Define the term emf of a battery.

Electromotive force (EMF) is the maximum potential difference that a battery or source of electrical energy can provide across its terminals when no current is flowing in the external circuit. It represents the total energy supplied per unit charge by the source.

When the switch S is closed, current flows through the external circuit. This current also flows through the internal resistance ($r$) of the battery, causing a potential drop ($Ir$) across the internal resistance. The voltmeter measures the terminal potential difference ($V_{terminal}$), which is given by $V_{terminal} = \varepsilon - Ir$. Since $Ir > 0$ when current flows, the terminal potential difference is less than the EMF, causing the voltmeter reading to decrease.

Resistors $R_2$ and $R_3$ are in parallel. Their equivalent resistance ($R_p$) is:

Resistor $R_1$ is in series with $R_p$. The total external resistance ($R_{ext}$) is:

The decrease in voltmeter reading ($1.5 \, \text{V}$) is due to the potential drop across the internal resistance ($Ir$).

Given the internal resistance $r = 0.5 \, \Omega$:

The EMF ($\varepsilon$) is given by the formula $\varepsilon = I(R_{ext} + r)$.

The statement is: "The current through resistor $R_3$ is larger than the current through resistor $R_2$."
Resistors $R_2$ ($25 \, \Omega$) and $R_3$ ($15 \, \Omega$) are connected in parallel, meaning the voltage across them is the same. According to Ohm's Law ($I = V/R$), current is inversely proportional to resistance. Since $R_3 < R_2$, the current through $R_3$ will be greater than the current through $R_2$.
Therefore, the statement is CORRECT.

The EMF of a battery is an intrinsic property of the battery itself, determined by its chemical composition and internal structure. It represents the maximum potential difference the battery can provide. Removing an external resistor changes the total external resistance of the circuit, which affects the total current and the terminal potential difference, but it does not change the fundamental EMF of the battery.
Therefore, the EMF of the battery REMAINS THE SAME.