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**Hint:**The ratio of the output divided by the input is known as amplifier gain.Gain is a ratio of no units. $\beta $ denotes the current gain in a common-emitter circuit. The relationship between collector current (output current) and base current is known as beta (input current).

**Complete step by step answer:**

Power gain = voltage gain $\times$ current gain –(i)

Voltage gain $({A_v}) = \beta \times \dfrac{{{R_2}}}{{{R_1}}}$

Here, ${R_2}$ is the output resistance, ${R_1}$ is the input resistance and $\beta $ is the current gain.

We know that $\beta = \dfrac{\alpha }{{1 - \alpha }}$

Its given in the question that $\alpha = 0.98$, substituting these values to the equation we get,

$\beta = \dfrac{{0.98}}{{1 - 0.98}} \\

\Rightarrow \beta = \dfrac{{0.98}}{{0.02}} \\

\Rightarrow \beta = 49 \\ $

We also know that ${R_2} = 500 \times {10^3}\Omega $, and power gain $ = 6.0625 \times {10^6}$. Substituting these values to the (i) equation we get,

Power gain = voltage gain $\times$ current gain

$6.0625 \times {10^6} = 49 \times \dfrac{{500 \times {{10}^3}}}{{{R_1}}} \times 49 \\

\Rightarrow {R_1} = 49 \times \dfrac{{500 \times {{10}^3}}}{{6.0625 \times {{10}^6}}} \times 49 \\

\therefore {R_1} = 198\Omega \\ $

**Hence, the correct option is A.**

**Note:**The transistor parameters that characterise the current gain in a transistor are $\alpha $ and $\beta $ . The ratio of the collector current to the emitter current is known as $\alpha $. $\beta $ is the current gain, which is calculated by dividing the collector current by the base current.

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