| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch trig curve and straight line, count intersections |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question requiring basic knowledge of cosine transformations and linear graphs. Part (i) involves simple range identification, part (ii) is routine sketching of a transformed trig function and a line, and part (iii) requires counting intersections from the sketch—all standard P1 techniques with no problem-solving insight needed. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.05f Trigonometric function graphs: symmetries and periodicities1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(3,\ -3\) | B1 | Accept \(\pm 3\) |
| \(-\frac{1}{2}\) | B1 | |
| \(2\frac{1}{2}\) | B1 | Condone misuse of inequality signs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 2 complete oscillations of a cosine curve starting with a maximum at \((0, a),\ a > 0\) | B1 | Only mark the curve from \(0 \rightarrow 2\pi\). If \(x\) axis is not labelled assume \(0 \rightarrow 2\pi\) is the range shown. Labels on axes are not required |
| Fully correct curve which must appear to level off at \(0\) and/or \(2\pi\) | B1 | |
| Line starting on positive \(y\) axis and finishing below the \(x\) axis at \(2\pi\). Must be straight | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(4\) | B1 |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $3,\ -3$ | B1 | Accept $\pm 3$ |
| $-\frac{1}{2}$ | B1 | |
| $2\frac{1}{2}$ | B1 | Condone misuse of inequality signs |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| 2 complete oscillations of a cosine curve starting with a maximum at $(0, a),\ a > 0$ | B1 | Only mark the curve from $0 \rightarrow 2\pi$. If $x$ axis is not labelled assume $0 \rightarrow 2\pi$ is the range shown. Labels on axes are not required |
| Fully correct curve which must appear to level off at $0$ and/or $2\pi$ | B1 | |
| Line starting on positive $y$ axis and finishing below the $x$ axis at $2\pi$. Must be straight | B1 | |
## Question 6(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $4$ | B1 | |6 The equation of a curve is $y = 3 \cos 2 x$ and the equation of a line is $2 y + \frac { 3 x } { \pi } = 5$.\\
(i) State the smallest and largest values of $y$ for both the curve and the line for $0 \leqslant x \leqslant 2 \pi$.\\
(ii) Sketch, on the same diagram, the graphs of $y = 3 \cos 2 x$ and $2 y + \frac { 3 x } { \pi } = 5$ for $0 \leqslant x \leqslant 2 \pi$.\\
(iii) State the number of solutions of the equation $6 \cos 2 x = 5 - \frac { 3 x } { \pi }$ for $0 \leqslant x \leqslant 2 \pi$.\\
\hfill \mbox{\textit{CAIE P1 2019 Q6 [7]}}