$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620 \, years$ and $810 \, years$. The time (in years) after which one-fourth of the material remains is

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$,while box $B$ contains one mole of helium at temperature $(7/3) T_0$. The boxes are then put into thermal contact with each other,and heat flows between them until the gases reach a common final temperature (Ignore the heat capacity of boxes). Then,the final temperature of gases,$T_f$,in terms of $T_0$ is

If the vectors $p\hat{i} + \hat{j} + \hat{k}$,$\hat{i} + q\hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + r\hat{k}$ $(p \neq 1, q \neq 1, r \neq 1)$ are coplanar,then find the value of $pqr - (p + q + r)$.

$A$ regular polygon of $n$ sides has $170$ diagonals,then $n$ is equal to

Consider the following reaction: $N_{2(g)} + 3H_{2(g)} \longrightarrow 2NH_{3(g)}$. The rate of this reaction in terms of $N_2$ at $T \ K$ is $-\frac{d[N_2]}{dt} = 0.02 \ mol \ L^{-1} \ s^{-1}$. What is the value of $-\frac{d[H_2]}{dt}$ (in units of $mol \ L^{-1} \ s^{-1}$) at the same temperature?

Which one of the following pairs is correctly matched with regard to the codon and the amino acid coded by it?