| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard harmonic form question with three routine parts: (a) converting to R cos(θ-α) using standard formulas, (b) solving a straightforward equation using the result from (a), and (c) finding maximum value of a rational expression by recognizing when the denominator is minimized. All parts follow predictable patterns taught in P2 with no novel insights required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(R = 25\) | B1 | |
| Use appropriate trigonometry to find \(\alpha\) | M1 | Allow if found in radians |
| Obtain \(\alpha = 73.74\) | A1 | Or greater accuracy |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use correct method to find one value of \(\theta\) | M1 | |
| Obtain 29.8 (or 117.7) | A1 | Or greater accuracy |
| Use correct method to find second value of \(\theta\) between 0 and 360 | M1 | |
| Obtain 117.7 (or 29.8) | A1 | Or greater accuracy; and no others between 0 and 360 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply expression is \(\frac{150}{25\cos(\frac{1}{2}\beta - 73.74) + 50}\) | B1 FT | Following *their* \(R\) and \(\alpha\) |
| Obtain \(V = 6\) | B1 | |
| Attempt complete method to find positive value from \(\cos(\frac{1}{2}\beta - 73.74) = -1\) | M1 | For *their* \(\alpha\) |
| Obtain 507.5 | A1 | Or greater accuracy |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $R = 25$ | B1 | |
| Use appropriate trigonometry to find $\alpha$ | M1 | Allow if found in radians |
| Obtain $\alpha = 73.74$ | A1 | Or greater accuracy |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct method to find one value of $\theta$ | M1 | |
| Obtain 29.8 (or 117.7) | A1 | Or greater accuracy |
| Use correct method to find second value of $\theta$ between 0 and 360 | M1 | |
| Obtain 117.7 (or 29.8) | A1 | Or greater accuracy; and no others between 0 and 360 |
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply expression is $\frac{150}{25\cos(\frac{1}{2}\beta - 73.74) + 50}$ | B1 FT | Following *their* $R$ and $\alpha$ |
| Obtain $V = 6$ | B1 | |
| Attempt complete method to find positive value from $\cos(\frac{1}{2}\beta - 73.74) = -1$ | M1 | For *their* $\alpha$ |
| Obtain 507.5 | A1 | Or greater accuracy |7
\begin{enumerate}[label=(\alph*)]
\item Express $7 \cos \theta + 24 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $\alpha$ correct to 2 decimal places.
\item Solve the equation $7 \cos \theta + 24 \sin \theta = 18$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
\item As $\beta$ varies, the greatest possible value of
$$\frac { 150 } { 7 \cos \frac { 1 } { 2 } \beta + 24 \sin \frac { 1 } { 2 } \beta + 50 }$$
is denoted by $V$.\\
Find the value of $V$ and determine the smallest positive value of $\beta$ (in degrees) for which the value of $V$ occurs.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q7 [11]}}