To determine the resistance of the galvanometer, we can use the concept of current division in a parallel circuit.

Let's assume the resistance of the galvanometer is G ohms. When the 12 ohm resistor is connected in parallel with the galvanometer, the total resistance in the circuit becomes:

1/R_total = 1/R_galvanometer + 1/12

Now, let's consider the deflection of the galvanometer. Initially, the deflection is 50 divisions, and when the 12 ohm resistor is added, the deflection reduces to 10 divisions.

From the concept of current division, we know that the deflection of the galvanometer is inversely proportional to the current passing through it. Therefore, we can write:

Deflection ∝ 1/Current

Now, let's assume the initial current passing through the galvanometer is I1, and the current passing through the galvanometer after adding the 12 ohm resistor is I2.

We can write the proportionality equation as:

I1/I2 = Deflection2/Deflection1

Substituting the given values, we have:

I1/I2 = 10/50
I1/I2 = 1/5

Now, let's consider the current division in the parallel circuit. The current passing through the galvanometer (I_galvanometer) can be written as:

I_galvanometer = I1 - I2

Substituting the value of I1/I2 from the proportionality equation, we have:

I_galvanometer = 5I2 - I2
I_galvanometer = 4I2

Now, let's consider the voltage division in the parallel circuit. The voltage across the galvanometer (V_galvanometer) can be written as:

V_galvanometer = Voltage across 12 ohm resistor

Using Ohm's law, we have:

V_galvanometer = I_galvanometer * G
12 = 4I2 * G

Simplifying the equation, we get:

3 = I2 * G

Now, we have two equations:

12 = 4I2 * G
3 = I2 * G

Dividing the two equations, we get:

4 = I2/I2
4 = 1

Since the ratio of I2/I2 is 1, we can conclude that I2 = I2.

Now, let's substitute the value of I2 in the equation 3 = I2 * G:

3 = 1 * G
3 = G

Therefore, the resistance of the galvanometer is 3 ohms, which is equal to option C in the given options.