List of top Mathematics Questions

Evaluate \[ \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7} \cos\frac{4\pi}{7}\cos\frac{5\pi}{7}\cos\frac{6\pi}{7}. \]

For \(\theta\in\left(0,\frac{\pi}{2}\right)\), if the complete range of \[ (\cot^2\theta-\cos^2\theta)(\tan^2\theta-\sin^2\theta) \] is \((\alpha,\beta]\), then \(\beta-\alpha=\)

One of the partial fractions of \[ \frac{2x^2+x-3}{(x^2+2)(3x-1)} \] is

If \(\alpha\) is a root of the equation \[ x^4-4x^3+16x-16=0 \] of multiplicity \(m\), then \[ m^2-4m+2= \]

A divisor of \[ 3^{6n}+56n-1,\qquad n\in\mathbb{N} \] is

If the point \(P\) represents a complex number \(z\) in the Argand diagram and \[ \frac{z-i}{z-1} \] is always purely imaginary, then the locus of \(P\) is

The solutions of the equation \[ 2\sqrt{2}\,x^4=(\sqrt{3}-1)+i(\sqrt{3}+1) \] are

If \(\alpha,\beta,\gamma,\delta\) are the roots of the equation \[ x^4+x^3-x-1=0, \] then \[ \alpha^3+\beta^3+\gamma^3+\delta^3= \]

If for all \(x\in\mathbb{R}\), \[ \frac{1}{3} < \frac{x^2-2x+4}{x^2+2x+4} <3, \] then the values of \[ \frac{9\cdot 3^{2x}+6\cdot 3^x+4} {9\cdot 3^{2x}-6\cdot 3^x+4} \] lie between

If the minimum value of \[ f(x)=x^2+2bx+2c^2 \] is greater than the maximum value of \[ g(x)=-x^2-2cx+b^2, \] \(x\) being real, then

If \(z_1=8+4i,\; z_2=6+4i\) and \[ \operatorname{Arg}\left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi}{4}, \] then \(z\) satisfies

If \( \theta_1 \) and \( \theta_2 \) are the values of \( \theta \in (0,\pi) \) for which the system of linear equations \[ x+3y+7z=0, \] \[ -x+4y+7z=0, \] \[ (\sin 3\theta)x+(\cos 2\theta)y+2z=0 \] has a non-trivial solution, then \( |\theta_1-\theta_2| \) is equal to

The set of all values of \( \theta \) satisfying \(0<\theta<\frac{\pi}{2}\) and \[ \begin{vmatrix} 1+\sin^2\theta & \cos^2\theta & 4\sin4\theta
\sin^2\theta & 1+\cos^2\theta & 4\sin4\theta
\sin^2\theta & \cos^2\theta & 1+4\sin4\theta \end{vmatrix} =0 \] is

If \[ \begin{bmatrix} 1 & -\tan\frac{\theta}{2}
[4pt] \tan\frac{\theta}{2} & 1 \end{bmatrix} \begin{bmatrix} 1 & \tan\frac{\theta}{2}
[4pt] -\tan\frac{\theta}{2} & 1 \end{bmatrix}^{-1} = ? \]

The sum \(S_n\) of the first \(n\) terms of the series \[ 1^2+2\times 2^2+3^2+2\times 4^2+5^2+2\times 6^2+\cdots \] is given by

In how many ways can 8 letters be posted in 5 letter boxes.

Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]