The efficiency of carnot's heat engine is 0.5 when the temperature of the source is `T_(1)` and that of sink is `T_(2)` .The efficiency of another carnot's heat engine is also 0.5.the temperature of source and sink of the second engine are respecitvely

Options

A. a. `2T_(1),2T_(2)`
B. b. `2T_(1),(T_(2))/(2)`
C. c. `T_(1)+5,T_(2)-5`
D. d. `T_(1)+10,T_(2)-10`

Explanation

To solve the problem, we need to analyze the efficiency of Carnot's heat engines based on the given temperatures of the source and sink.

### Step-by-Step Solution:

1. **Understanding Carnot's Efficiency Formula**:
The efficiency (η) of a Carnot engine is given by the formula:
\[ \eta = 1 - \frac{T_2}{T_1} \]
where \( T_1 \) is the temperature of the source and \( T_2 \) is the temperature of the sink.

2. **Setting Up the First Engine**:
For the first Carnot engine, we know that:
\[ \eta_1 = 0.5 \]
Substituting this into the efficiency formula:
\[ 0.5 = 1 - \frac{T_2}{T_1} \]
Rearranging gives:
\[ \frac{T_2}{T_1} = 0.5 \quad \Rightarrow \quad T_2 = 0.5 T_1 \]

3. **Setting Up the Second Engine**:
The second Carnot engine also has an efficiency of:
\[ \eta_2 = 0.5 \]
Using the same efficiency formula:
\[ 0.5 = 1 - \frac{T_2'}{T_1'} \]
Rearranging gives:
\[ \frac{T_2'}{T_1'} = 0.5 \quad \Rightarrow \quad T_2' = 0.5 T_1' \]

4. **Analyzing the Options**:
We need to check the options given for the second engine's temperatures \( T_1' \) and \( T_2' \) to see if they satisfy the condition \( T_2' = 0.5 T_1' \).

- **Option 1**: \( T_1' = 2T_1 \) and \( T_2' = 2T_2 \)
\[ T_2' = 2T_2 = 2(0.5T_1) = T_1 \quad \Rightarrow \quad 0.5T_1' = 0.5(2T_1) = T_1 \]
This option is correct.

- **Option 2**: \( T_1' = 2T_1 \) and \( T_2' = \frac{T_2}{2} \)
\[ T_2' = \frac{T_2}{2} = \frac{0.5T_1}{2} = 0.25T_1 \quad \Rightarrow \quad 0.5T_1' = 0.5(2T_1) = T_1 \]
This option is incorrect.

- **Option 3**: \( T_1' = T_1 + 5 \) and \( T_2' = T_2 - 5 \)
\[ T_2' = T_2 - 5 = 0.5T_1 - 5 \quad \Rightarrow \quad 0.5T_1' = 0.5(T_1 + 5) \]
This option does not satisfy the condition.

- **Option 4**: \( T_1' = T_1 + 10 \) and \( T_2' = T_2 - 10 \)
\[ T_2' = T_2 - 10 = 0.5T_1 - 10 \quad \Rightarrow \quad 0.5T_1' = 0.5(T_1 + 10) \]
This option also does not satisfy the condition.

5. **Conclusion**:
The only option that satisfies the condition \( T_2' = 0.5 T_1' \) is Option 1.

### Final Answer:
The temperatures of the source and sink of the second engine are respectively \( 2T_1 \) and \( 2T_2 \).