## Question 5:

### Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| Sketch of positive sine curve passing through O with at least one complete cycle from O | B1 | Correct shape with positive gradient through $O$; ignore any part of curve to left of origin |
| Correct shape with one and a half cycles (from O to $\frac{3\pi}{2}$) crossing $x$-axis at $\frac{\pi}{2}$ and $\pi$ | B1 | If curve extends beyond $x=\frac{3\pi}{2}$ then $x=\frac{3\pi}{2}$ must be labelled; labels for $\frac{\pi}{2}$ and $\pi$ may be on diagram or in text but not just in table; must be radians not degrees (allow awrt 1.57 and 3.14) |

### Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Table: $x: 0,\ \frac{\pi}{12},\ \frac{\pi}{6},\ \frac{\pi}{4}$; $y: 0,\ 0.5,\ 0.866,\ 1$ | — | — |
| Uses $\frac{1}{2}\times\frac{\pi}{12}$ | B1 | Need $\frac{1}{2}$ of $\frac{\pi}{12}$ or to see $\frac{\pi}{24}$; may be implied by $\frac{1}{2}h=\frac{1}{2}\!\left(\frac{\pi}{6}-\frac{\pi}{12}\right)=\frac{1}{2}(0.261...)$ |
| $\cdots\{(0+1)+2(0.5+0.866)\}$ | M1 | First bracket: first plus last values; second bracket: no additional values from table; if $x$ values used instead of $y$ values scores M0 |
| $0.4885176576\ldots$ awrt **0.49** | A1 | — |

**Special Case:** Bracketing mistake $\frac{\pi}{24}(0+1)+2(0.5+0.866)$ scores B1 M1 A0 unless final answer implies correct calculation. Need to see trapezium rule used so answer only (no working) is 0/3.

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