Three of the quantities defined below has the same dimensional formula. Identify the
i) `sqrt("Energy/mass")`
ii) `sqrt("Pressure/density")`
iii) `sqrt("force/linear density")`
iv) `sqrt("Angular frequency/radius")`

Options

A. ii only
B. ii and iii only
C. iii only
D. i, ii and iii only

Answer

To solve the problem, we need to determine the dimensional formulas for each of the given quantities and identify which three have the same dimensional formula.

### Step-by-Step Solution:

1. **Calculate the dimensional formula for `sqrt("Energy/mass")`:**
- The dimensional formula for energy (E) is given by:
\[ [E] = [M][L^2][T^{-2}] \]
- The dimensional formula for mass (m) is:
\[ [m] = [M] \]
- Therefore, the dimensional formula for `Energy/mass` is:
\[ \frac{[E]}{[m]} = \frac{[M][L^2][T^{-2}]}{[M]} = [L^2][T^{-2}] \]
- Taking the square root:
\[ \sqrt{\frac{[E]}{[m]}} = \sqrt{[L^2][T^{-2}]} = [L][T^{-1}] \]

2. **Calculate the dimensional formula for `sqrt("Pressure/density")`:**
- The dimensional formula for pressure (P) is given by:
\[ [P] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^2]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]
- The dimensional formula for density (ρ) is:
\[ [\rho] = \frac{[M]}{[L^3]} = [M][L^{-3}] \]
- Therefore, the dimensional formula for `Pressure/density` is:
\[ \frac{[P]}{[\rho]} = \frac{[M][L^{-1}][T^{-2}]}{[M][L^{-3}]} = [L^2][T^{-2}] \]
- Taking the square root:
\[ \sqrt{\frac{[P]}{[\rho]}} = \sqrt{[L^2][T^{-2}]} = [L][T^{-1}] \]

3. **Calculate the dimensional formula for `sqrt("Force/linear density")`:**
- The dimensional formula for force (F) is:
\[ [F] = [M][L][T^{-2}] \]
- The dimensional formula for linear density (λ) is:
\[ [\lambda] = \frac{[M]}{[L]} = [M][L^{-1}] \]
- Therefore, the dimensional formula for `Force/linear density` is:
\[ \frac{[F]}{[\lambda]} = \frac{[M][L][T^{-2}]}{[M][L^{-1}]} = [L^2][T^{-2}] \]
- Taking the square root:
\[ \sqrt{\frac{[F]}{[\lambda]}} = \sqrt{[L^2][T^{-2}]} = [L][T^{-1}] \]

4. **Calculate the dimensional formula for `sqrt("Angular frequency/radius")`:**
- The dimensional formula for angular frequency (ω) is:
\[ [\omega] = [T^{-1}] \]
- The dimensional formula for radius (r) is:
\[ [r] = [L] \]
- Therefore, the dimensional formula for `Angular frequency/radius` is:
\[ \frac{[\omega]}{[r]} = \frac{[T^{-1}]}{[L]} = [L^{-1}][T^{-1}] \]
- Taking the square root:
\[ \sqrt{\frac{[\omega]}{[r]}} = \sqrt{[L^{-1}][T^{-1}]} = [L^{-1/2}][T^{-1/2}] \]

### Conclusion:
The quantities `sqrt("Energy/mass")`, `sqrt("Pressure/density")`, and `sqrt("Force/linear density")` all have the same dimensional formula of \([L][T^{-1}]\). The fourth quantity, `sqrt("Angular frequency/radius")`, has a different dimensional formula.

### Final Answer:
The three quantities that have the same dimensional formula are:
- i) `sqrt("Energy/mass")`
- ii) `sqrt("Pressure/density")`
- iii) `sqrt("Force/linear density")`