To understand why the value of resistor R in the circuit is 100 ohms, let's analyze the given circuit diagram and the conditions mentioned in the question.

Circuit Diagram:
-------------------------------
A ---|--- R ---|--- B
|
G
-------------------------------

Given Conditions:
1. The galvanometer G shows zero deflection.
2. Batteries A and B have negligible internal resistance.

Explanation:

1. Galvanometer Deflection:
Since the galvanometer G shows zero deflection, it means that no current is passing through it. This implies that the potential difference across the terminals of the galvanometer is zero. Therefore, the potential difference across resistor R is equal to the potential difference across battery A.

2. Negligible Internal Resistance:
When batteries A and B have negligible internal resistance, it means that the potential difference across their terminals is equal to their electromotive force (emf). Hence, the potential difference across battery A is equal to the emf of battery A.

3. Applying Kirchhoff's Voltage Law:
Now, let's apply Kirchhoff's voltage law (KVL) to the loop containing battery A, resistor R, and battery B. According to KVL, the sum of the potential differences around a closed loop is zero.

Since the potential difference across the galvanometer is zero, the sum of the potential differences across resistor R and battery B must also be zero.

This can be expressed as:
Potential difference across resistor R + emf of battery B = 0

Since the potential difference across resistor R is equal to the emf of battery A (as mentioned in point 1), we can rewrite the above equation as:
Emf of battery A + emf of battery B = 0

4. Solving for R:
Since the emf of battery A is equal to the potential difference across resistor R, we can substitute the value of emf of battery A in the equation obtained in point 3.

Emf of battery A + emf of battery B = 0
Potential difference across resistor R + emf of battery B = 0
Potential difference across resistor R + potential difference across resistor R = 0
2 * potential difference across resistor R = 0
Potential difference across resistor R = 0

Since the potential difference across resistor R is zero, it implies that no current is passing through it. Thus, the value of resistor R must be infinite.

However, none of the given options (a, b, c, d) mention "infinity" as a possible value for resistor R. Therefore, none of the given options are correct.

Hence, the given options do not provide the correct answer for the value of resistor R.