| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Integrate using double angle |
| Difficulty | Standard +0.3 Part (a) is a standard algebraic manipulation using difference of squares and Pythagorean identity to derive a double angle formula. Part (b) applies this result with recognition that 4sin²θcos²θ = sin²2θ, requiring double angle formula for integration—a routine P3 technique. The question is slightly easier than average due to the guided structure and standard methods, though it does require connecting multiple ideas. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Factorise LHS using difference of 2 squares | *M1 | \(\left(\cos^2\theta - \sin^2\theta\right)\left(\cos^2\theta + \sin^2\theta\right)\) |
| Simplify | DM1 | \(\cos^2\theta + \sin^2\theta = 1\) must be seen or implied, e.g. \(\left(\cos^2\theta - \sin^2\theta\right)\left(\cos^2\theta + \sin^2\theta\right) = \left(\cos^2\theta - \sin^2\theta\right)\) |
| Obtain \(\cos^4\theta - \sin^4\theta \equiv \cos 2\theta\) from correct working | A1 | AG |
| Alternative Method 1: Use of correct rearrangements of double angle formulae | (*M1) | E.g. \(\left(\frac{1+\cos 2\theta}{2}\right)^2 - \left(\frac{1-\cos 2\theta}{2}\right)^2\). Only condone \(\left(\frac{1+\cos 2\theta}{2}\right)^2 - \left(\frac{\cos 2\theta - 1}{2}\right)^2\) if correct expression for \(\sin^2\theta\) seen. |
| Expand and simplify | (DM1) | Collect like terms. Condone recovery from missing brackets. |
| Obtain \(\cos^4\theta - \sin^4\theta \equiv \cos 2\theta\) from correct working | (A1) | AG |
| Alternative Method 2: Correct use of Pythagoras | (*M1) | E.g. \(\left(1-\sin^2\theta\right)^2 - \sin^4\theta\) or \(\cos^2\theta\left(1-\sin^2\theta\right) - \sin^2\theta\left(1-\cos^2\theta\right)\) |
| Expand and simplify | (DM1) | Condone recovery from missing brackets. |
| Obtain \(\cos^4\theta - \sin^4\theta \equiv \cos 2\theta\) from correct working | (A1) | AG |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use part (a) and correct double angle formula to obtain expression involving \(\int \sin^2 2\theta\,\mathrm{d}\theta\) or \(\int \cos^2 2\theta\,\mathrm{d}\theta\) | M1 | \(\int \cos^4\theta - \sin^4\theta + 4\sin^2\theta\cos^2\theta\,\mathrm{d}\theta = \int \cos 2\theta + \sin^2 2\theta\,\mathrm{d}\theta\). Allow BOD for \(2\sin^2 2\theta\) if \(\sin 2\theta = 2\sin\theta\cos\theta\) seen. |
| \(\int \cos 2\theta\,\mathrm{d}\theta = \frac{1}{2}\sin 2\theta\) | B1 | Seen or implied. |
| Use of correct double angle formula on second part of the integral to obtain a form that can be integrated directly | M1 | E.g. \(\int \sin^2 2\theta\,\mathrm{d}\theta = \int \frac{1-\cos 4\theta}{2}\,\mathrm{d}\theta\) |
| Obtain \(\frac{1}{2}\theta - \frac{1}{8}\sin 4\theta\) | A1 | Condone a mixture of \(x\) and \(\theta\). |
| Correct use of limits \(\pm\frac{\pi}{8}\) in an expression of the form \(p\theta + q\sin 2\theta + r\sin 4\theta\) and evaluate the trig | M1 | \(\left(2\left(\frac{1}{2}\times\frac{1}{\sqrt{2}} + \frac{\pi}{16} - \frac{1}{8}\right)\right)\) |
| Obtain \(\frac{1}{2}\sqrt{2} + \frac{1}{8}\pi - \frac{1}{4}\) | A1 | ISW. Or exact equivalent from exact working. |
| 6 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Factorise LHS using difference of 2 squares | *M1 | $\left(\cos^2\theta - \sin^2\theta\right)\left(\cos^2\theta + \sin^2\theta\right)$ |
| Simplify | DM1 | $\cos^2\theta + \sin^2\theta = 1$ must be seen or implied, e.g. $\left(\cos^2\theta - \sin^2\theta\right)\left(\cos^2\theta + \sin^2\theta\right) = \left(\cos^2\theta - \sin^2\theta\right)$ |
| Obtain $\cos^4\theta - \sin^4\theta \equiv \cos 2\theta$ from correct working | A1 | AG |
| **Alternative Method 1:** Use of correct rearrangements of double angle formulae | (*M1) | E.g. $\left(\frac{1+\cos 2\theta}{2}\right)^2 - \left(\frac{1-\cos 2\theta}{2}\right)^2$. Only condone $\left(\frac{1+\cos 2\theta}{2}\right)^2 - \left(\frac{\cos 2\theta - 1}{2}\right)^2$ if correct expression for $\sin^2\theta$ seen. |
| Expand and simplify | (DM1) | Collect like terms. Condone recovery from missing brackets. |
| Obtain $\cos^4\theta - \sin^4\theta \equiv \cos 2\theta$ from correct working | (A1) | AG |
| **Alternative Method 2:** Correct use of Pythagoras | (*M1) | E.g. $\left(1-\sin^2\theta\right)^2 - \sin^4\theta$ or $\cos^2\theta\left(1-\sin^2\theta\right) - \sin^2\theta\left(1-\cos^2\theta\right)$ |
| Expand and simplify | (DM1) | Condone recovery from missing brackets. |
| Obtain $\cos^4\theta - \sin^4\theta \equiv \cos 2\theta$ from correct working | (A1) | AG |
| | **3** | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use part (a) and correct double angle formula to obtain expression involving $\int \sin^2 2\theta\,\mathrm{d}\theta$ or $\int \cos^2 2\theta\,\mathrm{d}\theta$ | M1 | $\int \cos^4\theta - \sin^4\theta + 4\sin^2\theta\cos^2\theta\,\mathrm{d}\theta = \int \cos 2\theta + \sin^2 2\theta\,\mathrm{d}\theta$. Allow BOD for $2\sin^2 2\theta$ if $\sin 2\theta = 2\sin\theta\cos\theta$ seen. |
| $\int \cos 2\theta\,\mathrm{d}\theta = \frac{1}{2}\sin 2\theta$ | B1 | Seen or implied. |
| Use of correct double angle formula on second part of the integral to obtain a form that can be integrated directly | M1 | E.g. $\int \sin^2 2\theta\,\mathrm{d}\theta = \int \frac{1-\cos 4\theta}{2}\,\mathrm{d}\theta$ |
| Obtain $\frac{1}{2}\theta - \frac{1}{8}\sin 4\theta$ | A1 | Condone a mixture of $x$ and $\theta$. |
| Correct use of limits $\pm\frac{\pi}{8}$ in an expression of the form $p\theta + q\sin 2\theta + r\sin 4\theta$ and evaluate the trig | M1 | $\left(2\left(\frac{1}{2}\times\frac{1}{\sqrt{2}} + \frac{\pi}{16} - \frac{1}{8}\right)\right)$ |
| Obtain $\frac{1}{2}\sqrt{2} + \frac{1}{8}\pi - \frac{1}{4}$ | A1 | ISW. Or exact equivalent from exact working. |
| | **6** | |7
\begin{enumerate}[label=(\alph*)]
\item Show that $\cos ^ { 4 } \theta - \sin ^ { 4 } \theta \equiv \cos 2 \theta$.
\item Hence find the exact value of $\int _ { - \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta + 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \right) \mathrm { d } \theta$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q7 [9]}}