Answer by periblepsis for How to solve this question by nodal analysis?

Yes I am self learning.The question asks for Maximum power for theload resistance.So,using maximum power transfer theorem we have tofind the load resistance which should be equal to thevenins resistance

Hmm. But your title says:

How to solve this question by nodal analysis?

So I will take it that you understand already that the Thevenin impedance must be matched by the load. That's good. I'm glad you already understand that fact.

Here's an updated diagram for nodal analysis:

From this, we can form the following nodal equations:

$$\begin{align*}\frac{V_1}{R_1}+\frac{V_1}{R_2}+\frac{V_1}{R_3} &=\frac{V_0}{R_1}+\frac{V_2}{R_3}\\\\\frac{V_2}{R_3}+\frac{V_2}{R_4}&=\frac{V_1}{R_3}+\frac{V_3}{R_4}+I_0\\\\\frac{V_3}{R_4}&=\frac{V_2}{R_4}+I_1\end{align*}$$

We will use two values for an injected current, \$I_1\$: at \$0\:\text{A}\$ and \$1 \:\text{A}\$:

EQ1 = Eq( v1/r1 + v1/r2 + v1/r3, v0/r1 + v2/r3 )EQ2 = Eq( v2/r3 + v2/r4, v1/r3 + v3/r4 + i0 )EQ3 = Eq( v3/r4, i1 + v2/r4 )solve([ EQ1, EQ2, EQ3 ], [ v1, v2, v3 ])[v3].subs(    { v0:12, i0:2, r1:6, r2:12, r3:3, r4:2, i1:0 })22solve([ EQ1, EQ2, EQ3 ], [ v1, v2, v3 ])[v3].subs(    { v0:12, i0:2, r1:6, r2:12, r3:3, r4:2, i1:1 })31

From the above, you can see that the unloaded voltage is \$22\:\text{V}\$ and that after injecting \$1\:\text{A}\$ that the voltage moves to \$31\:\text{V}\$. So this says that the Thevenin impedance is \$9\:\Omega\$.

So that is the load impedance needed to maximize the power delivery into the load.

The above uses nodal analysis to reach this result, as requested.

As proof, here's the result from LTspice:

You can readily see that the peak power in the load occurs when the load impedance is \$9\:\Omega\$.