Moderate -0.3 This is a straightforward application of the binomial expansion for a fractional power combined with simple polynomial multiplication. It requires routine use of the formula (1+bx)^n with n=1/2, then multiplying by (3+x) and collecting terms—standard technique with no novel insight needed, slightly easier than average due to being a direct textbook-style question.
1 Expand \(( 3 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
State correct unsimplified first two terms of the expansion of \((1-2x)^{\frac{1}{2}}\), e.g. \(1+\frac{1}{2}(-2x)\)
B1
Symbolic coefficients are not sufficient. \(1-x\)
State correct unsimplified term in \(x^2\), e.g. \(\dfrac{\frac{1}{2}\left(\frac{1}{2}-1\right)(-2x)^2}{2!}\)
B1
Symbolic coefficients are not sufficient. \(-\frac{1}{2}x^2\)
Obtain sufficient terms of the product of \((3+x)\) and the expansion up to the term in \(x^2\)
M1
Obtain final answer \(3-2x-\dfrac{5}{2}x^2\)
A1
Total
4
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| State correct unsimplified first two terms of the expansion of $(1-2x)^{\frac{1}{2}}$, e.g. $1+\frac{1}{2}(-2x)$ | B1 | Symbolic coefficients are not sufficient. $1-x$ |
| State correct unsimplified term in $x^2$, e.g. $\dfrac{\frac{1}{2}\left(\frac{1}{2}-1\right)(-2x)^2}{2!}$ | B1 | Symbolic coefficients are not sufficient. $-\frac{1}{2}x^2$ |
| Obtain sufficient terms of the product of $(3+x)$ and the expansion up to the term in $x^2$ | M1 | |
| Obtain final answer $3-2x-\dfrac{5}{2}x^2$ | A1 | |
| **Total** | **4** | |
---
1 Expand $( 3 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.\\
\hfill \mbox{\textit{CAIE P3 2024 Q1 [4]}}