If one real root of the quadratic equation $81x^2 + kx + 256 = 0$ is the cube of the other root,then a value of $k$ is

The difference between two roots of the equation $x^3-13x^2+15x+189=0$ is $2$. Then the roots of the equation are:

In a triangle $PQR$,$\angle R = \frac{\pi}{2}$. If $\tan(\frac{P}{2})$ and $\tan(\frac{Q}{2})$ are the roots of the equation $ax^2 + bx + c = 0$ $(a \neq 0)$,then:

The expression $x^3 - 3x^2 - 9x + c$ can be written in the form $(x - a)^2 (x - b)$ if the value of $c$ is

Let $\alpha, \beta$ be the roots of $x^{2}-x-1=0$ and $S_{n}=\alpha^{n}+\beta^{n}$ for all integers $n \geq 1$. Then,for every integer $n \geq 2$,which of the following is true?

If $\alpha, \beta$ are the roots of $x^2 - x + p = 0$ and $\gamma, \delta$ are the roots of $x^2 - 4x + q = 0$,and if $\alpha, \beta, \gamma, \delta$ are in a geometric progression,then the values of $p$ and $q$ are respectively: