Which of the following allotropes of carbon is isomorphous with crystalline silicon?

$A$ solution which is $10^{-3} \ M$ each in $Mn^{2+}$,$Fe^{2+}$,$Zn^{2+}$ and $Hg^{2+}$ is treated with $10^{-16} \ M$ sulphide ion. If the $K_{sp}$ of $MnS$,$FeS$,$ZnS$ and $HgS$ are $10^{-15}$,$10^{-23}$,$10^{-20}$ and $10^{-54}$ respectively,which one will precipitate first?

Consider the following anions:
$(I)$ $CF_{3}SO_{3}^{-}$
$(II)$ $C_{6}H_{5}SO_{3}^{-}$
$(III)$ $C_{6}H_{5}O^{-}$
$(IV)$ $CH_{3}COO^{-}$
When attached to $sp^{3}$-hybridized carbon,their leaving group ability in nucleophilic substitution reaction decreases in the order:

Thermite is a mixture of iron oxide and which of the following?

If $n_1, n_2$,and $n_3$ are the fundamental frequencies of three segments into which a string is divided,then the original fundamental frequency $n$ of the string is given by

If $a_1, a_2, \dots, a_n$ are positive real numbers whose product is a fixed number $c$,then the minimum value of $a_1 + a_2 + \dots + a_{n-1} + 2a_n$ is