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Let's solve this step-by-step by matching the physical quantities to their corresponding dimensions from the options provided.

### List-I (Physical Quantities) and their dimensions:
1. **Planck's constant (P):**
   - Dimension: \([ML^2T^{-1}]\)
   - Planck's constant has units of action, which is energy (ML²T⁻²) multiplied by time (T), giving \( [ML^2T^{-1}] \).
   
2. **Surface tension (Q):**
   - Dimension: \([ML^0T^{-2}]\)
   - Surface tension is force per unit length. Force is \( [MLT^{-2}] \), and dividing by length (\(L\)) gives \( [ML^0T^{-2}] \).

3. **Coefficient of viscosity (R):**
   - Dimension: \([ML^{-1}T^{-1}]\)
   - The coefficient of viscosity has units of \( \frac{\text{Force} \times \text{time}}{\text{area}} \), giving it the dimension \( [ML^{-1}T^{-1}] \).

4. **Modulus of elasticity (S):**
   - Dimension: \([ML^{-1}T^{-2}]\)
   - Modulus of elasticity is stress, which is force per unit area. The dimension of force is \( [MLT^{-2}] \), and dividing by area (\(L^2\)) gives \( [ML^{-1}T^{-2}] \).

### Matching List-I to List-II:
- \( P \rightarrow 4 \) (\([ML^2T^{-1}]\))
- \( Q \rightarrow 5 \) (\([ML^0T^{-2}]\))
- \( R \rightarrow 1 \) (\([ML^{-1}T^{-1}]\))
- \( S \rightarrow 2 \) (\([ML^{-1}T^{-2}]\))

### Correct Option:
Looking at the choices:
- **Option D**: \( P \rightarrow 4, Q \rightarrow 5, R \rightarrow 1, S \rightarrow 2 \)

The correct answer is **Option D**.

Would you like more details on any of these physical quantities or how the dimensions are derived?

Here are 5 related questions to deepen your understanding:
1. What is the significance of Planck's constant in quantum mechanics?
2. How is surface tension measured, and what role does temperature play?
3. Can you explain how the coefficient of viscosity affects fluid flow?
4. How does the modulus of elasticity relate to material strength?
5. What are the units of energy and how do they relate to force and time?

**Tip:** Understanding dimensional analysis is key to checking the consistency of physical equations and simplifying unit conversions.

The List-I contains some physical quantity and List-II their dimension. Match the list correctly.

Solve the problem of matching physical quantities such as Planck's constant, surface tension, viscosity, and modulus of elasticity with their correct dimensional formulas. This question uses the principle of dimensional analysis to verify the correctness of physical quantities. Ideal for Grades 11-12 students.

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