| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Sum of first n terms |
| Difficulty | Moderate -0.3 This is a straightforward C2 geometric series question with standard applications. Part (i) uses basic sector formulas (A = ½r²θ), parts (ii)(a-b) apply the standard GP sum formula with clearly given ratio 3/5, and part (ii)(c) tests understanding of sum to infinity. All steps are routine with no novel insight required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<11.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2} \times r^2 \times 1.2 = 60\) | M1 | Attempt \(\frac{1}{2}r^2\theta = 60\) |
| \(r = 10\) | A1 | Obtain \(r = 10\) |
| \(r\theta = 10 \times 1.2 = 12\) | B1\(\checkmark\) | State or imply arc length is \(1.2r\), following their \(r\) |
| perimeter \(= 10 + 10 + 12 = 32\) cm | A1 (4) | Obtain \(32\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_5 = 60 \times 0.6^4 = 7.78\) | M1 | Attempt \(u_5\) using \(ar^4\), or list terms |
| A1 (2) | Obtain \(7.78\), or better |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_{10} = \dfrac{60(1 - 0.6^{10})}{1 - 0.6} = 149\) | M1 | Attempt use of correct sum formula for a GP, or sum terms |
| A1 (2) | Obtain \(149\), or better (allow \(149.0 - 149.2\) inclusive) |
| Answer | Marks | Guidance |
|---|---|---|
| Common ratio is less than 1, so series is convergent and hence sum to infinity exists | B1 | Series is convergent or \(-1 < r < 1\) (allow \(r < 1\)) or reference to areas getting smaller/adding on less each time |
| \(S_\infty = \dfrac{60}{1 - 0.6} = 150\) | M1 | Attempt \(S_\infty\) using \(\dfrac{a}{1-r}\) |
| A1 (3) | Obtain \(S_\infty = 150\) | |
| SR | B1 only for \(150\) with no method shown |
## Question 8:
### Part (i):
| $\frac{1}{2} \times r^2 \times 1.2 = 60$ | M1 | Attempt $\frac{1}{2}r^2\theta = 60$ |
|---|---|---|
| $r = 10$ | A1 | Obtain $r = 10$ |
| $r\theta = 10 \times 1.2 = 12$ | B1$\checkmark$ | State or imply arc length is $1.2r$, following their $r$ |
| perimeter $= 10 + 10 + 12 = 32$ cm | A1 **(4)** | Obtain $32$ |
### Part (ii)(a):
| $u_5 = 60 \times 0.6^4 = 7.78$ | M1 | Attempt $u_5$ using $ar^4$, or list terms |
|---|---|---|
| | A1 **(2)** | Obtain $7.78$, or better |
### Part (ii)(b):
| $S_{10} = \dfrac{60(1 - 0.6^{10})}{1 - 0.6} = 149$ | M1 | Attempt use of correct sum formula for a GP, or sum terms |
|---|---|---|
| | A1 **(2)** | Obtain $149$, or better (allow $149.0 - 149.2$ inclusive) |
### Part (c):
| Common ratio is less than 1, so series is convergent and hence sum to infinity exists | B1 | Series is convergent or $-1 < r < 1$ (allow $r < 1$) or reference to areas getting smaller/adding on less each time |
|---|---|---|
| $S_\infty = \dfrac{60}{1 - 0.6} = 150$ | M1 | Attempt $S_\infty$ using $\dfrac{a}{1-r}$ |
| | A1 **(3)** | Obtain $S_\infty = 150$ |
| | SR | B1 only for $150$ with no method shown |
---8
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_378_467_269_840}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Fig. 1 shows a sector $A O B$ of a circle, centre $O$ and radius $O A$. The angle $A O B$ is 1.2 radians and the area of the sector is $60 \mathrm {~cm} ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item Find the perimeter of the sector.
A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors. The first sector in the pattern is sector $A O B$ from Fig. 1, and the area of each successive sector is $\frac { 3 } { 5 }$ of the area of the previous one.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_362_1011_1263_568}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
\item (a) Find the area of the fifth sector in the pattern.\\
(b) Find the total area of the first ten sectors in the pattern.\\
(c) Explain why the total area will never exceed a certain limit, no matter how many sectors are used, and state the value of this limit.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2009 Q8 [11]}}