Two circular coils can be arranged in any of the three situations shown in the figure. Their mutual inductance will be:

Two ideal slits $S_1$ and $S_2$ are at a distance $d$ apart and are illuminated by light of wavelength $\lambda$ passing through an ideal source slit $S$ placed on the line through $S_2$ as shown. The distance between the plane of the slits and the source slit is $D$. $A$ screen is held at a distance $D$ from the plane of the slits. The minimum value of $d$ for which there is darkness at $O$ is

An uncharged capacitor is connected to a battery. On charging the capacitor,

The shortest distance between the lines $r=3i+5j+7k+\lambda(i+2j+k)$ and $r=-i-j-k+\mu(7i-6j+k)$ is

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x^2}-\sqrt{1-x+x^2}}{3^x-1}$ is equal to

Identify the correct gland on the basis of the following characteristics:
$(I)$ It is located on the dorsal side of the forebrain.
$(II)$ It plays a very important role in the regulation of the $24$-hour rhythm of our body.
$(III)$ It helps in maintaining the body temperature.
$(IV)$ Its hormone influences metabolism as well as defense capability.