Let $n \geq 3$ be an integer. For a permutation $\sigma=\left(a_1, a_2, \ldots, a_n\right)$ of $(1,2, \ldots, n)$ we let $f_\sigma(x)=a_n x^{n-1}+a_{n-1} x^{n-2}+\ldots a_2 x+a_1$. Let $S_\sigma$ be the sum of the roots of $f_\sigma(x)=0$ and let $S$ denote the sum over all permutations $\sigma$ of $(1,2, \ldots, n)$ of the numbers $S_\sigma$. Then,

If $p,q,r$ are in $G.P$ and ${\tan ^{ - 1}}p$, ${\tan ^{ - 1}}q,{\tan ^{ - 1}}r$ are in $A.P.$ then $p, q, r$ are satisfies the relation

The minimum value of the sum of real numbers $a^{-5}, a^{-4}, 3 a^{-3}, 1, a^8$ and $a^{10}$ with $a>0$ is

Let $b_i>1$ for $i=1,2, \ldots, 101$. Suppose $\log _e b_1, \log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression ($A.P$.) with the common difference $\log _e 2$. Suppose $a_1, a_2, \ldots, a_{101}$ are in $A.P$. such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\cdots+b_{51}$ and $s=a_1+a_2+\cdots+t_{65}$, then

If $a,\;b,\;c$ are the positive integers, then $(a + b)(b + c)(c + a)$ is

The common difference of an $A.P.$ whose first term is unity and whose second, tenth and thirty fourth terms are in $G.P.$, is