Three resistors `R_1, R_2` and `R_3` are to be combined to form an electrical circuit as shown in the figure. It is found that when `R_1 R_2 and R_3` are put respectively in positions A, B and C, the effective resistance of the circuit is 70 `Omega`When `R_2, R_3 and R_1` are put respectively in position A, B and C the effective resistance is 35`Omega` and when `R_3, R_1 and R_2` are respectively put in the position A, B and C, the effective resistance is `42 Omega`
If `R_(1), R_(2)` and` R_(3)`are put in series, the effective resistance will be

Options

A. `210Omega`
B. `100Omega`
C. `105Omega`
D. `11Omega`

Explanation

To solve the problem, we need to find the effective resistance when three resistors \( R_1, R_2, \) and \( R_3 \) are connected in series. We are given three different configurations of these resistors and their corresponding effective resistances. Let's denote the effective resistances in the three configurations as follows:

1. Configuration 1: \( R_1, R_2, R_3 \) in positions A, B, C results in \( R_{eff1} = 70 \, \Omega \)
2. Configuration 2: \( R_2, R_3, R_1 \) in positions A, B, C results in \( R_{eff2} = 35 \, \Omega \)
3. Configuration 3: \( R_3, R_1, R_2 \) in positions A, B, C results in \( R_{eff3} = 42 \, \Omega \)

Let's denote the resistances as follows:
- \( R_1 = x \)
- \( R_2 = y \)
- \( R_3 = z \)

From the configurations, we can set up the following equations based on the rules for resistors in series and parallel:

1. For Configuration 1:
\[ R_{eff1} = R_1 + \frac{1}{\frac{1}{R_2} + \frac{1}{R_3}} = 70 \]
This simplifies to:
\[ 70 = x + \frac{1}{\frac{1}{y} + \frac{1}{z}} \implies 70 = x + \frac{yz}{y + z} \]

2. For Configuration 2:
\[ R_{eff2} = R_2 + \frac{1}{\frac{1}{R_3} + \frac{1}{R_1}} = 35 \]
This simplifies to:
\[ 35 = y + \frac{1}{\frac{1}{z} + \frac{1}{x}} \implies 35 = y + \frac{zx}{z + x} \]

3. For Configuration 3:
\[ R_{eff3} = R_3 + \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}} = 42 \]
This simplifies to:
\[ 42 = z + \frac{1}{\frac{1}{x} + \frac{1}{y}} \implies 42 = z + \frac{xy}{x + y} \]

Now we have three equations:
1. \( 70 = x + \frac{yz}{y + z} \)
2. \( 35 = y + \frac{zx}{z + x} \)
3. \( 42 = z + \frac{xy}{x + y} \)

Next, we will solve these equations simultaneously to find the values of \( x, y, \) and \( z \).

### Step 1: Solve the equations

From the first equation, we can express \( \frac{yz}{y + z} \):
\[ \frac{yz}{y + z} = 70 - x \quad \text{(1)} \]

From the second equation:
\[ \frac{zx}{z + x} = 35 - y \quad \text{(2)} \]

From the third equation:
\[ \frac{xy}{x + y} = 42 - z \quad \text{(3)} \]

### Step 2: Substitute and solve for one variable

We can express \( z \) in terms of \( x \) and \( y \) using equation (3):
\[ z = 42 - \frac{xy}{x + y} \]

### Step 3: Substitute \( z \) back into equations (1) and (2)

Substituting \( z \) into equations (1) and (2) will give us two equations in terms of \( x \) and \( y \). This will allow us to solve for \( x \) and \( y \).

### Step 4: Solve for \( R_1, R_2, R_3 \)

After solving the equations, we will find the values of \( R_1, R_2, \) and \( R_3 \).

### Step 5: Calculate the effective resistance in series

Once we have the values of \( R_1, R_2, \) and \( R_3 \), the effective resistance \( R_{eff} \) when they are connected in series is given by:
\[ R_{eff} = R_1 + R_2 + R_3 \]

### Final Calculation

After substituting the values of \( R_1, R_2, \) and \( R_3 \) into the equation for \( R_{eff} \), we will arrive at the final answer.