Statement-1: Temperature of a system can be increased without supplying heat.
Statement-2: If the initial volume of system is equal to final volume in a process then work done is necessarily zero.
Statement-3: If an adiabatic curve for a gas intersect with an isothermal curve for the same gas then at the point of intersection, the ratio of slope of the adiabatic and the isothermal curve is `gamma`.

Options

A. FTF
B. TFT
C. TFF
D. TTT

Answer

To analyze the three statements provided in the question, we will evaluate each statement one by one based on the principles of thermodynamics.

### Step 1: Evaluate Statement 1
**Statement 1:** "Temperature of a system can be increased without supplying heat."

- In thermodynamics, an adiabatic process is one in which no heat is exchanged with the surroundings (Q = 0).
- According to the first law of thermodynamics:
\[ Q = \Delta U + W \]
If \( Q = 0 \), then:
\[ 0 = \Delta U + W \implies \Delta U = -W \]
- Here, \( \Delta U \) is the change in internal energy, and \( W \) is the work done on the system. If work is done on the system (compression), then \( W \) is negative, which means \( \Delta U \) is positive, leading to an increase in temperature.
- Therefore, it is possible to increase the temperature of a system without supplying heat by doing work on it.

**Conclusion:** Statement 1 is **True**.

### Step 2: Evaluate Statement 2
**Statement 2:** "If the initial volume of a system is equal to final volume in a process then work done is necessarily zero."

- Work done in a thermodynamic process is given by the area under the pressure-volume (P-V) curve.
- If the initial and final volumes are the same, it does not necessarily mean that the work done is zero. For example, in a cyclic process like the Carnot cycle, the system may return to the same volume but still perform work during the intermediate steps.
- Therefore, even if the initial and final volumes are the same, work can be done in the process.

**Conclusion:** Statement 2 is **False**.

### Step 3: Evaluate Statement 3
**Statement 3:** "If an adiabatic curve for a gas intersects with an isothermal curve for the same gas then at the point of intersection, the ratio of slope of the adiabatic and the isothermal curve is gamma."

- For an adiabatic process, the equation is given by:
\[ PV^\gamma = \text{constant} \]
Differentiating gives the slope:
\[ \frac{dP}{dV} = -\frac{\gamma P}{V} \]
- For an isothermal process, the equation is given by:
\[ PV = \text{constant} \]
Differentiating gives the slope:
\[ \frac{dP}{dV} = -\frac{P}{V} \]
- To find the ratio of the slopes at the point of intersection:
\[ \text{Ratio} = \frac{\text{slope of adiabatic}}{\text{slope of isothermal}} = \frac{-\frac{\gamma P}{V}}{-\frac{P}{V}} = \gamma \]
- This confirms that the ratio of the slopes at the intersection point is indeed gamma.

**Conclusion:** Statement 3 is **True**.

### Final Summary:
- Statement 1: **True**
- Statement 2: **False**
- Statement 3: **True**

### Answer:
The correct answer is that Statement 1 and Statement 3 are true, while Statement 2 is false.

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