1. Statement A

For small temperature difference between a body and surroundings, Newton’s law of cooling applies:

$\frac{dQ}{dt} \propto (T-T_s)$

where $T-T_s$ is the excess temperature.

So, if the temperature difference is doubled, the rate of loss of heat also doubles.

Hence, A is true.

2. Statement B

Thermal radiation emitted per unit time is given by Stefan’s law:

$P = \sigma A T^4$

Since the surface areas are equal,

$\frac{P_P}{P_Q} = \left(\frac{T_P}{T_Q}\right)^4$

Convert temperatures to kelvin:

$T_P = 10^\circ C = 283\,K, \qquad T_Q = 20^\circ C = 293\,K$

Now,

$\frac{P_Q}{P_P} = \left(\frac{293}{283}\right)^4$

$\frac{293}{283} \approx 1.0353$

$\left(1.0353\right)^4 \approx 1.15$

Thus, emitted radiations are approximately in the ratio

$P_P : P_Q = 1 : 1.15$

Hence, B is true.

3. Statement C

Efficiency of a Carnot engine is

$\eta = 1 - \frac{T_c}{T_h}$

Here,

$T_c = 100\,K, \qquad T_h = 400\,K$

So,

$\eta = 1 - \frac{100}{400} = 1 - \frac14 = \frac34 = 0.75$

Thus,

$\eta = 75\%$

Hence, C is true.

4. Statement D

Again using Newton’s law of cooling for small temperature differences,

$\frac{dQ}{dt} \propto (T-T_s)$

If the temperature difference is quadrupled, the rate of loss of heat should also become four times, not twice.

Hence, D is false.

5. Correct option

True statements are:

• A true
• B true
• C true
• D false

Therefore, the correct choice is:

$\boxed{\text{A: A, B, C only}}$