Let p be the sum of all possible determinants of order 2 having 0,1,2 and 3 as their four entries. Let α be the common root of the equations x2+ax+[m+1]=0 x2+bx+[m+4]=0 x2−cx+[m+15]=0 such that α>p and a+b+c=0. If m=limn→∞1n2n∑r=1r√n2+r2, then the value of α+p is ( [.] denotes the greatest integer function)

Let p be the sum of all possible determinants of order 2 having 0,1,2

1.	Let p be the sum of all possible determinants of order 2 having 0,1,2 and 3 as their four entries. Let α be the common root of the equations x2+ax+[m+1]=0 x2+bx+[m+4]=0 x2−cx+[m+15]=0 such that α>p and a+b+c=0. If m=limn→∞1n2n∑r=1r√n2+r2, then the value of α+p is ( [.] denotes the greatest integer function)
Answer» Let $p$ be the sum of all possible determinants of order $2$ having $0, 1, 2$ and $3$ as their four entries. Let $α$ be the common root of the equations $x^{2} + a x + [m + 1] = 0$ $x^{2} + b x + [m + 4] = 0$ $x^{2} - c x + [m + 15] = 0$ such that $α > p$ and $a + b + c = 0$ . If $m = lim n \to \infty \frac{1}{n} 2 n \sum r = 1 \frac{r}{\sqrt{n^{2} + r^{2}}}$ , then the value of $α + p$ is ( $[.]$ denotes the greatest integer function)