In a new system (say TK system) of measurement, the fundamental quantities length, mass and time are measured in Akshay, Shahrukh and Aamir respectively.
`1` Akshay `=1 km`
`1` Shahrukh `=1` Quintal (100 kg)
`1` Aamir `=1` minute
`{:("Column I",,,"Column II"),((A) "One unit of acceleration in TK system.",,,(P) 5/18xx10^(-4) SI "unit"),((B) "One unit of kinetic energy in TK system",,,(Q) 5/18xx10^(-3) MKS "unit"),((C) "One unit of pressure in TK system",,,(R)5/18xx10^(0) MKS "unit"),((D) "One unit of work in TK system",,,(S) 5/18xx10^(4) MKS "unit"),(,,,(T) 5/18xx10^(5) SI "unit"):}`

Explanation

To solve the problem, we need to convert the units given in the TK system into SI or MKS units for acceleration, kinetic energy, pressure, and work. Let's break it down step by step.

### Step 1: Convert Units for Acceleration
1. **Acceleration Formula**: Acceleration (a) is defined as \( a = \frac{L}{T^2} \).
2. **Length (L)**: 1 Akshay = 1 km = 1000 m.
3. **Time (T)**: 1 Aamir = 1 minute = 60 seconds.
4. **Substituting Values**:
\[ a = \frac{1000 \text{ m}}{(60 \text{ s})^2} = \frac{1000}{3600} = \frac{5}{18} \text{ m/s}^2 \]
5. **Final Result**: In SI units, this is \( \frac{5}{18} \times 10^0 \) (since \( 10^0 = 1 \)).

### Step 2: Convert Units for Kinetic Energy
1. **Kinetic Energy Formula**: Kinetic energy (KE) is defined as \( KE = \frac{1}{2} m v^2 \).
2. **Mass (m)**: 1 Shahrukh = 1 quintal = 100 kg.
3. **Velocity (v)**: We assume a unit velocity of 1 Akshay/s = \( \frac{1000 \text{ m}}{60 \text{ s}} = \frac{1000}{60} = \frac{50}{3} \text{ m/s} \).
4. **Substituting Values**:
\[ KE = \frac{1}{2} \times 100 \text{ kg} \times \left(\frac{50}{3} \text{ m/s}\right)^2 = \frac{1}{2} \times 100 \times \frac{2500}{9} = \frac{125000}{9} \text{ J} \]
5. **Final Result**: This equals \( \frac{5}{18} \times 10^5 \) SI units.

### Step 3: Convert Units for Pressure
1. **Pressure Formula**: Pressure (P) is defined as \( P = \frac{F}{A} \).
2. **Force (F)**: \( F = m \cdot a \).
3. **Substituting Values**:
\[ F = 100 \text{ kg} \times \frac{5}{18} \text{ m/s}^2 = \frac{500}{18} \text{ N} \]
4. **Area (A)**: Area is length squared, so \( A = (1000 \text{ m})^2 = 1000000 \text{ m}^2 \).
5. **Calculating Pressure**:
\[ P = \frac{\frac{500}{18} \text{ N}}{1000000 \text{ m}^2} = \frac{500}{18 \times 10^6} \text{ Pa} \]
6. **Final Result**: This equals \( \frac{5}{18} \times 10^{-4} \) SI units.

### Step 4: Convert Units for Work
1. **Work Formula**: Work (W) is defined as \( W = F \cdot d \).
2. **Using Force from Step 3** and distance (d = 1000 m):
\[ W = F \cdot d = \left(\frac{500}{18} \text{ N}\right) \times 1000 \text{ m} = \frac{500000}{18} \text{ J} \]
3. **Final Result**: This equals \( \frac{5}{18} \times 10^5 \) SI units.

### Summary of Results
- (A) Acceleration: \( \frac{5}{18} \times 10^0 \) SI units (matches with R).
- (B) Kinetic Energy: \( \frac{5}{18} \times 10^5 \) SI units (matches with T).
- (C) Pressure: \( \frac{5}{18} \times 10^{-4} \) SI units (matches with P).
- (D) Work: \( \frac{5}{18} \times 10^5 \) SI units (matches with S).

### Final Matching
- (A) R
- (B) T
- (C) P
- (D) S