The solution of the equation $x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}$ is:

The algebraic equation of degree $4$ whose roots are the translates of the roots of the equation $x^4+5x^3+6x^2+7x+9=0$ by $-1$ is

If $3$ is a root of $x^2+kx-24=0$,then it is also a root of which of the following equations?

The number of real values of $a$ satisfying the equation $a^2 - 2a\sin x + 1 = 0$ is

Solve the equation $x^{2}+x+\frac{1}{\sqrt{2}}=0$.

Let $p, q$ and $r$ be real numbers $(p \ne q, r \ne 0)$ such that the roots of the equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$ are equal in magnitude but opposite in sign. Then the sum of squares of these roots is equal to: