| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Convergence conditions |
| Difficulty | Standard +0.3 Part (a) is a routine arithmetic progression problem requiring a simple inequality. Part (b) requires understanding that convergence needs |r| < 1, finding r = (2cos θ)/√3, and solving an absolute value inequality with trigonometric functions. While part (b) involves multiple concepts, the steps are straightforward for A-level students familiar with GP convergence conditions and basic trigonometry, making this slightly easier than average overall. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2222/17\) (\(=131\) or \(130.7\)) | M1 | Ignore signs. Allow \(2239/17 \to 131.7\) or \(132\) |
| \(131 \times 17\) (\(=2227\)) | M1 | Ignore signs. Use 131. |
| \(-2222 + 2227 = 5\) | A1 [3] | 5 www gets 3/3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = \frac{2\cos\theta}{\sqrt{3}}\) soi oe | B1 | |
| \((-1<)\frac{2\cos\theta}{\sqrt{3}} < 1\) or \((0<)\frac{2\cos\theta}{\sqrt{3}} < 1\) soi | M1\(\checkmark\) | Ft on *their* \(r\). Ignore a 2nd inequality on LHS |
| \(\pi/6,\ 5\pi/6\) soi (but dep. on M1) | A1A1 | Allow 30°, 150°. Accept \(\leq\) |
| \(\pi/6 < \theta < 5\pi/6\) cao | A1 [5] |
## Question 9(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2222/17$ ($=131$ or $130.7$) | M1 | Ignore signs. Allow $2239/17 \to 131.7$ or $132$ |
| $131 \times 17$ ($=2227$) | M1 | Ignore signs. Use 131. |
| $-2222 + 2227 = 5$ | A1 [3] | 5 www gets 3/3 |
## Question 9(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = \frac{2\cos\theta}{\sqrt{3}}$ soi oe | B1 | |
| $(-1<)\frac{2\cos\theta}{\sqrt{3}} < 1$ or $(0<)\frac{2\cos\theta}{\sqrt{3}} < 1$ soi | M1$\checkmark$ | Ft on *their* $r$. Ignore a 2nd inequality on LHS |
| $\pi/6,\ 5\pi/6$ soi (but dep. on M1) | A1A1 | Allow 30°, 150°. Accept $\leq$ |
| $\pi/6 < \theta < 5\pi/6$ cao | A1 [5] | |
---9
\begin{enumerate}[label=(\alph*)]
\item The first term of an arithmetic progression is - 2222 and the common difference is 17 . Find the value of the first positive term.
\item The first term of a geometric progression is $\sqrt { } 3$ and the second term is $2 \cos \theta$, where $0 < \theta < \pi$. Find the set of values of $\theta$ for which the progression is convergent.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2015 Q9 [8]}}