Exercise 3.10 Bias issues within an antenna amplifier

a)

An answer:
With the assumptions described in the hint (ignore base current of the proceeding stage) a derivation could be as shown below. Note that the supply voltage in this particular circuit is denoted as $V_{B1}$...

$$\begin{array}{rcll}& I_{C2}& =\frac{\mathit{\beta fe}}{\mathit{\beta fe}+1}{I}_{E2}& \\ & & I_{E2}=\frac{V_{E2}}{R_{E2}}& \\ & & V_{E2}=V_{B2}-V_{\mathit{BE}2}& \\ & & V_{B2}=V_{B1}\frac{R_{B21}}{R_{B21}+R_{B22}}-I_{B2}\cdot \frac{R_{B21}{R}_{B22}}{R_{B21}+R_{B22}}& \\ & & I_{B2}=\frac{I_{C2}}{\mathit{\beta fe}}& \\ & & ⟹& \\ & I_{C2}& =\frac{\mathit{\beta fe}}{\mathit{\beta fe}+1}\frac{V_{B1}\frac{R_{B21}}{R_{B21}+R_{B22}}-\frac{I_{C2}}{\mathit{\beta fe}}\cdot \frac{R_{B21}{R}_{B22}}{R_{B21}+R_{B22}}-V_{\mathit{BE}2}}{R_{E2}}& \\ & & =\frac{\mathit{\beta fe}}{\mathit{\beta fe}+1}\frac{V_{B1}\frac{R_{B21}}{R_{B21}+R_{B22}}-V_{\mathit{BE}2}}{R_{E2}+\frac{1}{\mathit{\beta fe}+1}\cdot \frac{R_{B21}{R}_{B22}}{R_{B21}+R_{B22}}}& \end{array}$$

Also $V_{E2}=\frac{\mathit{\beta fe}+1}{\mathit{\beta fe}}{I}_{C2}{R}_{E2}$

For the derivation of $I_{C3}$:

$$\begin{array}{rcll}& I_{C3}& =\frac{\mathit{\beta fe}}{\mathit{\beta fe}+1}{I}_{E3}& \\ & & I_{E3}=\frac{V_{E3}}{R_{E3}}& \\ & & V_{E3}=V_{E2}-V_{\mathit{BE}3}& \\ & & ⟹& \\ & I_{C3}& =\frac{\mathit{\beta fe}}{\mathit{\beta fe}+1}\frac{V_{E2}-V_{\mathit{BE}3}}{R_{E3}}& \end{array}$$

Where you may substitute the earlier found relation for $V_{E2}$.

b)

An answer:

$$\begin{array}{rcll}{I}_{C2}\approx 2.05\mathit{mA}& & & \\ I_{C3}\approx 1.6\mathit{mA}& & & \end{array}$$

c)

An answer:
This relates to small signal properties, in this case to voltage gain of the stages. See therefore chapters 4 and 5.

$R_9$ is decoupled because any (signal induced) change in the base voltage appears in full across $V_{\mathit{BE}}$ of $Q_3$ which results in maximum sensitivity. For biasing purposes, the circuit should be insensitive to changes: at DC we want to have a relatively high ohmic emitter series resistance.

If $R_6$ would be similarly decoupled there would be no signal present at the base of $Q_3$. Then the voltage gain of the stage with $Q3$ would be zero.