Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative indices. Students need to factor out the constant (3^{-3}), then apply the standard expansion of (1+y)^{-3} with y=2x/3, and simplify coefficients—a routine procedure requiring only formula recall and careful arithmetic, making it easier than average.
State a correct unsimplified version of the \(x\) or \(x^2\) term in the expansion of \(\left(1+\frac{2}{3}x\right)^{-3}\) or \((3+2x)^{-3}\)
(M1)
Symbolic binomial coefficients, e.g. \(\binom{-3}{2}\), are not sufficient for M1
State correct first term \(\frac{1}{27}\)
B1
Obtain term \(-\frac{2}{27}x\)
A1
Obtain term \(\frac{8}{81}x^2\)
A1
*OR:* Differentiate expression and evaluate \(f(0)\) and \(f'(0)\), where \(f'(x) = k(3+2x)^{-4}\)
(M1)
State correct first term \(\frac{1}{27}\)
B1
Obtain term \(-\frac{2}{27}x\)
A1
Obtain term \(\frac{8}{81}x^2\)
A1
Total: 4
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| State a correct unsimplified version of the $x$ or $x^2$ term in the expansion of $\left(1+\frac{2}{3}x\right)^{-3}$ or $(3+2x)^{-3}$ | (M1) | Symbolic binomial coefficients, e.g. $\binom{-3}{2}$, are not sufficient for M1 |
| State correct first term $\frac{1}{27}$ | B1 | |
| Obtain term $-\frac{2}{27}x$ | A1 | |
| Obtain term $\frac{8}{81}x^2$ | A1 | |
| *OR:* Differentiate expression and evaluate $f(0)$ and $f'(0)$, where $f'(x) = k(3+2x)^{-4}$ | (M1) | |
| State correct first term $\frac{1}{27}$ | B1 | |
| Obtain term $-\frac{2}{27}x$ | A1 | |
| Obtain term $\frac{8}{81}x^2$ | A1 | |
| **Total: 4** | | |
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2 Expand $( 3 + 2 x ) ^ { - 3 }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$, simplifying the coefficients.\\
\hfill \mbox{\textit{CAIE P3 2017 Q2 [4]}}