| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find function constants from given conditions |
| Difficulty | Moderate -0.3 Part (i) is a straightforward system of two linear equations using exact values (cos(π/3)=1/2, cos(π)=-1), requiring only substitution and basic algebra. Part (ii) requires understanding that the range of a cosine function is [a-|b|, a+|b|], which is standard A-level knowledge. The question tests routine application of trigonometric properties with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05g Exact trigonometric values: for standard angles |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | Use correct double angle formulae and express LHS in terms of | |
| cos x and sin x | M1 | 2sinx−2sinxcosx |
| Answer | Marks |
|---|---|
| Obtain a correct expression | A1 |
| Answer | Marks |
|---|---|
| to remove a factor of 1−cosx | M1 |
| Answer | Marks |
|---|---|
| OR (working R to L): | A1 |
| Answer | Marks |
|---|---|
| 2−2cos2 x | Given answer so check working carefully |
| Answer | Marks | Guidance |
|---|---|---|
| 4(ii) | State integral of the form a ln(1+cosx) | M1* |
| Answer | Marks |
|---|---|
| Obtain integral − ln(1+cosx) | A1 |
| Substitute correct limits in correct order | M1(dep)* |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | Use correct double angle formulae and express LHS in terms of
cos x and sin x | M1 | 2sinx−2sinxcosx
( )
1− 2cos2x−1
Obtain a correct expression | A1
Complete method to get correct denominator e.g. by factorising
to remove a factor of 1−cosx | M1
Obtain the given RHS correctly
OR (working R to L): | A1
sinx 1−cosx sinx−sinxcosx
× = M1A1
1+cosx 1−cosx 1−cos2 x
2sinx−2sinxcosx
=
2−2cos2 x | Given answer so check working carefully
2sinx−sin2x
= M1A1
1−cos2x
4
--- 4(ii) ---
4(ii) | State integral of the form a ln(1+cosx) | M1* | If they use the substitution u=1+cosx
allow M1A1 for −lnu
Obtain integral − ln(1+cosx) | A1
Substitute correct limits in correct order | M1(dep)*
( )
3
Obtain answerln , or equivalent
2 | A1
4
Question | Answer | Marks | GuidanceThe function f is such that $\mathrm{f}(x) = a + b \cos x$ for $0 \leqslant x \leqslant 2\pi$. It is given that $\mathrm{f}\left(\frac{1}{3}\pi\right) = 5$ and $\mathrm{f}(\pi) = 11$.
\begin{enumerate}[label=(\roman*)]
\item Find the values of the constants $a$ and $b$. [3]
\item Find the set of values of $k$ for which the equation $\mathrm{f}(x) = k$ has no solution. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q4 [6]}}