The number of solutions of the equation $x^2 + y^2 = a^2 + b^2 + c^2$,where $x, y, a, b, c$ are all prime numbers,is

If $a, b, c$ are real and $x^3 - 3b^2x + 2c^3$ is divisible by $x - a$ and $x - b$,then:

If $\alpha$ is a root of the equation $x^2-x+1=0$,then $\left(\alpha+\frac{1}{\alpha}\right)^3+\left(\alpha^2+\frac{1}{\alpha^2}\right)^3+\left(\alpha^3+\frac{1}{\alpha^3}\right)^3+\left(\alpha^4+\frac{1}{\alpha^4}\right)^3=$

If $x = \sqrt[3]{{\sqrt{2} + 1}} - \sqrt[3]{{\sqrt{2} - 1}}$,then ${x^3} + 3x = $

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos \theta)x - 1 = 0$,where $\theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4 + \beta_\theta^4$,then $16(M + m)$ equals:

For real values of $x$ and $a$,if the expression $\frac{x+a}{2 x^2-3 x+1}$ assumes all real values,then