| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Integrate using double angle |
| Difficulty | Moderate -0.3 This is a straightforward application of the double angle formula cos²θ = (1+cos2θ)/2 to simplify before differentiating and integrating. Part (i) requires chain rule differentiation and evaluating at a specific point. Part (ii) is a standard integration after using the identity. Both parts are routine techniques covered extensively in P3 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain derivative of form \(k\cos 3x\sin 3x\), any constant \(k\) | M1 | |
| Obtain \(-24\cos 3x\sin 3x\) or unsimplified equivalent | A1 | |
| Obtain \(-6\sqrt{3}\) or exact equivalent | A1 | [3] |
| (ii) Express integrand in the form \(a + b\cos 6x\), where \(ab \neq 0\) | M1 | |
| Obtain \(2 + 2\cos 6x\) o.e. | A1 | |
| Obtain \(2x + \frac{1}{3}\sin 6x\) or equivalent, conditioning absence of \(+ c\), ft on \(a, b\) | A1 | [3] |
**(i)** Obtain derivative of form $k\cos 3x\sin 3x$, any constant $k$ | M1 |
Obtain $-24\cos 3x\sin 3x$ or unsimplified equivalent | A1 |
Obtain $-6\sqrt{3}$ or exact equivalent | A1 | [3]
**(ii)** Express integrand in the form $a + b\cos 6x$, where $ab \neq 0$ | M1 |
Obtain $2 + 2\cos 6x$ o.e. | A1 |
Obtain $2x + \frac{1}{3}\sin 6x$ or equivalent, conditioning absence of $+ c$, ft on $a, b$ | A1 | [3]4 It is given that $\mathrm { f } ( x ) = 4 \cos ^ { 2 } 3 x$.\\
(i) Find the exact value of $\mathrm { f } ^ { \prime } \left( \frac { 1 } { 9 } \pi \right)$.\\
(ii) Find $\int \mathrm { f } ( x ) \mathrm { d } x$.
\hfill \mbox{\textit{CAIE P3 2010 Q4 [6]}}