October 2021 Paper 1 Q6
6
| Scheme | Marks | AO |
|---|---|---|
| \(\left(1 - \frac{3}{8}x\right)^{\frac{1}{3}} = 1 + \left(\frac{1}{3}\right)\left(-\frac{3}{8}x\right) + \dfrac{\left(\frac{1}{3}\right)\left(-\frac{2}{3}\right)\left(-\frac{3}{8}x\right)^2}{2!}\) | B1 M1 | 1.1 1.1 |
| \(= 1 - \frac{1}{8}x - \frac{1}{64}x^2\) | A1 | 1.1 |
| \((8 - 3x)^{\frac{1}{3}} = 8^{\frac{1}{3}}\left(1 - \frac{3}{8}x\right)^{\frac{1}{3}} = 2\left(1 - \frac{3}{8}x\right)^{\frac{1}{3}}\) \((8 - 3x)^{\frac{1}{3}} = 2 - \frac{1}{4}x - \frac{1}{32}x^2\) | B1FT | 1.1 |
| [4] |
Notes
B1: Obtain correct first two terms
Allow unsimplified second term, including product of two fractions
M1: Attempt third term in expansion of \(\left(1 - \frac{3}{8}x\right)^{\frac{1}{3}}\)
Allow BOD if no brackets, even if never recovered
Allow BOD if no negative sign
A1: Correct third term
Allow unsimplified fraction as coefficient, but must be single term
B1FT: Correct expansion of \((8 - 3x)^{\frac{1}{3}}\)
FT as 2 x their expansion (at least two terms)
Bracket expanded and fractions simplified
| Scheme | Marks | AO |
|---|---|---|
| \(|x| \lt \frac{8}{3}\) | B1 | 1.2 |
| [1] |
Notes
B1: Allow any equivalent eg \(-\frac{8}{3} \lt x \lt \frac{8}{3}\)
Must be strict inequality
Must be condition for \(x\), so B0 for \(|3x| \lt 8\)
| Scheme | Marks | AO |
|---|---|---|
| \((1 + 2x)^{-2} = 1 + (-2)(2x) + \dfrac{(-2)(-3)}{2!}(2x)^2\) | M1 | 3.1a |
| \(= 1 - 4x + 12x^2\) | A1 | 1.1 |
| \((2 \times 12) + \left(-\frac{1}{4} \times -4\right) + \left(-\frac{1}{32} \times 1\right)\) | M1 | 1.1 |
| \(\frac{799}{32}\) or \(24\frac{31}{32}\) | A1 | 1.1 |
| [4] |
Notes
M1: Attempt first three terms of expansion
Must be expanding \((1 + 2x)^{-2}\)
Allow BOD if no brackets on \(2x\), even if never recovered
A1: Obtain correct first three terms
Allow unsimplified fraction for coeff of third term
M1: Attempt all 3 relevant products
Finding 3 appropriate terms from the product of two 3-term quadratics
If part of full expansion then M1 when reqd 3 products and no others are combined
A1: Any exact equivalent, including 24.96875
Condone \(x^2\) still present