Let $a$ ,$b$, $c$ , $d$ , $e$ be five numbers satisfying the system of equations

                            $2a + b + c + d + e = 6$
                            $a + 2b + c + d + e = 12$
                            $a + b + 2c + d + e = 24$
                            $a + b + c + 2d + e = 48$
                            $a + b + c + d + 2e = 96$ ,

then $|c|$ is equal to 

The number of pairs of reals $(x, y)$ such that $x=x^2+y^2$ and $y=2 x y$ is

The complete solution of the inequation ${x^2} - 4x < 12\,{\rm{ is}}$

Let $r_1, r_2, r_3$ be roots of equation $x^3 -2x^2 + 4x + 5074 = 0$, then the value of $(r_1 + 2)(r_2 + 2)(r_3 + 2)$ is

If $a, b, c, d$ are four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$, then the minimum value of $\frac{a}{b}+\frac{c}{d}$ is

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 $0 \leq \alpha+\beta+\alpha \beta \leq 8$