| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 Part (i) requires recognizing and applying the sum of cubes factorization (a³+b³=(a+b)(a²-ab+b²)), which is a standard algebraic identity. Part (ii) involves substituting the proven identity and solving a cubic equation in sin x, requiring careful algebraic manipulation but following predictable steps. This is slightly easier than average as it's a guided two-part question with a clear pathway once the key identity is recognized. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin^2 x = 1 - \cos^2 x\) and \(\cos^2 x = 1 - \sin^2 x \rightarrow \sin^2 x + \cos^2 x\) | M1, M1, A1 [3] | Needs 4 terms from the product. Needs to be used once. All ok. |
| Answer | Marks | Guidance |
|---|---|---|
| Uses part (i) \(\rightarrow 8 \sin^3 x = \cos^3 x \rightarrow \tan^3 x = \frac{1}{8} \rightarrow \tan x = \frac{1}{2} \rightarrow x = 26.6°\) and \(206.6°\) | M1, A1 B1√ [3] | Uses \(\tan x = \sin x \div \cos x \rightarrow \tan^3 x = k\). Co. √ for 180° + first answer and providing there are no other answers in range. |
**(i)** $(\sin x + \cos x)(1 - \sin x \cos x) = \sin x + \cos x - \sin^2 x \cos x - \cos^2 x \sin x$
$\sin^2 x = 1 - \cos^2 x$ and $\cos^2 x = 1 - \sin^2 x \rightarrow \sin^2 x + \cos^2 x$ | M1, M1, A1 [3] | Needs 4 terms from the product. Needs to be used once. All ok.
**(ii)** $(\sin x + \cos x)(1 - \sin x \cos x) = 9 \sin^3 x$
Uses part (i) $\rightarrow 8 \sin^3 x = \cos^3 x \rightarrow \tan^3 x = \frac{1}{8} \rightarrow \tan x = \frac{1}{2} \rightarrow x = 26.6°$ and $206.6°$ | M1, A1 B1√ [3] | Uses $\tan x = \sin x \div \cos x \rightarrow \tan^3 x = k$. Co. √ for 180° + first answer and providing there are no other answers in range.
**Total: [6]**
---5 (i) Prove the identity $( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x$.\\
(ii) Solve the equation $( \sin x + \cos x ) ( 1 - \sin x \cos x ) = 9 \sin ^ { 3 } x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2009 Q5 [6]}}