| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Convergence conditions |
| Difficulty | Standard +0.3 This is a straightforward two-part question testing standard formulas for arithmetic and geometric progressions. Part (a) requires routine application of the AP sum formula with basic trigonometric manipulation. Part (b)(i) tests the standard convergence condition |r| < 1 for a GP, requiring solving an inequality with tan²θ. Part (b)(ii) is direct substitution into the sum to infinity formula. All techniques are standard textbook exercises with no novel insight required, making it slightly easier than average. |
| 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|<11.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S_{10} = \frac{10}{2[2 + 9(\cos^2 x - 1)]}\) | M1 | Correct formula with \(d = \pm(\cos^2 x - 1)\) |
| \(S_{10} = 5[2 - 9\sin^2 x]\) | M1 | Use of \(c^2 + s^2 = 1\) in a correct \(S_{10}\) |
| \(S_{10} = 10 - 45\sin^2 x\) | A1 [3] | Or \(a = 10\), \(b = 45\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((0 <) \frac{1}{3}\tan^2\theta < 1\) oe | M1 | Allow \(<\) |
| \((0 <) \theta < \frac{\pi}{3}\) | A1 [2] | cao. Allow \(<\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(S_\infty = \frac{1}{1 - \frac{1}{3}\tan^2\frac{\pi}{6}}\) | M1 | |
| \(S_\infty = \frac{9}{8}\) or \(1.125\) | A1 [2] | cao |
## Question 7:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S_{10} = \frac{10}{2[2 + 9(\cos^2 x - 1)]}$ | M1 | Correct formula with $d = \pm(\cos^2 x - 1)$ |
| $S_{10} = 5[2 - 9\sin^2 x]$ | M1 | Use of $c^2 + s^2 = 1$ in a correct $S_{10}$ |
| $S_{10} = 10 - 45\sin^2 x$ | A1 [3] | Or $a = 10$, $b = 45$ |
### Part (b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(0 <) \frac{1}{3}\tan^2\theta < 1$ oe | M1 | Allow $<$ |
| $(0 <) \theta < \frac{\pi}{3}$ | A1 [2] | cao. Allow $<$ |
### Part (b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $S_\infty = \frac{1}{1 - \frac{1}{3}\tan^2\frac{\pi}{6}}$ | M1 | |
| $S_\infty = \frac{9}{8}$ or $1.125$ | A1 [2] | cao |
---7
\begin{enumerate}[label=(\alph*)]
\item The first two terms of an arithmetic progression are 1 and $\cos ^ { 2 } x$ respectively. Show that the sum of the first ten terms can be expressed in the form $a - b \sin ^ { 2 } x$, where $a$ and $b$ are constants to be found.
\item The first two terms of a geometric progression are 1 and $\frac { 1 } { 3 } \tan ^ { 2 } \theta$ respectively, where $0 < \theta < \frac { 1 } { 2 } \pi$.
\begin{enumerate}[label=(\roman*)]
\item Find the set of values of $\theta$ for which the progression is convergent.
\item Find the exact value of the sum to infinity when $\theta = \frac { 1 } { 6 } \pi$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q7 [7]}}