| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Trigonometric arithmetic progression |
| Difficulty | Standard +0.3 This is a straightforward application of arithmetic and geometric progression formulas with basic trigonometric manipulation. Part (a) requires finding the common ratio and applying the sum to infinity formula. Part (b) involves finding the common difference and calculating S₁₆, with simple substitution of θ = π/3. The trigonometric identities needed (sin²θ + cos²θ = 1) are standard, and all steps follow routine procedures with no novel insight required. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a): \(r = \cos^2\theta\) | M1 | SOI |
| \(S_\infty = \frac{\sin^2\theta}{1-\cos^2\theta}\) | M1 | |
| \(1\) | A1 | |
| 8(b)(i): \(d = \sin^2\theta\cos^2\theta - \sin^2\theta\) | M1 | |
| \(\sin^2\theta(\cos^2\theta - 1)\) | M1 | |
| \(-\sin^4\theta\) | A1 | |
| 8(b)(ii): Use of \(S_{16} = \frac{16}{2}[2a + 15d]\) | M1 | |
| With both \(a = \frac{3}{4}\) and \(d = -\frac{9}{16}\) | A1 | |
| \(S_{16} = -55\frac{1}{2}\) | A1 |
## Question 8:
**8(a):** $r = \cos^2\theta$ | M1 | SOI
$S_\infty = \frac{\sin^2\theta}{1-\cos^2\theta}$ | M1 |
$1$ | A1 |
**8(b)(i):** $d = \sin^2\theta\cos^2\theta - \sin^2\theta$ | M1 |
$\sin^2\theta(\cos^2\theta - 1)$ | M1 |
$-\sin^4\theta$ | A1 |
**8(b)(ii):** Use of $S_{16} = \frac{16}{2}[2a + 15d]$ | M1 |
With both $a = \frac{3}{4}$ and $d = -\frac{9}{16}$ | A1 |
$S_{16} = -55\frac{1}{2}$ | A1 |
---8 The first term of a progression is $\sin ^ { 2 } \theta$, where $0 < \theta < \frac { 1 } { 2 } \pi$. The second term of the progression is $\sin ^ { 2 } \theta \cos ^ { 2 } \theta$.
\begin{enumerate}[label=(\alph*)]
\item Given that the progression is geometric, find the sum to infinity.\\
It is now given instead that the progression is arithmetic.
\item \begin{enumerate}[label=(\roman*)]
\item Find the common difference of the progression in terms of $\sin \theta$.
\item Find the sum of the first 16 terms when $\theta = \frac { 1 } { 3 } \pi$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q8 [9]}}