Let $x, y, z$ be positive reals. Which of the following implies $x=y=z$ ?

$I.$ $x^3+y^3+z^3=3 x y z$

$II.$ $x^3+y^2 z+y z^2=3 x y z$

$III.$ $x^3+y^2 z+z^2 x=3 x y z$

$IV.$ $(x+y+z)^3=27 x y z$

Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?

Suppose the quadratic polynomial $p(x)=a x^2+b x+c$ has positive coefficient $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$ then what could be the possible value of $\alpha+\beta+\alpha \beta$ if $0 \leq \alpha+\beta+\alpha \beta \leq 8$

The set of values of $x$ which satisfy $5x + 2 < 3x + 8$ and $\frac{{x + 2}}{{x - 1}} < 4,$ is

The number of real solutions of the equation $e ^{4 x }+4 e ^{3 x }-58 e ^{2 x }+4 e ^{ x }+1=0$ is..........

If the sum of two of the roots of ${x^3} + p{x^2} + qx + r = 0$ is zero, then $pq =$