The $H.M.$ between the roots of the equation $x^2 - 10x + 11 = 0$ is

The geometric and harmonic means of two numbers $x_1$ and $x_2$ are $18$ and $16\frac{8}{13}$ respectively. The value of $|x_1 - x_2|$ is

If the $2^{nd}, 5^{th}, \text{and } 9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$, then the common ratio of this $G.P.$ is:

If ${S_k}$ denotes the sum of the first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively,then ${S_{kn}}/{S_n}$ is independent of $n$ if:

How many terms of the $A.P.$ $1, 4, 7, \ldots$ are needed to give the sum $715$?

If $T_n = (n^2 + 1)n!$ and $S_n = T_1 + T_2 + T_3 + ...... + T_n$. Let $\frac{T_{10}}{S_{10}} = \frac{a}{b}$,where $a$ and $b$ are relatively prime natural numbers,then the value of $(b - a)$ is: