| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find sum to infinity |
| Difficulty | Moderate -0.3 Part (a) requires solving simultaneous equations using standard AP formulas (a+4d=18, S_5=75), which is routine. Part (b) involves finding the common ratio from given terms (r³=27/64, so r=3/4), then applying the sum to infinity formula S=a/(1-r). Both parts are straightforward applications of standard formulas with no conceptual challenges, making this slightly easier than average. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a + 4d = 18\) | \(B1\) | co or \(75 = \frac{5}{2}(a+18) \rightarrow a = 12\) etc |
| \(\frac{5}{2}(2a+4d) = 75\) | \(B1\) | co |
| Solution \(\rightarrow a = 12,\ d = 1\frac{1}{2}\) | \(M1\ A1\) | Solution of simultaneous equations, co for both |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 16\) and \(ar^3 = \frac{27}{4}\) | \(B1\) | Needs both of these |
| \(r = \frac{3}{4}\) | ||
| Sum to infinity \(= 64\) | \(M1\ A1\) | Correct formula and \( |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a + 4d = 18$ | $B1$ | co or $75 = \frac{5}{2}(a+18) \rightarrow a = 12$ etc |
| $\frac{5}{2}(2a+4d) = 75$ | $B1$ | co |
| Solution $\rightarrow a = 12,\ d = 1\frac{1}{2}$ | $M1\ A1$ | Solution of simultaneous equations, co for both |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 16$ and $ar^3 = \frac{27}{4}$ | $B1$ | Needs both of these |
| $r = \frac{3}{4}$ | | |
| Sum to infinity $= 64$ | $M1\ A1$ | Correct formula and $|r| < 1$ |
---6
\begin{enumerate}[label=(\alph*)]
\item The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75 . Find the first term and the common difference.
\item The first term of a geometric progression is 16 and the fourth term is $\frac { 27 } { 4 }$. Find the sum to infinity of the progression.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2010 Q6 [7]}}