If the roots of the equation $x^3 - ax^2 + bx - c = 0$ are in $GP$,then $\frac{b^3}{a^3}$ is equal to:

Let $n (> 1)$ be a positive integer. The largest integer $m$ such that $(n^m + 1)$ divides $(1 + n + n^2 + \dots + n^{127})$ is:

If $a, b, c$ are in $G.P.$,then

The interior angles of an $n$-sided convex polygon are in $G.P.$ The smallest angle is $1^\circ$ and the common ratio is $2$. Then the number of possible values of $n$ is:

If $x, 2x + 2, 3x + 3$ are in $G.P.$,then the fourth term is

For the functions $f(\theta) = \alpha \tan^2 \theta + \beta \cot^2 \theta$ and $g(\theta) = \alpha \sin^2 \theta + \beta \cos^2 \theta$, where $\alpha > \beta > 0$, let $\min_{0 < \theta < \pi/2} f(\theta) = \max_{0 < \theta < \pi} g(\theta)$. If the first term of a $G$.$P$. is $(\frac{\alpha}{2\beta})$, its common ratio is $(\frac{2\beta}{\alpha})$ and the sum of its first $10$ terms is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $m+n$ is equal to . . . . . . .