Standard +0.8 This question requires expanding √(1+4x) using the generalised binomial theorem with fractional index 1/2, multiplying by (3+x), and collecting terms—a multi-step problem requiring careful algebraic manipulation. While the technique is standard for P3, the combination of fractional binomial expansion with polynomial multiplication and coefficient extraction makes it moderately challenging, above average difficulty.
State unsimplified term in \(x^3\), or its coefficient, in the expansion of \((1+4x)^{\frac{1}{2}}\)
B1
\(\frac{\frac{1}{2} \times \frac{-1}{2} \times \frac{-3}{2}}{6}(4x)^3 (=4)\) Must expand binomial coefficient
State unsimplified term in \(x^2\), or its coefficient, in the expansion of \((1+4x)^{\frac{1}{2}}\)
B1
\(\frac{\frac{1}{2} \times \frac{-1}{2}}{2}(4x)^2 (=-2)\) Must expand binomial coefficient
Multiply by \((3+x)\) and combine terms in \(x^3\), or their coefficients
M1
\((3\times4 - 1\times2)\); Allow if they expanded with \(x\) rather than \(4x\)
Obtain answer \(10\)
A1
Accept \(10x^3\)
4
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| State unsimplified term in $x^3$, or its coefficient, in the expansion of $(1+4x)^{\frac{1}{2}}$ | B1 | $\frac{\frac{1}{2} \times \frac{-1}{2} \times \frac{-3}{2}}{6}(4x)^3 (=4)$ Must expand binomial coefficient |
| State unsimplified term in $x^2$, or its coefficient, in the expansion of $(1+4x)^{\frac{1}{2}}$ | B1 | $\frac{\frac{1}{2} \times \frac{-1}{2}}{2}(4x)^2 (=-2)$ Must expand binomial coefficient |
| Multiply by $(3+x)$ and combine terms in $x^3$, or their coefficients | M1 | $(3\times4 - 1\times2)$; Allow if they expanded with $x$ rather than $4x$ |
| Obtain answer $10$ | A1 | Accept $10x^3$ |
| | **4** | |
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