If α, β, and γ are the roots of the polynomial equation ax³+ bx²+ cx + d = 0, find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

If p and q are the roots of the equation 1x²+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must  be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).

Solve the equation 9x³- 36x²+ 44x -16 = 0 if the roots form an arithmetic progression.

Solve the equation 3x³-26x²+52x - 24 = 0 if its roots form a geometric progression.

Find the number of real solutions of sin (e^x) -5^x + 5^-x

Find the number of positive integral solutions of (pairs of positive integers satisfying) x² - y² = 353702.

Solve: 2x+2^x-1+2^x-2 = 7x+7^x-1+7^x-2

Solve: (x-1)⁴+(x-5)⁴ = 82

Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)