The sum of all real values of 'm' for which the polynomial `f(x)={x^(2)-2mx-4(m^(2)+1)} {x^(2)-4x-2m(m^(2)+1)}` has exactly three distinct real linear factors is

Consider f(x)=(x^(2)-2mx4(m^(2)+1))(x^(2)-4x-2m(m^(2)+1)), For m=k,f(x)=0 has three differentreal roots,then k is :

Find the set of real values of 'm' for which the equation ((x)/(1+x^(2)))^(2)-(m-3)((x)/(1+x^(2)))+m=0 has real roots.

The number of positive integral values of m , m le 16 for which the equation (x^(2) +x+1) ^(2) - (m-3)(x^(2) +x+1) +m=0, has 4 distinct real root is:

Absolute sum of integral values of m for which the equation (m^(2)+2m+1)x^(2)-2(m+1)x+(m+1)^(4)-9=0 has rational roots is

The number of integral values of m, for which the root of x^(2)-2mx+m^(2)-1=0 will lie between -2 and 4

The number of integral values of m for which the equation (1+m^(2)) x^(2) - 2(1+3m)x+(1+8m) = 0 , has no real roots is

The one of possible real value of m for which the circles, x^(2)+y^(2)+4x+2(m^(2)+m)y+6=0 and x^(2)+y^(2)+(2y+1)(m^(2)+m)=0 intersect orthogonally is

The one of possible real value of m for which the circles, x^(2)+y^(2)+4x+2(m^(2)+m)y+6=0 and x^(2)+y^(2)+(2y+1)(m^(2)+m)=0 intersect orthogonally is

If all the real values of m for which the function f(x)=(x^(3))/(3)-(m-3)(x^(2))/(2)+mx-2013 is strictly increasing in x in[0,oo) is [0, k] ,then the value of k is,

The least value of the quadratic polynomial, f(x) = (2p^(2) + 1) x^(2) + 2 (4p^(2) - 1) x + 4(2p^(2)+1) for real values of p and x is