List of top Mathematics Questions

The number of all possible positive integral solutions of the equation \(xyz = 30\) is:

The number of ways of arranging all the letters of the word PERFECTION such that there must be exactly two consonants between any two vowels is:

If \(\alpha\) and \(\beta\) (\(\alpha>\beta\)) are the multiple roots of the equation \[ 4x^4 + 4x^3 - 23x^2 - 12x + 36 = 0, \] then find \(2\alpha - \beta\).

If the harmonic mean of the roots of the equation \[ \sqrt{2} x^2 - b x + \left( 8 - 2\sqrt{5} \right) = 0 \] is 4, then the value of \( b \) is:

All the values of \(k\) such that the quadratic expression \(2kx^2 - (4k+1)x + 2\) is negative for exactly three integral values of \(x\), lie in the interval:

\[ \left( \sqrt{2} + 1 + i \sqrt{2} - 1 \right)^8 = ? \]

If \( z_1 \) and \( z_2 \) are two of the \( n^{th} \) roots of unity such that the line segment joining them subtends a right angle at the origin, then for a positive integer \( k \), \( n \) takes the form:

If \[ A = \begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & -3 \\ 4 & -4 & 5 \end{bmatrix} \] and \( A^T \) represents the transpose of \( A \), then calculate \( AA^T - A - A^T \).

The domain and range of a real valued function \( f(x) = \cos (x-3) \) are respectively.

If \( A \) and \( B \) are both \( 3 \times 3 \) matrices, then which of the following statements are true? \[ \begin{cases}

For all \( n \in \mathbb{N} \), if \( 1^3 + 2^3 + 3^3 + \cdots + n^3>x \), then a value of \( x \) among the following is:

If $ S_n = 1^3 + 2^3 + \ldots + n^3 $ and $ T_n = 1 + 2 + \ldots + n $, then

The domain of the real valued function $ f(x) = \frac{3}{4 - x^2} + \log_{10}(x^3 - x) $ is

Find the general solution of: $$ (2x - y)^2 dy - 2(2x - y)^2 dx - 2 dx = 0 $$

Find the general solution of the differential equation: $$ x \log x \, dy = (x \log x - y) dx $$

Evaluate: $$ \int_{-\pi/2}^{\pi/2} \sin \left(x - [x]\right) dx $$ where $[x]$ is the greatest integer function.

If $$ y = A t^2 + \frac{B}{t} \quad (A, B \text{ constants}) $$ is a general solution of the differential equation $$ f(t) y'' + g(t) y' + h(t) y = 0, $$ then find the relation between $ g(t), f(t), h(t) $.

Evaluate the integral: $$ \int_0^2 x^2 (2 - x)^5 \, dx $$

If $ f(x) = \max \{ x^3 - 4, x^4 - 4 \} $ and $ g(x) = \min \{ x^2, x^3 \} $, evaluate: $$ \int_{-1}^1 (f(x) - g(x)) \, dx $$

Evaluate: $$ \int_0^1 x \sin^{-1} x \, dx $$

If Lagrange's mean value theorem is applied to the function $$ f(x) = e^x $$ defined on the interval $ [1, 2] $ and the value of $ c \in (1, 2) $ is $ k $, then find $ e^{k-1} $.

If $$ \int \frac{dx}{1 - \sin^4 x} = A \tan x + B \tan^{-1}(\sqrt{2} \tan x) + C, $$ then find $ A^2 - B^2 $.

If $$ \int \frac{a \cos x + 3 \sin x}{5 \cos x + 2 \sin x} dx = \frac{26}{29} x - \frac{k}{29} \log |5 \cos x + 2 \sin x| + c, $$ then find $ |a + k| $.

If the function $$ f(x) = \sin x - \cos^2 x $$ is defined on the interval $ [-\pi, \pi] $, then $ f $ is strictly increasing in the interval: