We match each statement in Column I with the relevant quantum numbers from Column II.

Column II:

• $(P)$ Principal quantum number $n$
• $(Q)$ Azimuthal quantum number $l$
• $(R)$ Magnetic quantum number $m_l$
• $(S)$ Spin quantum number $m_s$

1. (A) Orbital angular momentum of the electron in a hydrogen-like atomic orbital

The magnitude of orbital angular momentum is $L = \sqrt{l(l+1)}\,\frac{h}{2\pi}$ So it depends only on the azimuthal quantum number $l$.

However, among the given options, all choices include both $q,r$ or something else; in standard matching for an orbital, the angular momentum vector specification is associated with:

• $l$ determining magnitude
• $m_l$ determining its orientation

So the intended match is: $(A) \to (Q,R)$

2. (B) A hydrogen-like one-electron wave function obeying Pauli's principle

A complete one-electron state is specified by four quantum numbers: $n,\; l,\; m_l,\; m_s$ Pauli's principle applies to the complete set of these quantum numbers.

Hence, $(B) \to (P,Q,R,S)$

3. (C) Shape, size and orientation of hydrogen-like atomic orbitals

For an atomic orbital:

• size depends mainly on $n$
• shape depends on $l$
• orientation depends on $m_l$

Therefore, $(C) \to (P,Q,R)$

4. (D) Probability density of electron at the nucleus in hydrogen-like atom

Probability density at nucleus depends on whether the orbital has nonzero value at $r=0$. For hydrogen-like orbitals, only $s$-orbitals have nonzero electron density at nucleus, so this depends on:

• $n$
• $l$

It does not depend on $m_l$ or spin $m_s$.

Thus, $(D) \to (P,Q)$

5. Compare with options

We obtained:

• $(A) \to q,r$
• $(B) \to p,q,r,s$
• $(C) \to p,q,r$
• $(D) \to p,q$

This matches Option D.

Final Answer

$\boxed{\text{D}}$