## Question 3:

### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt relevant term | M1 | Must be an attempt at a product involving a binomial coeff of 20 (not just $^6C_3$ unless later seen as 20), $4^3$ and an intention to cube $ax$ (but allow for $ax^3$). Could come from $4^6(1 + \frac{ax}{4})^6$ as long as done correctly. Ignore any other terms if fuller expansion attempted |
| $20 \times 4^3 \times a^3 = 160$; Obtain correct $1280a^3$, or unsimplified equiv | A1 | Allow $1280a^3x^3$, or $1280(ax)^3$, but not $1280ax^3$ unless $a^3$ subsequently seen, or implied by working |
| Equate to 160 and attempt to solve for $a$ | M1 | Must be equating coeffs — allow if $x^3$ present on both sides (but not just one) as long as they both go at same point. Allow for their coeff of $x^3$, as long as two or more parts of product are attempted eg $20ax^3 / 64ax^3$. Allow M1 for $1280a = 160$ (giving $a = 0.125$). M0 for incorrect division (eg giving $a^3 = 8$) |
| $1280a^3 = 160$, $a^3 = \frac{1}{8}$, $a = \frac{1}{2}$; Obtain $a = \frac{1}{2}$ | A1 | Allow 0.5, but not an unsimplified fraction. Answer only gets full credit, as does T&I. **SR: max of 3 marks** for $a = 0.5$ from incorrect algebra |
| **[4]** | | |

### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4^6 + 6 \times 4^5 \times \frac{1}{2} = 4096 + 3072x$; State 4096 | B1 | Allow $4^6$ if given as final answer. Mark final answer — so do not isw if a constant term is subsequently added to 4096 from an incorrect attempt at second term eg using sum rather than product |
| State $3072x$, or ($6144 \times$ their $a)x$ | B1FT | Must follow a numerical value of $a$, from attempt in part (i). Must be of form $kx$ so just stating coeff of $x$ is B0. Mark final answer. B2 can still be awarded if two terms are not linked by a '+' sign — could be a comma, 'and', or just two separate terms. **SR: B1** can be awarded if both terms seen as correct, but then 'cancelled' by a common factor |
| **[2]** | | |

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