The equation \(16x^4 + 16x^3 – 4x – 1 = 0\) has a multiple root. If \(\alpha, \beta, \gamma, \delta\) are the roots of this equation, then \(\frac{1}{\alpha^4} + \frac{1}{\beta^4} + \frac{1}{\gamma^4} + \frac{1}{\delta^4} =\)

If \(\alpha, \beta, \gamma\) are the roots of the equation \(4x^3 – 3x^2 + 2x – 1 = 0\), then \(\alpha^3 + \beta^3 + \gamma^3 =\)

\(\alpha, \beta\) are the real roots of the equation \(x^2 + ax + b = 0\). If \(\alpha + \beta = \frac{1}{2}\) and \(\alpha^3 + \beta^3 = \frac{37}{8}\), then \(a – \frac{1}{b} =\)

If \(f(x)\) is a quadratic function such that \(f(x)f(\frac{1}{x}) = f(x) + f(\frac{1}{x})\), then
\(\sqrt{f(\frac{2}{3}) + f(\frac{3}{2})} =\)

With respect to the roots of the equation \(3x^3 + bx^2 + bx + 3 = 0\), match the items of List-I with those of List-II.

The roots of the equation \(x^3 – 3x^2 + 3x + 7 = 0\) are \(\alpha, \beta, \gamma\) and \(\omega, \omega^2\) are complex cube roots of unity. If the terms containing \(x^2\) and \(x\) are missing in the transformed equation when each one of these roots is decreased by \(h\), then \(\frac{\alpha-h}{\beta-h} + \frac{\beta-h}{\gamma-h} + \frac{\gamma-h}{\alpha-h} =\)

If \(\frac{2x-1}{2x^2-x-6} = ax + b + \frac{A}{px-2} + \frac{B}{2x+q}\), then \(51apB =\)

\(\alpha, \beta, \gamma\) are the roots of the equation \(8x^3 – 42x^2 + 63x – 27 = 0\). If \(\beta < \gamma < \alpha\) and \(\beta, \gamma, \alpha\) are in geometric progression, then the extreme value of the expression \(\gamma x^2 + 4\beta x + \alpha\) is

\(\alpha, \beta, \gamma\) are the roots of the equation \(x^3 + 3x^2 – 10x – 24 = 0\).
If \(\alpha > \beta > \gamma\) and \(\alpha^3 + 3\beta^2 – 10\gamma – 24 = 11k\), then \(k =\)

If the expression \(7 + 6x – 3x^2\) attains its extreme value \(\beta\) at \(x = \alpha\), then the sum of the squares of the roots of the equation \(x^2 + \alpha x – \beta = 0\) is