The reciprocal of $x+\frac{1}{x}$ is

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $\frac{b^2}{ac} + \frac{bc}{a^2} = $

If $x^{2}+x=5$,then find the value of $(x+3)^{3}+\frac{1}{(x+3)^{3}}$.

Find the combined product of $(a^{4}+b^{4})(a^{2}+b^{2})(a+b)(a-b)$.

The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is

The number of integral values of $m$ for which the quadratic expression $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m)$ is always positive for all $x \in R$ is: