If x = 0.001, prove that \(\frac { { \left( 1-2x \right) }^{ \frac { 2 }{ 3 } }{ \left( 4+5x \right) }^{ \frac { 3 }{ 2 } } }{ \sqrt { 1-x } } \) = 8.01 up to two places of decimals 

If A and G be respectively the A. M and G. M between two positive numbers, find the numbers

If \(\alpha ,\beta \)are the roots of the equation x²-px + q = 0, then prove that \(\log { (1+px+q{ x }^{ 2 }) } =(\alpha +\beta )x=\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } { x }^{ 2 }+\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 3 } { x }^{ 3 }-....\infty \)

Find the value of \((a^{2}+\sqrt{a^{2}-1})^{4}+(a^{2}-\sqrt{a^{2}-1})^{4}\)

If sum of the n terms of a G.P be S, their product P and the sum of their reciprocals R, then prove that \(P^{2}=(\frac{S}{R})^{n}\)