In the given circuit 'a' is an arbitrary constant. The value of $$m$$ for which the equivalent circuit resistance is minimum, will be $$\sqrt{\frac{x}{2}}$$. The value of $$x$$ is ______.

Step 1: Equivalent Resistance

• Left block (3 resistors ma in parallel):

$$R_1=\frac{ma}{3}$$

• Right block (2 resistors $$\frac{a}{m}$$​ in parallel):

$$R_2=\frac{a}{2m}$$

Total resistance:

$$R=\frac{ma}{3}+\frac{a}{2m}$$

Step 2: Condition for minimum RRR

$$\frac{dR}{dm}=\frac{a}{3}-\frac{a}{2m^2}=0$$

$$\frac{a}{3}=\frac{a}{2m^2}$$

$$\Rightarrow m^2=\frac{3}{2}$$​

$$m=\sqrt{\ \ \frac{\ 3}{2}}$$

Step 3: Compare with given form

$$m=\sqrt{\frac{x}{2}}\Rightarrow x=3$$

Final Answer:

3