List of top Mathematics Questions

Let \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \), \( \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k} \), and \( \vec{c} = \hat{i} - 2\hat{j} + 3\hat{k} \) be three vectors. If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b} \) and \( \vec{r} . \vec{c} = 18 \), then the magnitude of the orthogonal projection of \( 4\hat{i} + 3\hat{j} - \hat{k} \) on \( \vec{r} \) is:
The equation \[ \cos^{-1}(1 - x) - 2 \cos^{-1} x = \frac{\pi}{2} \] has:
In \( \triangle ABC \), if A, B, C are in arithmetic progression, then \[ \sqrt{a^2 - ac + c^2} . \cos\left(\frac{A - C}{2}\right) =\ ? \]
If \( \sinh^{-1}(2) + \sinh^{-1}(3) = \alpha \), then \( \sinh\alpha = \) ?
If in \( \triangle ABC \), \( B = 45^\circ \), \( a = 2(\sqrt{3} + 1) \) and area of \( \triangle ABC \) is \( 6 + 2\sqrt{3} \) sq. units, then the side \( b = \ ? \)
If \[ \frac{x^2}{(x^2 + 2)(x^4 - 1)} = \frac{A}{x^2 - 1} + \frac{B}{x^2 + 1} + \frac{C}{x^2 + 2}, \text{ then } A + B - C =\ ? \]
If \( x \) is a positive real number and the first negative term in the expansion of \[ (1 + x)^{27/5} \text{ is } t_k, \text{ then } k =\ ? \]
An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is
A string of letters is to be formed by using 4 letters from all the letters of the word “MATHEMATICS”. The number of ways this can be done such that two letters are of same kind and the other two are of different kind is
\[ \sum_{r=1}^{15} r^2 \left( \frac{{}^{15}C_r}{{}^{15}C_{r-1}} \right) =\ ? \]
Evaluate the following expression: \[ \frac{1}{81^n} - \binom{2n}{1} . \frac{10}{81^n} + \binom{2n}{2} . \frac{10^2}{81^n} - .s + \frac{10^{2n}}{81^n} = ? \]
If \( \alpha, \beta, \gamma \) are the roots of the equation \[ x^3 + px^2 + qx + r = 0, \] then \[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) =\ ? \]
If \( \alpha, \beta \) are the roots of \( x^2 - 5x - 68 = 0 \) and \( \gamma, \delta \) are the roots of \( x^2 - 5\alpha x - 6\beta = 0 \), then \( \alpha + \beta + \gamma + \delta = \) ?
The equation \[ x^{\frac{3}{4}(\log_{x} x)^2 + \log_{x} x^{-\frac{5}{4}}} = \sqrt{2} \] has
If \( \omega_1 \) and \( \omega_2 \) are two non-zero complex numbers and \( a, b \) are non-zero real numbers such that \[ |a\omega_1 + b\omega_2| = |a\omega_1 - b\omega_2|, \] then \( \dfrac{\omega_1}{\omega_2} \) is:
If \( \alpha \) is the common root of the quadratic equations \( x^2 - 5x + 4a = 0 \) and \( x^2 - 2ax - 8 = 0 \), where \( a \in \mathbb{R} \), then the value of \( \alpha^4 - \alpha^3 + 68 \) is:
Let \( g(x) = 1 + x - \lfloor x \rfloor \) and \[ f(x) = \begin{cases} -1, & x<0\\ 0, & x = 0 \\ 1, & x>0 \end{cases} \] where \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). Then for all \( x \), \( f(g(x)) = \)
If the system of equations \( 2x + py + 6z = 8 \), \( x + 2y + qz = 5 \) and \( x + y + 3z = 4 \) has infinitely many solutions, then \( p = \)?
If \( f : \mathbb{R} \to A \), defined by \( f(x) = \cos x + \sqrt{3}\sin x - 1 \), is an onto function, then \( A = \)
The remainder obtained when \( (2m + 1)^{2n} \), \( m, n \in \mathbb{N} \) is divided by 8 is
A value of \( \theta \) lying between \( 0 \) and \( \dfrac{\pi}{2} \) and satisfying \[ \begin{vmatrix} 1 + \sin^2 \theta & \cos^2 \theta & 4\sin 4\theta \\ \sin^2 \theta & 1 + \cos^2 \theta & 4\sin 4\theta \\ \sin^2 \theta & \cos^2 \theta & 1 + 4\sin 4\theta \end{vmatrix} = 0 \] is:
The general solution of the differential equation \((1 + \sin^2 x) \, \frac{dy}{dx} + \sin 2x = 0\) is?
Evaluate \[ \int \frac{1}{x^4 + 1} \, dx = ? \]
Evaluate \[ \int_0^\pi \left( \sin^3 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^3 x \right) dx = ? \]
Evaluate \[ \lim_{n \to \infty} \frac{1}{2n} \left( \sin \frac{\pi}{2n} + \sin \frac{\pi}{n} + \sin \frac{2\pi}{2n} + \dots \right) = ? \]