List of top Mathematics Questions

The remainder when the polynomial \( 2x^5 - 3x^4 + 5x^3 - 3x^2 + 7x - 9 \) is divided by \( x^2 - x - 3 \) is:
Let \( f(x) = x^2 + \frac{1}{x^2} \) and \( g(x) = x - \frac{1}{x} \) for \( x \in \mathbb{R} \setminus \{-1, 0, 1\} \). Then, the local minimum of \( \frac{f(x)}{g(x)} \) is:
If \( 1, \omega, \omega^2 \) are the cube roots of unity, then evaluate: \[ (2 - \omega)^2 (2 - \omega^2)^2 (2 - \omega^{10})^2 (2 - \omega^{11})^2 \]
Which of the following quadratic equations whose real roots \( x_1, x_2 \) satisfy the conditions \( x_1^2 + x_2^2 = 5 \), \( 3(x_1^5 + x_2^5) = 11(x_1^3 + x_2^3) \)?
If \( ^nC_{r-1} = 36 \), \( ^nC_r = 84 \), and \( ^nC_{r+1} = 126 \), then the value of \( nr^2 \) is:
The total number of positive integral solutions \((x, y, z)\) of \( xyz = 24 \) is:
Let \( z = x + iy \) be a complex number with \( x, y \in \mathbb{Z} \). Then the area (in square units) of the rectangle whose vertices are the roots of the equation \( \bar{z} \cdot z^3 + z \cdot \bar{z}^3 = 350 \) is:
Let \( z \) and \( w \) be two complex numbers such that \( \bar{z} + i\bar{w} = 0 \) and \( \text{Arg}(zw) = \pi \). Then \( \text{Arg}(z) = \)
If the complex numbers \( z_1, z_2, 0 \) are vertices of an equilateral triangle, then \( z_1^2 + z_2^2 = \)
If \(A = \begin{bmatrix}x & 2 & 1 \\ 2 & x & 1 \\ 2 & 1 & 0\end{bmatrix}\) and \(\det(A) = 125\), then \(x =\)
Let \(A = \begin{bmatrix}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{bmatrix}\) and \(B = \begin{bmatrix}0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0\end{bmatrix}\). Then \(A^2 + B^2 + AB =\)
Let \( A = \begin{bmatrix} n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0 \end{bmatrix} \). Then \( A^2 + B^2 + AB = \)
The domain of the real-valued function \(f(x) = \frac{\sqrt{\log_{0.5}(x-3)}}{\sqrt{x-1}}\) is:
If the system of equations \(x+y-z=6\), \(3x-y+z=2\), and \(x+ky+z=-8\) has a unique solution \(x=2\), \(y=0\), \(z=-4\), then the value of \(k\) satisfies which quadratic equation?
If a function \( f \) satisfies \( f(x+1) + f(x-1) = \sqrt{2}f(x) \), then \( f(x+2) + f(x-2) = \)
The maximum value of \( f(x) = \frac{x}{1 + 4x + x^2} \) is
The minimum value of \( f(x) = x + \frac{4}{x + 2} \) is
If \( f(x) = \max \{3 - x, 3 + x, 6\} \) is not differentiable at \( x = a \) and \( x = b \), then \( |a| + |b| = \)
At any point \( (x, y) \) on a curve, if the length of the subnormal is \( (x - 1) \) and the curve passes through \( (1, 2) \), then the curve is a conic. A vertex of the curve is:
\( \lim_{n \to \infty} \left( \frac{1}{1 + n^5} + \frac{2^4}{2^5 + n^5} + \frac{3^4}{3^5 + n^5} + \ldots + \frac{n^4}{n^5 + n^5} \right) = \)
The function \( y = Ae^x + Be^{-2x} \) satisfies which of the following differential equations?
If the solution of \( \frac{dy}{dx} = y \log_e 0.5 = 0 \), \( y(0) = 1 \), and \( y(x) \to k \) as \( x \to \infty \), then \( k = \)
Assertion (A): \( I_n = \int \cot^n x \, dx \), then \( I_6 + I_4 = -\frac{\cot^5 x}{5} \)
Reason (R): \( \int \cot^n x \, dx = \frac{-\cot^{n-1} x}{n - 1} - \int \cot^{n - 2} x \, dx \)
\( \int \frac{e^{2x}}{\sin^2 x} \left( 2 \log \csc x + \sin 2x \right) dx = \)
If \( I_n = \int \tan^n x \, dx \), and \( I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \), then \( n = \)