| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Solve trigonometric equation with exact values |
| Difficulty | Moderate -0.3 This question tests understanding of transformed trig graphs and solving trig equations with standard techniques. Part (i) requires identifying max/min from sin² range [0,1]; part (ii) involves counting intersections from a given graph; part (iii) is a routine solve using sin²2x substitution and inverse trig. All steps are standard A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| 9(i) | q(cid:45)f ( x ) (cid:45) p+q | B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 9(ii) | (a) 2 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (b) 3 | B1 | π |
| Answer | Marks | Guidance |
|---|---|---|
| (c) 4 | B1 | π 3π 5π 7π |
| Answer | Marks |
|---|---|
| 9(iii) | 2 |
| Answer | Marks |
|---|---|
| 3 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 3 | A1 | OR Implied by at least one correct value for x. Allow sin–1 form |
| (2x =) at least two of 0.955(3), 2.18(6), 4.09(7) , 5.32(8) | A1 | Can be implied by corresponding values of x below |
| Answer | Marks | Guidance |
|---|---|---|
| (x =) 0.478, 1.09, 2.05, 2.66. | A1A1 | Allow 0.152π, 0.348π, 0.652π, 0.848π |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(i) ---
9(i) | q(cid:45)f ( x ) (cid:45) p+q | B1B1 | B1 each inequality – allow two separate statements
Accept < , (q, p + q), [q, p + q]
Condone y or x or f in place of f(x)
2
Question | Answer | Marks | Guidance
--- 9(ii) ---
9(ii) | (a) 2 | B1 | π 3π
Allow ,
4 4
(b) 3 | B1 | π
Allow 0, , π
2
(c) 4 | B1 | π 3π 5π 7π
Allow , , ,
8 8 8 8
3
--- 9(iii) ---
9(iii) | 2
3sin22x+2=4 → sin22x= soi
3 | M1
2 2
sin2x=(±) 2x=sin−1(±)
Sin2x = (±)0.816(5). Allow or
3 3 | A1 | OR Implied by at least one correct value for x. Allow sin–1 form
(2x =) at least two of 0.955(3), 2.18(6), 4.09(7) , 5.32(8) | A1 | Can be implied by corresponding values of x below
Allow for at least two of 0.304π, 0.696π, 1.30(4)π, 1.69(6)π
OR at least two of 54.7(4)°, 125.2(6)°, 234.7(4)°, 305.2(6)°
(x =) 0.478, 1.09, 2.05, 2.66. | A1A1 | Allow 0.152π, 0.348π, 0.652π, 0.848π
SC A1 for 2 or 3 correct.
SC A1 for all of 27.4º, 62.6º, 117.4º, 152.6º
2
Sin2x = ± → x = 0.365,1.21,1.94,2.78 scores SC M1A0A0A1
3
5
Question | Answer | Marks | Guidance\includegraphics{figure_9}
The function f : $x \mapsto p \sin^2 2x + q$ is defined for $0 \leqslant x \leqslant \pi$, where $p$ and $q$ are positive constants. The diagram shows the graph of $y = \text{f}(x)$.
\begin{enumerate}[label=(\roman*)]
\item In terms of $p$ and $q$, state the range of f. [2]
\item State the number of solutions of the following equations.
\begin{enumerate}[label=(\alph*)]
\item $\text{f}(x) = p + q$ [1]
\item $\text{f}(x) = q$ [1]
\item $\text{f}(x) = \frac{1}{2}p + q$ [1]
\end{enumerate}
\item For the case where $p = 3$ and $q = 2$, solve the equation $\text{f}(x) = 4$, showing all necessary working. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q9 [10]}}