| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Derive triple angle then evaluate integral |
| Difficulty | Standard +0.3 Part (i) is a standard derivation of the triple angle formula using addition formulae—routine for P3 students. Part (ii) requires substituting the identity to create a cubic, but it factors easily. Part (iii) applies the rearranged identity to integrate cos³x, which is a direct application. All parts follow predictable patterns with no novel insight required, making this slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(\cos(A+B)\) formula to express \(\cos3x\) in terms of trig functions of \(2x\) and \(x\) | M1 | |
| Use double angle formulae and Pythagoras to obtain an expression in terms of \(\cos x\) only | M1 | |
| Obtain a correct expression in terms of \(\cos x\) in any form | A1 | |
| Obtain \(\cos3x\equiv 4\cos^3x-3\cos x\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use identity and solve cubic \(4\cos^3x=-1\) for \(x\) | M1 | \(\cos x = -0.6299...\) |
| Obtain answer \(2.25\) and no other in the interval | A1 | Accept \(0.717\pi\); M1A0 for \(129.0°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain indefinite integral \(\frac{1}{12}\sin3x+\frac{3}{4}\sin x\) | B1 + B1 | |
| Substitute limits in an indefinite integral of the form \(a\sin3x+b\sin x\), where \(ab\neq0\) | M1 | \(\frac{1}{4}\left[\frac{1}{3}\sin\pi+3\sin\frac{\pi}{3}-\frac{1}{3}\sin\frac{\pi}{2}-3\sin\frac{\pi}{6}\right]\) |
| Obtain answer \(\frac{1}{24}(9\sqrt{3}-11)\), or exact equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int\cos x(1-\sin^2x)\,dx = \sin x - \frac{1}{3}\sin^3x\,(+C)\) | B1 + B1 | |
| Substitute limits in an indefinite integral of the form \(a\sin x+b\sin^3x\) where \(ab\neq0\) | M1 | \(\left(\frac{\sqrt{3}}{2}-\frac{1}{2}-\frac{1}{4}\cdot\frac{\sqrt{3}}{2}+\frac{1}{24}\right)\) |
| Obtain answer \(\frac{1}{24}(9\sqrt{3}-11)\), or exact equivalent | A1 |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $\cos(A+B)$ formula to express $\cos3x$ in terms of trig functions of $2x$ and $x$ | M1 | |
| Use double angle formulae and Pythagoras to obtain an expression in terms of $\cos x$ only | M1 | |
| Obtain a correct expression in terms of $\cos x$ in any form | A1 | |
| Obtain $\cos3x\equiv 4\cos^3x-3\cos x$ | A1 | AG |
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use identity and solve cubic $4\cos^3x=-1$ for $x$ | M1 | $\cos x = -0.6299...$ |
| Obtain answer $2.25$ and no other in the interval | A1 | Accept $0.717\pi$; M1A0 for $129.0°$ |
## Question 9(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain indefinite integral $\frac{1}{12}\sin3x+\frac{3}{4}\sin x$ | B1 + B1 | |
| Substitute limits in an indefinite integral of the form $a\sin3x+b\sin x$, where $ab\neq0$ | M1 | $\frac{1}{4}\left[\frac{1}{3}\sin\pi+3\sin\frac{\pi}{3}-\frac{1}{3}\sin\frac{\pi}{2}-3\sin\frac{\pi}{6}\right]$ |
| Obtain answer $\frac{1}{24}(9\sqrt{3}-11)$, or exact equivalent | A1 | |
**Alternative method for question 9(iii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int\cos x(1-\sin^2x)\,dx = \sin x - \frac{1}{3}\sin^3x\,(+C)$ | B1 + B1 | |
| Substitute limits in an indefinite integral of the form $a\sin x+b\sin^3x$ where $ab\neq0$ | M1 | $\left(\frac{\sqrt{3}}{2}-\frac{1}{2}-\frac{1}{4}\cdot\frac{\sqrt{3}}{2}+\frac{1}{24}\right)$ |
| Obtain answer $\frac{1}{24}(9\sqrt{3}-11)$, or exact equivalent | A1 | |9 (i) By first expanding $\cos ( 2 x + x )$, show that $\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$.\\
(ii) Hence solve the equation $\cos 3 x + 3 \cos x + 1 = 0$, for $0 \leqslant x \leqslant \pi$.\\
(iii) Find the exact value of $\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 3 } x \mathrm {~d} x$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q9 [10]}}