If $a, b, c$ are real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1$,then $(3a+5b-8c)^2+(-8a+3b+5c)^2+(5a-8b+3c)^2$ is equal to

The sum of two roots of the equation $x^4-x^3-16x^2+4x+48=0$ is zero. If $\alpha, \beta, \gamma, \delta$ are the roots of this equation,then $\alpha^4+\beta^4+\gamma^4+\delta^4=$

The number of triples $(x, y, z)$ of real numbers satisfying the equation $x^4+y^4+z^4+1=4xyz$ is

The polynomial equation of degree $5$ whose roots are the translates of the roots of $x^5-2x^4+3x^3-4x^2+5x-6=0$ by $-2$ is:

Let $S = \{x \in [-6, 3] \setminus \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$ and $T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$. Then the number of elements in $S \cap T$ is $....$

If the lengths of two sides of a triangle are the roots of the equation $x^2-2 \sqrt{3} x+2=0$ and the angle between these sides is $\frac{\pi}{3}$,then the perimeter of the triangle is