In a series LCR circuit, the net impedance ($Z$) is determined by the resistance ($R$), inductive reactance ($X_L$), and capacitive reactance ($X_C$) using the phasor relationship: $Z = \sqrt{R^2 + (X_L - X_C)^2}$. For this specific problem from NEET 2023, the circuit parameters are $R = 10 \ \Omega$, $X_L = 10 \ \Omega$, and $X_C = 5 \ \Omega$. Substituting these into the formula gives $Z = \sqrt{10^2 + (10 - 5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \ \Omega$.