## Question 1:

### Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^n \rightarrow x^{n-1}$ or $5 \rightarrow 0$ | M1 | Method mark for differentiation |
| $\frac{dy}{dx} = \frac{3x^2}{3} - 2 \times 2x + 3$ | M1A1 | Any 3 of the 4 terms differentiated correctly; could be 2 terms correct and $5 \rightarrow 0$; allow simplified or unsimplified including $3x^0$ for 3 |
| $\frac{dy}{dx} = x^2 - 4x + 3$ | A1 | All 3 terms correct, simplified, on same line, no $+0$. Do not allow $1x^2$ for $x^2$, $x^1$ for $x$, or $3x^0$ for $3$. Condone omission of $dy/dx =$ or use of $y=$ |

**Note:** Candidates who multiply by 3 before differentiating e.g. $\left(\frac{x^3}{3} - 2x^2 + 3x + 5\right) \times 3 = x^3 - 6x^2 + 9x + 15 \Rightarrow \frac{dy}{dx} = 3x^2 - 12x + 9$ score M1A0A0 but could recover in (a) if they divide by 3. If they persist with $\frac{dy}{dx} = 3x^2 - 12x + 9$ in (b), allow full recovery in (b).

---

### Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 4x + 3 = 0 \Rightarrow x = 1, \ 3$ | M1A1 | M1: Attempt to solve their 3TQ from part (a) as far as $x = \ldots$; if no working shown and roots incorrect for their 3TQ, score M0; A1: Correct values (may be implied by inequalities) |
| $x <$ "1", $x >$ "3" | M1 | Chooses outside region ($x <$ their lower limit, $x >$ their upper limit). Do not award simply for diagram or table. |
| $x < 1, \ x > 3$ | A1 | Correct answer. Allow regions separated by comma or written separately e.g. $(-\infty, 1) \cup (3, \infty)$. Do not allow $1 > x > 3$ or $x < 1$ **and** $x > 3$. Answers using $\leq, \geq$ lose final mark as would $[-\infty,1] \cup [3,\infty]$ |

---