The number of ordered pairs $(x, y)$ of real numbers that satisfy the simultaneous equations $x+y^2=x^2+y=12$ is

The integer $'k'$, for which the inequality $x^{2}-2(3 k-1) x+8 k^{2}-7>0$ is valid for every $x$ in $R ,$ is

The solution set of the equation $pq{x^2} - {(p + q)^2}x + {(p + q)^2} = 0$ is

Let $a$ , $b$ , $c$ are roots of equation $x^3 + 8x + 1 = 0$ ,then the value of 

 $\frac{{bc}}{{(8b + 1)(8c + 1)}} + \frac{{ac}}{{(8a + 1)(8c + 1)}} + \frac{{ab}}{{(8a + 1)(8b + 1)}}$ is equal to

The number of distinct real roots of $x^4-4 x^3+12 x^2+x-1=0$ is