If $$a,\,b,c$$ are positive real numbers. Then prove that $$${\left( {a + 1} \right)^7}{\left( {b + 1} \right)^7}{\left( {c + 1} \right)^7} > {7^7}\,{a^4}{b^4}{c^4}$$$

If $${x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0$$ where $$a,\,b\, \in \,R$$ then find the values of a for which equation has unequal real roots for all values of $$b$$.

Let $$a,\,b,\,c$$ be real numbers with $$a \ne 0$$ and let $$\alpha ,\,\beta $$ be the roots of the equation $$a{x^2} + bx + c = 0$$. Express the roots of $${a^3}{x^2} + abcx + {c^3} = 0$$ in terms of $$\alpha ,\,\beta \,$$.

If $$\alpha ,\,\beta $$ are the roots of $$a{x^2} + bx + c = 0$$, $$\,\left( {a \ne 0} \right)$$ and $$\alpha + \delta ,\,\,\beta + \delta $$ are the roots of $$A{x^2} + Bx + c = 0,$$ $$\left( {A \ne 0\,} \right)\,$$ for some contant $$\delta $$, then prove that $${{{b^2} - 4ac} \over {{a^2}}} = {{{B^2} - 4Ac} \over {{A^2}}}$$.