| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch single standard trig graph (sin/cos/tan) |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question on basic trig graphs requiring only routine skills: sketching y=2cos(x), finding coordinates, calculating distance between two points, and finding line intercepts. All techniques are standard AS-level procedures with no problem-solving insight needed, making it easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.05a Sine, cosine, tangent: definitions for all arguments1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| One complete cycle of \(y = 2\cos x\) | B1 | One whole cycle – starts and finishes at negative value |
| Smooth curve, flattens at ends and middle, shows \((0, 2)\) | DB1 | Smooth curve, flattens at ends and middle. Shows \((0, 2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P\left(\frac{\pi}{3}, 1\right)\), \(Q(\pi, -2)\) | ||
| \(PQ^2 = \left(\frac{2\pi}{3}\right)^2 + 3^2 \rightarrow PQ = 3.7\) | M1 A1 | Pythagoras on their coordinates, must be correct, OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Equation of \(PQ\): \(y - 1 = -\frac{9}{2\pi}\left(x - \frac{\pi}{3}\right)\) | M1 | Correct form of line equation or simultaneous equations from their \(P\) and \(Q\) |
| If \(y = 0 \rightarrow h = \frac{5\pi}{9}\) | A1 | AG, condone \(x = \frac{5\pi}{9}\) |
| If \(x = 0 \rightarrow k = \frac{5}{2}\) | A1 | SR: non-exact solutions A1 for both |
## Question 5:
**Part 5(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| One complete cycle of $y = 2\cos x$ | B1 | One whole cycle – starts and finishes at negative value |
| Smooth curve, flattens at ends and middle, shows $(0, 2)$ | DB1 | Smooth curve, flattens at ends and middle. Shows $(0, 2)$ |
**Part 5(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P\left(\frac{\pi}{3}, 1\right)$, $Q(\pi, -2)$ | | |
| $PQ^2 = \left(\frac{2\pi}{3}\right)^2 + 3^2 \rightarrow PQ = 3.7$ | M1 A1 | Pythagoras on their coordinates, must be correct, OE |
**Part 5(iii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Equation of $PQ$: $y - 1 = -\frac{9}{2\pi}\left(x - \frac{\pi}{3}\right)$ | M1 | Correct form of line equation or simultaneous equations from their $P$ and $Q$ |
| If $y = 0 \rightarrow h = \frac{5\pi}{9}$ | A1 | AG, condone $x = \frac{5\pi}{9}$ |
| If $x = 0 \rightarrow k = \frac{5}{2}$ | A1 | SR: non-exact solutions A1 for both |
---5 The equation of a curve is $y = 2 \cos x$.\\
(i) Sketch the graph of $y = 2 \cos x$ for $- \pi \leqslant x \leqslant \pi$, stating the coordinates of the point of intersection with the $y$-axis.
Points $P$ and $Q$ lie on the curve and have $x$-coordinates of $\frac { 1 } { 3 } \pi$ and $\pi$ respectively.\\
(ii) Find the length of $P Q$ correct to 1 decimal place.\\
The line through $P$ and $Q$ meets the $x$-axis at $H ( h , 0 )$ and the $y$-axis at $K ( 0 , k )$.\\
(iii) Show that $h = \frac { 5 } { 9 } \pi$ and find the value of $k$.\\
\hfill \mbox{\textit{CAIE P1 2017 Q5 [7]}}