Determine the value of R in the figure such that the current...

The given circuit is a Wheatstone bridge.

The condition for zero current through the central resistor $R$ is that the bridge must be balanced.

Checking the ratio of resistances of the arms:

$\dfrac{10}{10} = 1$

$\dfrac{5}{5} = 1$

Since the ratios are equal, the Wheatstone bridge is balanced.

In a balanced Wheatstone bridge, the potential difference across the central resistor is zero, so no current flows through it regardless of its resistance.

Therefore, $R$ can have any value.