Standard +0.3 This is a straightforward application of the binomial expansion for negative/fractional powers. Students need to expand (2+x)^{-2} = 2^{-2}(1+x/2)^{-2}, multiply by (1-x), and collect terms up to x². While it requires careful algebraic manipulation and knowledge of the generalized binomial theorem, it follows a standard procedure with no novel insight required, making it slightly easier than average.
Dealing correctly with the 2. Need not use negative powers for this mark. Note: direct expansion \((2+x)^{-2} = 2^{-2} - 2\times 2^{-3}x + 3\times 2^{-4}x^2\). Award B1M1 if seen
# Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| **AG** | | |
| $\dfrac{1}{(2+x)^2} = \dfrac{1}{4\left(1+\frac{x}{2}\right)^2} = \left[\dfrac{1}{4}\left(1+\dfrac{x}{2}\right)^{-2}\right]$ | B1 | Dealing correctly with the 2. Need not use negative powers for this mark. Note: direct expansion $(2+x)^{-2} = 2^{-2} - 2\times 2^{-3}x + 3\times 2^{-4}x^2$. Award B1M1 if seen |
| $= \dfrac{1}{4}\left(1 + (-2)\left(\dfrac{x}{2}\right) + \dfrac{(-2)(-3)}{2!}\left(\dfrac{x}{2}\right)^2 + \ldots\right)$ | M1 | Allow for expanding $(1+kx)^{-2}$ even where the B mark is not awarded |
| $\dfrac{1-x}{(2+x)^2} \approx \dfrac{(1-x)}{4}\left(1 - x + \dfrac{3}{4}x^2\right)$ | M1 | Attempt to multiply their expansion by the numerator |
| $\approx \dfrac{1}{4} - \dfrac{1}{2}x + \dfrac{7}{16}x^2$ | A1 | Convincing argument |
| **[4]** | | |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{1}{(2+x)^2} = \dfrac{3}{(2+x)^2} - \dfrac{1}{2+x}$ | M1 | Using partial fractions – allow an arithmetic slip |
| $\dfrac{3}{(2+x)^2} = \dfrac{3}{4}\left(1+(-2)\left(\dfrac{x}{2}\right) + \dfrac{(-2)(-3)}{2!}\left(\dfrac{x}{2}\right)^2 + \ldots\right)$ | B1 | Dealing correctly with the 2. Need not use negative powers for this mark |
| $-\dfrac{1}{2+x} = -\dfrac{1}{2}\left(1 - \dfrac{x}{2} + \dfrac{x^2}{4}\cdots\right)$ | M1 | Allow for expanding both $(1+kx)^{-2}$ and $(1+kx)^{-1}$ even where the B mark is not awarded |
| $\approx \dfrac{1}{4} - \dfrac{1}{2}x + \dfrac{7}{16}x^2$ | A1 | Adding terms to complete a convincing argument |
| **[4]** | | |
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