Bohr Model Dependencies: Radius and Angular Momentum Matching

According to Bohr's model for a one-electron hydrogen-like atom with nuclear charge $Z$:

1. Radius of the $n^{\text{th}}$ orbit ($r_n$):
   Centripetal force is provided by the electrostatic attraction:
   $\frac{m v_n^2}{r_n} = \frac{Z e^2}{4\pi\epsilon_0 r_n^2}$
   From Bohr's quantization condition for angular momentum:
   $m v_n r_n = \frac{n h}{2\pi} \implies v_n = \frac{n h}{2\pi m r_n}$
   Substituting $v_n$:
   $r_n = \frac{\epsilon_0 h^2}{\pi m Z e^2} n^2 \implies r_n \propto n^2$
   Thus, (I) corresponds to (T).

2. Angular momentum of the electron in the $n^{\text{th}}$ orbit ($L_n$):
   $L_n = m v_n r_n = \frac{n h}{2\pi} \implies L_n \propto n^1$
   Thus, (II) corresponds to (S).

Checking the given options for Question 15:

• (A) (I), (P) $\rightarrow$ Incorrect ($r_n \propto n^2$, not $n^{-2}$)
• (B) (I), (T) $\rightarrow$ Correct ($r_n \propto n^2$)
• (C) (II), (Q) $\rightarrow$ Incorrect ($L_n \propto n^1$, not $n^{-1}$)
• (D) (II), (R) $\rightarrow$ Incorrect ($L_n \propto n^1$, not $n^0$)

Therefore, the correct option is B.