$A$ cylindrical conductor of radius $R$ carries a current $i$. The value of the magnetic field at a point which is $R/4$ distance inside from the surface is $10 \ T$. Find the value of the magnetic field at a point which is $4R$ distance outside from the surface.

$A$ circular disc of radius $R$ and thickness $\frac{R}{6}$ has a moment of inertia $I$ about an axis passing through its center and perpendicular to its plane. It is melted and recast into a solid sphere. The moment of inertia of this sphere about its diameter is:

There are three lumps of a given radioactive substance. Their activity is in the ratio of $1 : 2 : 3$ now. What will be the ratio of their activities at any further date?

Which of the following statements are true regarding Bohr's model of the hydrogen atom?
$(I)$ Orbiting speed of the electron decreases as it shifts to discrete orbits away from the nucleus.
$(II)$ Radii of allowed orbits of the electron are proportional to the principal quantum number.
$(III)$ Frequency with which electrons orbit around the nucleus in discrete orbits is inversely proportional to the cube of the principal quantum number.
$(IV)$ Binding force with which the electron is bound to the nucleus increases as it shifts to outer orbits.
Select the correct answer using the codes given below.

Bulk of carbon dioxide $(CO_2)$ released from body tissues into the blood is present as

At $90 \, ^\circ C$,the concentration of $[H_3O^+]$ in pure water is $10^{-6} \, M$. What is the value of $K_w$ at this temperature?