| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - trigonometric functions |
| Difficulty | Standard +0.3 This is a straightforward multi-part differentiation question requiring product rule, chain rule, and basic trigonometric identities. Part (i) is routine differentiation, part (ii) guides students through algebraic manipulation with a helpful hint, and part (iii) involves standard equation solving. The question requires multiple techniques but follows a well-signposted path with no novel insights needed, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate derivative to zero and use correct \(\cos(A+B)\) formula | M1 | |
| Obtain the given equation | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to solve for \(x\) | M1 | |
| Obtain answer, e.g. \(x = \frac{1}{12}\pi\) | A1 | |
| Obtain second answer, e.g. \(\frac{7}{12}\pi\), and no other | A1 | |
| Total: 3 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| **Total: 2** | | |
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate derivative to zero and use correct $\cos(A+B)$ formula | M1 | |
| Obtain the given equation | A1 | |
| **Total: 2** | | |
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to solve for $x$ | M1 | |
| Obtain answer, e.g. $x = \frac{1}{12}\pi$ | A1 | |
| Obtain second answer, e.g. $\frac{7}{12}\pi$, and no other | A1 | |
| **Total: 3** | | |7 The curve $y = \sin \left( x + \frac { 1 } { 3 } \pi \right) \cos x$ has two stationary points in the interval $0 \leqslant x \leqslant \pi$.\\
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) By considering the formula for $\cos ( A + B )$, show that, at the stationary points on the curve, $\cos \left( 2 x + \frac { 1 } { 3 } \pi \right) = 0$.\\
(iii) Hence find the exact $x$-coordinates of the stationary points.\\
\hfill \mbox{\textit{CAIE P3 2019 Q7 [7]}}