June 2023 Paper 1 Q8
8
Determine the possible values of the constant \(a\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(\left(1 + \frac{3}{4}x\right)^{\frac{3}{2}} = 1 + \left(\frac{3}{2}\right)\left(\frac{3}{4}x\right)\) | B1 | 1.1 |
| \(+\,\dfrac{\left(\frac{3}{2}\right)\left(\frac{1}{2}\right)}{2}\left(\frac{3}{4}x\right)^2\) | M1 | 1.1 |
| A1 | 1.1 | |
| \((4 + 3x)^{\frac{3}{2}} = 8\left(1 + \frac{3}{4}x\right)^{\frac{3}{2}} = 8 + 9x + \frac{27}{16}x^2\) | B1FT | 1.1a |
| [4] |
Notes
B1: Correct first two terms
Allow unsimplified
Expect \(1 + \frac{9}{8}x\)
M1: Attempt third term
Condone lack of brackets when attempting to square ie \(\frac{3}{4}x^2\)
Coefficient must be \(\dfrac{\left(\frac{3}{2}\right)\left(\frac{1}{2}\right)}{2}\) or equiv
A1: Obtain correct third term
Allow unsimplified
\(\frac{3}{4}x^2\) is A0 unless recovered by later work
Expect \(\frac{27}{128}x^2\)
B1FT: Multiply their 3 term expansion by 8
Bracket expanded and coefficients simplified
If B1M1A1 awarded, but attempt to simplify then goes wrong, B1FT is not also awarded
ISW once correct expansion seen
| Scheme | Marks | AO |
|---|---|---|
| \(|x| \lt \frac{4}{3}\) or \(-\frac{4}{3} \lt x \lt \frac{4}{3}\) | B1 | 1.1 |
| [1] |
Notes
B1: Could also be \(|x| \leqslant \frac{4}{3}\) or \(-\frac{4}{3} \leqslant x \leqslant \frac{4}{3}\), as \(n \gt 0\)
Must be condition for \(x\), not \(kx\)
| Scheme | Marks | AO |
|---|---|---|
| \(\left(8 + 9x + \frac{27}{16}x^2\right)\left(1 + 2ax + a^2x^2\right)\) coeff of \(x^2\) is \(8a^2 + 18a + \frac{27}{16}\) | M1 | 3.1a |
| M1 | 1.1 | |
| \(8a^2 + 18a + \frac{27}{16} = \frac{107}{16}\) \(8a^2 + 18a - 5 = 0\) | A1 | 3.1a |
| \((2a + 5)(4a - 1) = 0\) \(a = -\frac{5}{2}\) and \(a = \frac{1}{4}\) | A1 | 1.1 |
| [4] |
Notes
M1: Expand \((1 + ax)^2\) and attempt at least one coeff of \(x^2\)
Allow \(ax\) as middle term, and/or \(ax^2\) as third term
Attempt at \(x^2\) term could be part of a fuller expansion
M1: Attempt all three coeff of \(x^2\), and no others
If part of fuller expansion then M1 awarded when only three relevant terms used
A1: Equate to \(\frac{107}{16}\) to obtain correct quadratic
aef, including unsimplified
A0 if a mix of terms and coefficients, but can be recovered
A1: Solve quadratic, possibly BC, to obtain \(a = -\frac{5}{2}\) and \(a = \frac{1}{4}\)