| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Range of squared harmonic expression |
| Difficulty | Moderate -0.3 Part (a) is a standard harmonic form conversion requiring Pythagorean theorem and inverse tangent (R=√34, α≈0.54), which is routine A-level material. Part (b) simply requires recognizing that squaring the range [-R, R] gives [0, R²], making this a straightforward application with minimal problem-solving beyond the textbook procedure. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(R = \sqrt{34}\) | B1 | |
| Use trig formulae to find \(\alpha\) | M1 | \(\tan\alpha = \frac{3}{5}\) or \(\sin\alpha = \frac{3}{\sqrt{34}}\) or \(\cos\alpha = \frac{5}{\sqrt{34}}\) |
| Obtain \(\alpha = 0.54\) | A1 | \(30.96°\) scores M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State greatest value \(34\) | B1 FT | Their \(R^2\) |
| State least value \(0\) | B1 |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $R = \sqrt{34}$ | B1 | |
| Use trig formulae to find $\alpha$ | M1 | $\tan\alpha = \frac{3}{5}$ or $\sin\alpha = \frac{3}{\sqrt{34}}$ or $\cos\alpha = \frac{5}{\sqrt{34}}$ |
| Obtain $\alpha = 0.54$ | A1 | $30.96°$ scores M1A0 |
## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State greatest value $34$ | B1 FT | Their $R^2$ |
| State least value $0$ | B1 | |2
\begin{enumerate}[label=(\alph*)]
\item Express $5 \sin x - 3 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Give the exact value of $R$ and give $\alpha$ correct to 2 decimal places.
\item Hence state the greatest and least possible values of $( 5 \sin x - 3 \cos x ) ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q2 [5]}}