List of top Mathematics Questions

For \(\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), if \(2\cos \theta + \sin \theta = 1\), and \(7\cos \theta + 6 \sin \theta = k\), then the possible values of \(k\) are:

Evaluate \[ \sum_{k=0}^{12} \sin\left( (k+1) \frac{\pi}{6} + \frac{\pi}{4} \right) \sin \left( \frac{k \pi}{6} + \frac{\pi}{4} \right) \]

Number of solutions of the equation \[ 2 \sin^2 \theta - 3 \cos^2 \theta = \sin \theta \cos \theta \] in the interval \((- \pi, \pi)\) is:

Evaluate \[ \tan^{-1} \frac{\sqrt{8} - 2\sqrt{15}}{\sqrt{15} + 1} + \tan^{-1} \frac{1}{\sqrt{5}} \]

If \(x\) is so large that terms containing \(x^{-3}\), \(x^{-4}\), \(x^{-5}\), \ldots can be neglected, then the approximate value of \[ \left(\frac{3x - 5}{4x^2 + 3}\right)^{-4/5} \] is:

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:

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

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

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

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

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

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

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

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

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:

If $$ \int \frac{x^2 - x + 2}{x^2 + x + 2} dx = x - \log(f(x)) + \frac{2}{\sqrt{7}} \tan^{-1}(g(x)) + c, $$ then find $$ f(-1) + \sqrt{7} g(-1). $$

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} $.

Evaluate the integral: $$ \int \sec \left(x - \frac{\pi}{3}\right) \sec \left(x + \frac{\pi}{6}\right) dx $$

If $$ \int \frac{x^4 + 1}{x^2 + 1} dx = Ax^3 + Bx^2 + Cx + D \tan^{-1} x + E, $$ then find $ A + B + C + D $.

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| $.