If p, q, r are + ve and are on A.P., the roots of quadratic equation $$p{x^2} + qx + r = 0$$ are all real for

Let $$p,q \in \left\{ {1,2,3,4} \right\}\,$$. The number of equations of the form $$p{x^2} + qx + 1 = 0$$ having real roots is

Let $$\alpha \,,\,\beta $$ be the roots of the equation (x - a) (x - b) = c, $$c \ne 0$$. Then the roots of the equation $$(x - \alpha \,)\,(x - \beta ) + c = 0$$ are

The product of $$n$$ positive numbers is unity. Then their sum is