ are the vertices of a triangle, then the distance between its circumcentre and orthocentre is
In a triangle ABC, if \((r_1 - r_3)(r_1 - r_2) - 2r_2r_3 = 0\), then \(a^2 - b^2 =\)
In triangle $ ABC $, if $ a = 13 $, $ b = 8 $, $ c = 7 $, then $ \cos(B+C) = $
If $ \alpha, \beta, \gamma $ are the roots of the equation $ x^3 + px^2 + qx + r = 0 $, then $ \alpha^3 + \beta^3 + \gamma^3 = $
The set of all real values of $x$ such that \[ f(x) = \frac{[x] - 1}{\sqrt{[x]^2 - [x] - 6}} \] is a real valued function is
If a function $f : \mathbb{Z} \to \mathbb{Z}$ is defined by $f(x) = x - (-1)^x$, then $f(x)$ is