Two spherical planets $P$ and $Q$ have the same uniform density $\rho$, masses $M_P$ and $M_Q$ and surface areas $A$ and $4A$ respectively. A spherical planet $R$ also has uniform density $\rho$ and its mass is $(M_P+M_Q)$. The escape velocities from the planets $P$, $Q$ and $R$ are $V_P$, $V_Q$ and $V_R$, respectively. Then

Step 1: Relate the radii of planets $P$ and $Q$.

The surface area of a spherical planet is given by $A = 4\pi R^2$. Given that $A_P = A$ and $A_Q = 4A$, we can find the ratio of their radii.

$$\frac{A_Q}{A_P} = \frac{4\pi R_Q^2}{4\pi R_P^2} = 4$$ $$R_Q = 2R_P$$

Step 2: Express escape velocity in terms of radius and density.

The escape velocity from a spherical planet of mass $M$ and radius $R$ is given by the formula:

$$v_{\text{esc}} = \sqrt{\frac{2GM}{R}}$$

Since the planets have uniform density $\rho$, the mass is related to the radius by:

$$M = \frac{4}{3}\pi R^3 \rho$$

Substituting the mass into the escape velocity formula yields:

$$v_{\text{esc}} = \sqrt{\frac{2G}{R} \left(\frac{4}{3}\pi R^3 \rho\right)}$$ $$v_{\text{esc}} = \sqrt{\frac{8}{3}\pi G \rho} \, R$$

For a constant density $\rho$, the escape velocity is directly proportional to the radius:

$$v_{\text{esc}} \propto R$$

Step 3: Find the ratio of escape velocities $V_P$ and $V_Q$.

Using the proportionality relation $V \propto R$:

$$\frac{V_P}{V_Q} = \frac{R_P}{R_Q}$$ $$\frac{V_P}{V_Q} = \frac{R_P}{2R_P} = \frac{1}{2}$$

Step 4: Determine the radius of planet $R$.

The mass of planet $R$ is the sum of the masses of planets $P$ and $Q$:

$$M_R = M_P + M_Q$$

Since $M \propto R^3$ for constant density:

$$R_R^3 = R_P^3 + R_Q^3$$

Substitute $R_Q = 2R_P$ into the equation:

$$R_R^3 = R_P^3 + (2R_P)^3$$ $$R_R^3 = 9R_P^3$$ $$R_R = 9^{1/3} R_P \approx 2.08 R_P$$

Step 5: Compare the escape velocities of the three planets.

Using the proportionality $V \propto R$, we express the escape velocities in terms of $V_P$:

$$V_Q = 2V_P$$ $$V_R = 9^{1/3} V_P \approx 2.08 V_P$$

Comparing these values gives:

$$V_R > V_Q > V_P$$

The ratio of $V_R$ to $V_P$ is:

$$\frac{V_R}{V_P} = 9^{1/3} \approx 2.08 \neq 3$$

Answer: The correct relations are $V_R > V_Q > V_P$ and $\dfrac{V_P}{V_Q} = \dfrac{1}{2}$.