| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Express cos²x or sin²x in terms of cos 2x |
| Difficulty | Standard +0.3 This is a straightforward application of double angle formulae requiring students to recognize that sin²x cos²x = (sin 2x)²/4, then apply the cos 2θ identity. The integration in part (ii) follows directly from part (i). While it requires knowing multiple identities and their manipulation, it's a standard textbook exercise with clear signposting and no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8\sin^2x\cos^2x = 2(1-\cos2x)(1+\cos2x)\) | M1 | Using a double angle formula |
| \(= 2(1-\cos^22x) = 2-(1+2\cos4x)\) | M1 | Second use of a double angle formula |
| \(= 1-\cos4x\) | E1 [3] | Clearly shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8\sin^2x\cos^2x = 2(2\sin x\cos x)^2\) | M1 | Using a double angle formula |
| \(= 2\sin^22x\) | M1 | Another use of a double angle formula |
| \([=1-\cos2(2x)] = 1-\cos4x\) | E1 [3] | Clearly shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1-\cos4x = 1-(1-2\sin^22x)\) | M1 | Using a double angle formula |
| \(= 2\sin^22x = 2(2\sin x\cos x)^2\) | M1 | Another use of a double angle formula |
| \(= 8\sin^2x\cos^2x\) | E1 [3] | Clearly shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int\sin^2x\cos^2x\,\text{d}x = \dfrac{1}{8}\int 1-\cos4x\,\text{d}x\) | M1 | Attempt to integrate both terms |
| A1 | \(\dfrac{1}{4}\sin4x\) seen or implied | |
| \(= \dfrac{1}{8}x - \dfrac{1}{32}\sin4x + c\) | A1 [3] | All correct; must include \(+c\) |
## Question 8(i):
**EITHER method 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8\sin^2x\cos^2x = 2(1-\cos2x)(1+\cos2x)$ | M1 | Using a double angle formula |
| $= 2(1-\cos^22x) = 2-(1+2\cos4x)$ | M1 | Second use of a double angle formula |
| $= 1-\cos4x$ | E1 [3] | Clearly shown |
**OR method 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8\sin^2x\cos^2x = 2(2\sin x\cos x)^2$ | M1 | Using a double angle formula |
| $= 2\sin^22x$ | M1 | Another use of a double angle formula |
| $[=1-\cos2(2x)] = 1-\cos4x$ | E1 [3] | Clearly shown |
**OR method 3:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1-\cos4x = 1-(1-2\sin^22x)$ | M1 | Using a double angle formula |
| $= 2\sin^22x = 2(2\sin x\cos x)^2$ | M1 | Another use of a double angle formula |
| $= 8\sin^2x\cos^2x$ | E1 [3] | Clearly shown |
## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int\sin^2x\cos^2x\,\text{d}x = \dfrac{1}{8}\int 1-\cos4x\,\text{d}x$ | M1 | Attempt to integrate both terms |
| | A1 | $\dfrac{1}{4}\sin4x$ seen or implied |
| $= \dfrac{1}{8}x - \dfrac{1}{32}\sin4x + c$ | A1 [3] | All correct; must include $+c$ |
---8 (i) Show that $8 \sin ^ { 2 } x \cos ^ { 2 } x$ can be written as $1 - \cos 4 x$.\\
(ii) Hence find $\int \sin ^ { 2 } x \cos ^ { 2 } x \mathrm {~d} x$.
\hfill \mbox{\textit{OCR MEI Paper 1 2018 Q8 [6]}}