If $m = -4$ and $n = -2$,then the value of $m^{3} - 3m^{2} + 3m + 3n + 3n^{2} + n^{3}$ is

If the equation $\frac{1}{x} + \frac{1}{x - 1} + \frac{1}{x - 2} = 3x^3$ has $k$ real roots,then $k$ is equal to -

If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0$ and $\gamma, \delta$ are the roots of $x^2 - 12x + b = 0$,and the numbers $\alpha, \beta, \gamma, \delta$ (in that order) form an increasing $G.P.$,then:

The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if

The roots of the equation $x^2 + ax + b = 0$ are $p$ and $q$. Then,the equation whose roots are $p^2q$ and $pq^2$ will be:

If the equation $\frac{x^2 + 5}{2} = x - 2\cos(ax + b)$ has at least one solution,then $(b + a)$ can be equal to