The effective resistance between A and B is  $3\mathrm{\Omega }$ Then the value of R is 
 

Explanation:

To determine the value of resistance R that results in an effective resistance of 3Ω, we can consider the common scenarios of series and parallel resistor combinations.

Case 1: Series Combination

In a series combination, the total or effective resistance R_total is the sum of individual resistances:

R_total = R₁ + R₂ + ...

If there are multiple resistors in series, the effective resistance increases.

Case 2: Parallel Combination

In a parallel combination, the total or effective resistance is given by:

1/R_total = 1/R₁ + 1/R₂ + ...

The effective resistance in parallel is always smaller than the smallest individual resistance.

Assumption: Two Resistors in Parallel

For two resistors R connected in parallel, the formula for the total resistance R_total is:

1/R_total = 1/R + 1/R

Solving for R_total = 3Ω

1/3 = 2/R

Solving for R gives:

R = 6Ω

Final Answer

If the two resistors are in parallel, the value of R is 6Ω.