Three unequal resistor in parallel are equivalent to a resistance `1` ohm If two of them are in the ratio `1:2` and if no resistance value is fractional the largest of three resistance in ohm is

Options

A. 4
B. 6
C. 8
D. 12

Answer

To solve the problem, we need to find the largest of three unequal resistors in parallel, given that their equivalent resistance is 1 ohm and two of them are in the ratio of 1:2. Let's denote the resistances as \( R_1 \), \( R_2 \), and \( R_3 \).

### Step-by-Step Solution:

1. **Assign Variables Based on the Ratio**:
Since \( R_1 \) and \( R_2 \) are in the ratio of 1:2, we can express them as:
\[ R_1 = x \quad \text{and} \quad R_2 = 2x \]
where \( x \) is a positive integer.

2. **Use the Formula for Equivalent Resistance in Parallel**:
The formula for the equivalent resistance \( R_{eq} \) of resistors in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]
Substituting the values we have:
\[ \frac{1}{1} = \frac{1}{x} + \frac{1}{2x} + \frac{1}{R_3} \]

3. **Combine the Terms**:
To combine the fractions on the right-hand side, we need a common denominator. The common denominator for \( x \) and \( 2x \) is \( 2x \):
\[ 1 = \frac{2}{2x} + \frac{1}{2x} + \frac{1}{R_3} \]
This simplifies to:
\[ 1 = \frac{3}{2x} + \frac{1}{R_3} \]

4. **Rearrange the Equation**:
Rearranging the equation gives:
\[ 1 - \frac{3}{2x} = \frac{1}{R_3} \]
Taking the reciprocal:
\[ R_3 = \frac{1}{1 - \frac{3}{2x}} = \frac{2x}{2x - 3} \]

5. **Ensure No Fractional Values**:
Since all resistance values must be integers and not fractional, the expression \( 2x - 3 \) must divide \( 2x \) evenly. This means \( 2x - 3 \) must be a factor of \( 2x \).

6. **Set Conditions for Integer Values**:
For \( R_3 \) to be an integer, \( 2x - 3 \) must be a divisor of \( 2x \). Therefore, we can set:
\[ 2x - 3 = k \quad \text{(where \( k \) is an integer)} \]
Rearranging gives:
\[ 2x = k + 3 \quad \Rightarrow \quad x = \frac{k + 3}{2} \]
For \( x \) to be an integer, \( k + 3 \) must be even, which implies \( k \) must be odd.

7. **Finding Possible Values**:
Testing integer values for \( k \):
- If \( k = 1 \): \( x = 2 \), then \( R_1 = 2 \), \( R_2 = 4 \), \( R_3 = \frac{4}{1} = 4 \) (not unequal).
- If \( k = 3 \): \( x = 3 \), then \( R_1 = 3 \), \( R_2 = 6 \), \( R_3 = \frac{6}{3} = 2 \) (unequal).
- If \( k = 5 \): \( x = 4 \), then \( R_1 = 4 \), \( R_2 = 8 \), \( R_3 = \frac{8}{5} = 1.6 \) (not integer).

8. **Conclusion**:
The only valid case that satisfies all conditions is when \( x = 3 \):
\[ R_1 = 3 \, \text{ohm}, \quad R_2 = 6 \, \text{ohm}, \quad R_3 = 2 \, \text{ohm} \]
Thus, the largest resistance is:
\[ \text{Largest Resistance} = R_2 = 6 \, \text{ohm} \]

### Final Answer:
The largest of the three resistances is **6 ohms**.