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

The number of solutions of the equation sin$${(e)^x} = {5^x} + {5^{ - x}}$$ is

If $$a,\,b,\,c,\,d$$ and p are distinct real numbers such that $$$\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left( {ab + bc + cd} \right)p + \left( {{b^2} + {c^2} + {d^2}} \right) \le 0$$$
then $$a,\,b,\,c,\,d$$