| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Solve reciprocal trig equation |
| Difficulty | Challenging +1.2 This is a multi-part harmonic form question requiring standard techniques: (i) converting to R cos(θ+α) form using standard identities, (ii) manipulating a reciprocal trig equation to use part (i), and (iii) integrating using substitution and the sec² identity. While it requires multiple steps and careful algebraic manipulation across three parts, all techniques are standard A-level further maths content with no novel insights required. The reciprocal trig manipulation and integration push it slightly above average difficulty. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(R = \sqrt{29}\) or \(5.385\ldots\) | B1 | |
| Use appropriate trigonometry to find \(\alpha\) | M1 | Allow M1 for \(\tan\alpha = \pm\frac{2}{5}\) or \(\pm\frac{5}{2}\) oe |
| Obtain \(0.3805\) with no errors seen | A1 | Or greater accuracy \(0.3805063\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that equation is \(5\cos\theta - 2\sin\theta = 4\) | B1 | |
| Evaluate \(\cos^{-1}(k/R) - \alpha\) to find one value of \(\theta\) | M1 | Allow M1 from their \(\sqrt{29}\cos(\theta \pm \alpha)\) |
| Obtain \(0.353\) | A1 | Or greater accuracy \(0.35307\ldots\) |
| Carry out correct method to find second value | M1 | |
| Obtain \(5.17\) and no extra solutions in the range | A1 | Or greater accuracy \(5.16909\ldots\) If working consistently in degrees, then no A marks are available, B1, M1, M1 max |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State integrand as \(\frac{1}{29}\sec^2\!\left(\frac{1}{2}x + 0.3805\right)\) | B1 FT | Following their answer from part (i), must be in the form \(R\cos(\theta \pm \alpha)\) |
| Integrate to obtain form \(k\tan\!\left(\frac{1}{2}x + \text{their } \alpha\right)\) | M1 | |
| Obtain \(\frac{2}{29}\tan\!\left(\frac{1}{2}x + 0.3805\right) + c\) | A1 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $R = \sqrt{29}$ or $5.385\ldots$ | B1 | |
| Use appropriate trigonometry to find $\alpha$ | M1 | Allow M1 for $\tan\alpha = \pm\frac{2}{5}$ or $\pm\frac{5}{2}$ oe |
| Obtain $0.3805$ with no errors seen | A1 | Or greater accuracy $0.3805063\ldots$ |
**Total: 3 marks**
---
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that equation is $5\cos\theta - 2\sin\theta = 4$ | B1 | |
| Evaluate $\cos^{-1}(k/R) - \alpha$ to find one value of $\theta$ | M1 | Allow M1 from their $\sqrt{29}\cos(\theta \pm \alpha)$ |
| Obtain $0.353$ | A1 | Or greater accuracy $0.35307\ldots$ |
| Carry out correct method to find second value | M1 | |
| Obtain $5.17$ and no extra solutions in the range | A1 | Or greater accuracy $5.16909\ldots$ If working consistently in degrees, then no A marks are available, B1, M1, M1 max |
**Total: 5 marks**
---
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State integrand as $\frac{1}{29}\sec^2\!\left(\frac{1}{2}x + 0.3805\right)$ | B1 FT | Following their answer from part (i), must be in the form $R\cos(\theta \pm \alpha)$ |
| Integrate to obtain form $k\tan\!\left(\frac{1}{2}x + \text{their } \alpha\right)$ | M1 | |
| Obtain $\frac{2}{29}\tan\!\left(\frac{1}{2}x + 0.3805\right) + c$ | A1 | |
**Total: 3 marks**7 (i) Express $5 \cos \theta - 2 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Give the value of $\alpha$ correct to 4 decimal places.\\
(ii) Using your answer from part (i), solve the equation
$$5 \cot \theta - 4 \operatorname { cosec } \theta = 2$$
for $0 < \theta < 2 \pi$.\\
(iii) Find $\int \frac { 1 } { \left( 5 \cos \frac { 1 } { 2 } x - 2 \sin \frac { 1 } { 2 } x \right) ^ { 2 } } \mathrm {~d} x$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P2 2018 Q7 [11]}}