The number of different possible values for the sum $x+y+z$,where $x, y, z$ are real numbers such that $x^4+4y^4+16z^4+64=32xyz$ is

If the equation $x^3-7x^2+14x-8=0$ is transformed to $y^3+py-\frac{20}{27}=0$ when its roots are diminished by $k$,then $p=$

The sum of the squares of all the roots of the equation $x^2+|2x-3|-4=0$ is:

Let $\phi(x)=\frac{x}{(x^2+1)(x+1)}$. If $a, b$ and $c$ are the roots of the equation $x^3-3x+\lambda=0, (\lambda \neq 0)$,then $\phi(a) \phi(b) \phi(c) =$

The equation $\left(x^4+1\right)=\frac{1}{a}(x+1)^4$ is a reciprocal equation:

Let $a, b \in \mathbb{R}$ be such that the equation $ax^{2}-2bx+15=0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-2bx+21=0$,then $\alpha^{2}+\beta^{2}$ is equal to