| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation only (no integral) |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of the compound angle formula for cosine, requiring routine algebraic simplification. Part (ii) applies the result from (i) with substitution (θ=2x and θ=x), simplifies to cos(2x)/cos(x)=3, then uses the double angle formula to solve a quadratic equation. This is a standard multi-step question with clear signposting and no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
| Answer | Marks |
|---|---|
| Use \(\cos(A+B)\) correctly at least once | M1 |
| State correct complete expansion | A1 |
| Confirm given answer \(\cos\theta\) with explicit use of \(\cos 60° = \frac{1}{2}\) | A1 |
| SR: "correct" answer from sign errors in both expansions is B1 only |
| Answer | Marks | Guidance |
|---|---|---|
| Use correct \(\cos A + \cos B\) formula | M1 | |
| State correct result e.g. \(2\cos\left(\frac{2\theta}{2}\right)\cos\left(\frac{-120}{2}\right)\) | A1 | |
| Confirm given answer \(\cos\theta\) with explicit use of \(\cos(\pm 60°) = \frac{1}{2}\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(\frac{\cos 2x}{\cos x} = 3\) | B1 | |
| Obtain equation \(2\cos^2 x - 3\cos x - 1 = 0\) | B1 | |
| Solve a three-term quadratic equation for \(\cos x\) | M1 | |
| Obtain \(\frac{1}{4}(3 - \sqrt{17})\) or exact equivalent and, finally, no other | A1 | [4] |
**(i) Either route:**
Use $\cos(A+B)$ correctly at least once | M1 |
State correct complete expansion | A1 |
Confirm given answer $\cos\theta$ with explicit use of $\cos 60° = \frac{1}{2}$ | A1 |
SR: "correct" answer from sign errors in both expansions is B1 only | |
**Or route:**
Use correct $\cos A + \cos B$ formula | M1 |
State correct result e.g. $2\cos\left(\frac{2\theta}{2}\right)\cos\left(\frac{-120}{2}\right)$ | A1 |
Confirm given answer $\cos\theta$ with explicit use of $\cos(\pm 60°) = \frac{1}{2}$ | A1 | [3]
**(ii)**
State or imply $\frac{\cos 2x}{\cos x} = 3$ | B1 |
Obtain equation $2\cos^2 x - 3\cos x - 1 = 0$ | B1 |
Solve a three-term quadratic equation for $\cos x$ | M1 |
Obtain $\frac{1}{4}(3 - \sqrt{17})$ or exact equivalent and, finally, no other | A1 | [4]4 (i) Show that $\cos \left( \theta - 60 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right) \equiv \cos \theta$.\\
(ii) Given that $\frac { \cos \left( 2 x - 60 ^ { \circ } \right) + \cos \left( 2 x + 60 ^ { \circ } \right) } { \cos \left( x - 60 ^ { \circ } \right) + \cos \left( x + 60 ^ { \circ } \right) } = 3$, find the exact value of $\cos x$.
\hfill \mbox{\textit{CAIE P3 2014 Q4 [7]}}