Moderate -0.8 This is a straightforward application of the binomial expansion formula for fractional powers. Students need only substitute n=2/3 and b=3 into the standard formula and simplify coefficients—pure recall with minimal algebraic manipulation, making it easier than average.
1 Expand \(( 1 + 3 x ) ^ { \frac { 2 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
State a correct unsimplified version of the \(x^2\) or \(x^3\) term
M1
Symbolic binomial coefficients are not sufficient for the M mark
Obtain the next term \(-x^2\)
A1
Obtain the final term \(\frac{4}{3}x^3\)
A1
Total
4
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| State correct first two terms $1 + 2x$ | B1 | |
| State a correct unsimplified version of the $x^2$ or $x^3$ term | M1 | Symbolic binomial coefficients are not sufficient for the M mark |
| Obtain the next term $-x^2$ | A1 | |
| Obtain the final term $\frac{4}{3}x^3$ | A1 | |
| **Total** | **4** | |
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1 Expand $( 1 + 3 x ) ^ { \frac { 2 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.\\
\hfill \mbox{\textit{CAIE P3 2021 Q1 [4]}}