A technician has only two resistance coils. By using them in series or in parallel he is able to obtain the resistance 3,4,12 and 16 ohm. The resistance of two coils are

Options

A. 6,10
B. 4,12
C. 7,9
D. 4,16

Explanation

To find the resistance of the two coils, let's denote the resistances of the two coils as \( R_1 \) and \( R_2 \). We know that the technician can achieve resistances of 3, 4, 12, and 16 ohms by using these coils in series and parallel combinations.

### Step 1: Determine the maximum resistance in series
When two resistors are connected in series, the equivalent resistance \( R_s \) is given by:
\[ R_s = R_1 + R_2 \]
The maximum resistance from the given options is 16 ohms. Therefore, we can write:
\[ R_1 + R_2 = 16 \quad \text{(Equation 1)} \]

### Step 2: Determine the minimum resistance in parallel
When two resistors are connected in parallel, the equivalent resistance \( R_p \) is given by:
\[ \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} \]
This can be rearranged to:
\[ R_p = \frac{R_1 R_2}{R_1 + R_2} \]
The minimum resistance from the given options is 3 ohms. Therefore, we can write:
\[ R_p = 3 \quad \text{(Equation 2)} \]

### Step 3: Relate the two equations
From Equation 1, we have:
\[ R_1 + R_2 = 16 \]
From Equation 2, substituting \( R_1 + R_2 \) into the parallel resistance formula gives:
\[ 3 = \frac{R_1 R_2}{16} \]
Multiplying both sides by 16:
\[ R_1 R_2 = 48 \quad \text{(Equation 3)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( R_1 + R_2 = 16 \)
2. \( R_1 R_2 = 48 \)

We can solve these equations simultaneously. Let’s express \( R_2 \) in terms of \( R_1 \):
\[ R_2 = 16 - R_1 \]
Substituting this into Equation 3:
\[ R_1(16 - R_1) = 48 \]
Expanding this gives:
\[ 16R_1 - R_1^2 = 48 \]
Rearranging it leads to:
\[ R_1^2 - 16R_1 + 48 = 0 \]

### Step 5: Use the quadratic formula
We can use the quadratic formula \( R_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -16, c = 48 \):
\[ R_1 = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 48}}{2 \cdot 1} \]
Calculating the discriminant:
\[ R_1 = \frac{16 \pm \sqrt{256 - 192}}{2} \]
\[ R_1 = \frac{16 \pm \sqrt{64}}{2} \]
\[ R_1 = \frac{16 \pm 8}{2} \]
This gives us two possible values for \( R_1 \):
\[ R_1 = \frac{24}{2} = 12 \quad \text{or} \quad R_1 = \frac{8}{2} = 4 \]

### Step 6: Find \( R_2 \)
If \( R_1 = 12 \), then:
\[ R_2 = 16 - 12 = 4 \]
If \( R_1 = 4 \), then:
\[ R_2 = 16 - 4 = 12 \]
Thus, the two resistances are \( R_1 = 12 \, \Omega \) and \( R_2 = 4 \, \Omega \).

### Conclusion
The resistances of the two coils are:
\[ R_1 = 12 \, \Omega \quad \text{and} \quad R_2 = 4 \, \Omega \]