If $$\lambda $$ $$ \in $$ R is such that the sum of the cubes of the roots of the equation,
x² + (2 $$-$$ $$\lambda $$) x + (10 $$-$$ $$\lambda $$) = 0 is minimum, then the magnitude of the difference of the roots of this equation is :

If tanA and tanB are the roots of the quadratic equation, 3x² $$-$$ 10x $$-$$ 25 = 0, then the value of 3 sin²(A + B) $$-$$ 10 sin(A + B).cos(A + B) $$-$$ 25 cos²(A + B) is :

The sum of all the real values of x satisfying the equation
2^{(x$$-$$1)(x2 + 5x $$-$$ 50)} = 1 is :

Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x$$-$$ 1 and it leaves remainder 6 when divided by x + 1; then :