$A$ 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

$A$ hoop of radius $r$ and mass $m$ rotating with an angular velocity $\omega_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

The length of the subtangent at any point of the curve $x^m y^n = a^{m+n}$ is proportional to -

Two unknown resistances are connected in two gaps of a meter-bridge. The null point is obtained at $40 \ cm$ from left end. A $30 \Omega$ resistance is connected in series with the smaller of the two resistances, the null point shifts by $20 \ cm$ to the right end. The value of smaller resistance in $\Omega$ is

Two oxides of nitrogen,$NO$ and $NO_2$,react together at $253 \ K$ to form a compound of nitrogen $X$. $X$ reacts with water to yield another compound of nitrogen $Y$. The shape of the anion of $Y$ molecule is:

In the reaction $\underset{\text{(Vapour)}}{C_2H_5OH} \xrightarrow[300^{\circ} C]{Cu} X$. The molecular formula of $X$ is