**Question 1 (Illustrative Example):**

$$\int\left(\frac{2}{3}x^3 - 6\sqrt{x} + 1\right)dx$$

| Working | Marks | Guidance |
|---|---|---|
| $\frac{2}{3} \cdot \frac{x^4}{4} - 6x^{\frac{3}{2}} \cdot \frac{2}{3} + x + c = \frac{1}{6}x^4 - 4x^{\frac{3}{2}} + x + c$ | M1 A1 | Attempt 1 — correct integration method, first term correct |
| $\frac{2}{3} \times 3x^2 - 6 \times \frac{1}{2}x^{-\frac{1}{2}} + x = 2x^2 - 3x^{-\frac{1}{2}} + x$ | M1 | Attempt 2 — differentiation applied instead |

> **Note:** The actual mark scheme questions are not present in the uploaded pages. Only the preamble/guidance pages were provided.

# Question 1:

## Part (a):

| Working/Answer | Mark | Guidance |
|---|---|---|
| $2x+4y-3=0 \Rightarrow y = \mp\frac{2}{4}x+...$ , Gradient of perpendicular $= \pm\frac{4}{2}$ | M1 | Attempts to set given equation in form $y=ax+b$ with $a=\mp\frac{2}{4}$; deduces $m=-\frac{1}{a}$. Condone errors on "$+b$". Alternative: find both intercepts to get gradient $l_1=\pm\frac{0.75}{1.5}$ |
| Either $m=2$ or $y=2x+7$ | A1 | Accept either. Must be simplified; not left as $m=\frac{4}{2}$ unless $y=2x+7$ seen |

## Part (b):

| Working/Answer | Mark | Guidance |
|---|---|---|
| Combines $y=2x+7$ with $2x+4y-3=0 \Rightarrow 2x+4(2x+7)-3=0$ | M1 | Substitutes their $y=mx+7$ into $2x+4y-3=0$; condoning slips. Alternatively equates $y=-\frac{1}{2}x+\frac{3}{4}$ with $y=mx+7$ |
| $x=-2.5$ oe | A1 | Answer alone scores both marks if both equations correct and no incorrect working |

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