• The time period $T$ of the ac source with frequency $\nu = 50\,\text{Hz}$ is:
  $$T = \frac{1}{\nu} = \frac{1}{50}\,\text{s}$$
• An alternating voltage is symmetrical and oscillates sinusoidally: $V(t) = V_0 \sin(\omega t)$.
• The average value of a sinusoidal wave over one complete period $T$ is identically zero because the positive half-cycle and negative half-cycle cancel each other exactly:
  $$V_{\text{avg}} = \frac{1}{T} \int_0^T V_0 \sin(\omega t)\,dt = 0\,\text{V}$$

In alternating current circuits:

• In a purely resistive circuit, current and voltage are in phase ($\phi = 0$).
• In a purely inductive circuit, voltage leads current by $\pi/2$ (or current lags voltage by $\pi/2$).
• In a purely capacitive circuit, current leads voltage by $\pi/2$ radians ($90^\circ$), described by $i = I_0 \sin(\omega t + \pi/2)$ for $v = V_0 \sin(\omega t)$.

The frequency dependence of AC opposition parameters is:

• Resistance $R$: Independent of AC frequency $\nu$ ($R = \text{constant}$, horizontal line) $\rightarrow$ (A) matches (IV).
• Inductive reactance $X_L = 2\pi \nu L$: Linearly increases with frequency ($X_L \propto \nu$, straight line passing through origin) $\rightarrow$ (B) matches (III).
• Capacitive reactance $X_C = \frac{1}{2\pi \nu C}$: Inversely proportional to frequency ($X_C \propto 1/\nu$, rectangular hyperbola) $\rightarrow$ (C) matches (II).
• Impedance of series LCR circuit $Z = \sqrt{R^2 + (X_L - X_C)^2}$: Passes through a minimum ($Z = R$) at resonance frequency $\nu_0$ $\rightarrow$ (D) matches (I).
  Correct match: (A) - (IV), (B) - (III), (C) - (II), (D) - (I).

Transformer cores undergo repeated cycles of magnetization and demagnetization at the supply frequency (e.g., 50 Hz):

• High magnetic permeability ($\mu_r$) is required so that large magnetic flux density is induced for a small magnetizing current, ensuring efficient flux linkage between primary and secondary windings.
• Low hysteresis loss (narrow hysteresis loop area) and low coercivity are required to minimize heat generation and energy dissipation during continuous cyclic magnetization.
  Soft iron or silicon steel is selected for transformer cores because it satisfies both criteria.

Hence, option 2 is the correct answer.