Let $f: R \rightarrow R$ be such that for all $x \in R$,the terms $(2^{1+x}+2^{1-x})$,$f(x)$,and $(3^x+3^{-x})$ are in $A.P.$. Then the minimum value of $f(x)$ is:

If the sum of $n$ terms of an $A$.$P$. is $3n^2 + 5n$ and its $m$th term is $164$,then the value of $m$ is

If the ratio of the sum of $n$ terms of two arithmetic progressions is $(7n + 1) : (4n + 27)$,then what is the ratio of their $11^{th}$ terms?

If the sum of $n$ terms of an $A.P.$ is $3n^{2} + 5n$ and its $m^{\text{th}}$ term is $164$,find the value of $m$.

Let $a, b, c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5 b^3 c^2 d$ is $3750 \beta$,then the value of $\beta$ is

$150$ workers were engaged to finish a job in a certain number of days. $4$ workers dropped out on the second day,$4$ more workers dropped out on the third day,and so on. It took $8$ more days to finish the work. Find the number of days in which the work was completed.