$A$ tangent is drawn to the ellipse $\frac{x^2}{27} + y^2 = 1$ at the point $(3\sqrt{3} \cos \theta, \sin \theta)$,where $\theta \in (0, \pi/2)$. The value of $\theta$ for which the sum of the intercepts on the axes made by this tangent is minimum,is:

Two slabs $A$ and $B$ of equal surface area are placed one over the other such that their surfaces are completely in contact. The thickness of slab $A$ is twice that of $B$. The coefficient of thermal conductivity of slab $A$ is twice that of $B$. The first surface of slab $A$ is maintained at $100^{\circ} C$,while the second surface of slab $B$ is maintained at $25^{\circ} C$. The temperature at the contact of their surfaces is (in $^{\circ} C$)

In the $NO_3^-$ ion,what are the number of bond pairs and lone pairs of electrons on the $N$ atom,respectively?

The fundamental frequency of a sonometer wire is $n$. If the length and diameter of the wire are doubled while keeping the tension constant,what is the new fundamental frequency?

$A$ particle of mass $m$ moves along line $PC$ with velocity $v$ as shown. What is the angular momentum of the particle about $O$?

For the reactions:
$2NO + O_2 \rightleftharpoons 2NO_2 ; K_1$
$4NO + 2Cl_2 \rightleftharpoons 4NOCl ; K_2$
$NO_2 + 1/2 Cl_2 \rightleftharpoons NOCl + 1/2 O_2 ; K_3$
where $K_1, K_2, K_3$ are equilibrium constants,then $K_3^2$ is equal to: