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To solve the given problem involving the binomial expansion of \(\left(x^2 + \frac{3}{x}\right)^{12}\), we will use the Binomial Theorem, which states:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case:
- \(a = x^2\)
- \(b = \frac{3}{x}\)
- \(n = 12\)

The general term (the \(k\)-th term) in the expansion is given by:

\[
T_k = \binom{12}{k} \left(x^2\right)^{12-k} \left(\frac{3}{x}\right)^k
\]

This simplifies to:

\[
T_k = \binom{12}{k} x^{2(12-k)} \cdot \frac{3^k}{x^k} = \binom{12}{k} \cdot 3^k \cdot x^{24 - 3k}
\]

Now, let's solve each part of the problem:

### a) Determine the sixth term.
The sixth term corresponds to \(k = 5\) (since \(k\) starts at 0 for the first term).

\[
T_6 = \binom{12}{5} \cdot 3^5 \cdot x^{24 - 3(5)} = \binom{12}{5} \cdot 3^5 \cdot x^{9}
\]

Calculating this:

\[
\binom{12}{5} = 792, \quad 3^5 = 243
\]

\[
T_6 = 792 \cdot 243 \cdot x^9 = 192456 \cdot x^9
\]

So, the sixth term is:

\[
192456x^9
\]

### b) Determine the independent term.
The independent term is the term where the exponent of \(x\) is zero, i.e., \(24 - 3k = 0\).

Solving for \(k\):

\[
24 = 3k \quad \Rightarrow \quad k = 8
\]

Substitute \(k = 8\) into the general term:

\[
T_{\text{independent}} = \binom{12}{8} \cdot 3^8 \cdot x^0 = \binom{12}{8} \cdot 6561
\]

\(\binom{12}{8} = \binom{12}{4} = 495\), and \(3^8 = 6561\).

\[
T_{\text{independent}} = 495 \cdot 6561 = 3246195
\]

So, the independent term is:

\[
3246195
\]

### c) Determine the central term(s).
For \(n = 12\), the central term(s) occur at \(k = 6\) and \(k = 7\) (since the expansion has an even number of terms).

#### For \(k = 6\):

\[
T_7 = \binom{12}{6} \cdot 3^6 \cdot x^{24 - 3(6)} = \binom{12}{6} \cdot 729 \cdot x^6
\]

\(\binom{12}{6} = 924\), and \(3^6 = 729\).

\[
T_7 = 924 \cdot 729 \cdot x^6 = 673596 \cdot x^6
\]

#### For \(k = 7\):

\[
T_8 = \binom{12}{7} \cdot 3^7 \cdot x^{24 - 3(7)} = \binom{12}{7} \cdot 2187 \cdot x^3
\]

\(\binom{12}{7} = 792\), and \(3^7 = 2187\).

\[
T_8 = 792 \cdot 2187 \cdot x^3 = 1730304 \cdot x^3
\]

So, the central terms are:

\[
673596x^6 \quad \text{and} \quad 1730304x^3
\]

---

### Do you want more details or have any questions? 

Here are 8 related questions to expand your understanding:

1. How is the Binomial Theorem derived?
2. How do you calculate binomial coefficients efficiently?
3. What is the significance of the independent term in a binomial expansion?
4. How would the solution change if the exponent in the binomial expression was different?
5. What happens if one of the terms in the binomial is negative?
6. Can the binomial theorem be applied to non-integer exponents?
7. How can you use Pascal's Triangle to find binomial coefficients?
8. What is the general term formula for a trinomial expansion?

**Tip:** When solving problems involving binomial expansions, always pay attention to whether the problem asks for specific terms or properties of the expansion, as this can simplify your calculations.

Learn how to solve the binomial expansion problem for (x^2 + 3/x)^12 using the Binomial Theorem. This detailed analysis covers finding specific terms like the sixth term, independent term, and central terms. Suitable for advanced high school students and those interested in advanced algebraic concepts.

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