| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Trigonometric arithmetic progression |
| Difficulty | Standard +0.8 This question combines arithmetic progressions with trigonometry, requiring students to set up equations using the AP property (equal common differences), solve a trigonometric equation involving both sin and cos in a restricted domain, then apply the sum formula. The algebraic manipulation and trigonometric solving elevate this above a routine AP question, but it follows a clear method once the AP property is recognized. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([(3^\text{rd}\text{ term}-1^\text{st}\text{ term})=(5^\text{th}\text{ term}-3^\text{rd}\text{ term})\) leading to...] \(-6\sqrt{3}\sin x-2\cos x=10\cos x+6\sqrt{3}\sin x\) [leading to \(-12\sqrt{3}\sin x=12\cos x\)] OR \([(1^\text{st}\text{ term}+5^\text{th}\text{ term})=2\times3^\text{rd}\text{ term}\) leading to...] \(12\cos x=-12\sqrt{3}\sin x\) | *M1 | OE. From the given terms, obtain 2 expressions relating to the common difference of the arithmetic progression, attempt to solve them simultaneously and achieve an equation just involving \(\sin x\) and \(\cos x\) |
| Elimination of \(\sin x\) and \(\cos x\) to give an expression in \(\tan x\): \(\left[\tan x=-\frac{1}{\sqrt{3}}\right]\) | DM1 | For use of \(\frac{\sin x}{\cos x}=\tan x\) |
| \(\left[x=\right]\dfrac{5\pi}{6}\) only | A1 | CAO. Must be exact |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(d=2\cos x\) or \(d=2\cos(\textit{their}\,x)\) | B1 FT | Or an equivalent expression involving \(\sin x\) and \(\cos x\) e.g. \(-3\sqrt{3}\sin(\textit{their}\,x)-\cos(\textit{their}\,x)\left[=-\sqrt{3}\right]\). FT for *their* \(x\) from (a) only. If not \(\pm\sqrt{3}\), must see unevaluated form |
| \(S_{25}=\frac{25}{2}\Big(2\times(2\cos(\textit{their}\,x))+(25-1)\times(\textit{their}\,d)\Big)\) \(\left[=12.5\Big(2\times(-\sqrt{3})+24(-\sqrt{3})\Big)\right]\) | M1 | Using the correct sum formula with \(\frac{25}{2}\), \((25-1)\) and with \(a\) replaced by either \(2\cos(\textit{their}\,x)\) or \(\pm\sqrt{3}\) and \(d\) replaced by either \(2\cos(\textit{their}\,x)\) or \(\pm\sqrt{3}\) |
| \(-325\sqrt{3}\) | A1 | Must be exact |
| Total | 3 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[(3^\text{rd}\text{ term}-1^\text{st}\text{ term})=(5^\text{th}\text{ term}-3^\text{rd}\text{ term})$ leading to...] $-6\sqrt{3}\sin x-2\cos x=10\cos x+6\sqrt{3}\sin x$ [leading to $-12\sqrt{3}\sin x=12\cos x$] OR $[(1^\text{st}\text{ term}+5^\text{th}\text{ term})=2\times3^\text{rd}\text{ term}$ leading to...] $12\cos x=-12\sqrt{3}\sin x$ | *M1 | OE. From the given terms, obtain 2 expressions relating to the common difference of the arithmetic progression, attempt to solve them simultaneously and achieve an equation just involving $\sin x$ and $\cos x$ |
| Elimination of $\sin x$ and $\cos x$ to give an expression in $\tan x$: $\left[\tan x=-\frac{1}{\sqrt{3}}\right]$ | DM1 | For use of $\frac{\sin x}{\cos x}=\tan x$ |
| $\left[x=\right]\dfrac{5\pi}{6}$ only | A1 | CAO. Must be exact |
| **Total** | **3** | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $d=2\cos x$ or $d=2\cos(\textit{their}\,x)$ | B1 FT | Or an equivalent expression involving $\sin x$ and $\cos x$ e.g. $-3\sqrt{3}\sin(\textit{their}\,x)-\cos(\textit{their}\,x)\left[=-\sqrt{3}\right]$. FT for *their* $x$ from (a) only. If not $\pm\sqrt{3}$, must see unevaluated form |
| $S_{25}=\frac{25}{2}\Big(2\times(2\cos(\textit{their}\,x))+(25-1)\times(\textit{their}\,d)\Big)$ $\left[=12.5\Big(2\times(-\sqrt{3})+24(-\sqrt{3})\Big)\right]$ | M1 | Using the correct sum formula with $\frac{25}{2}$, $(25-1)$ and with $a$ replaced by either $2\cos(\textit{their}\,x)$ or $\pm\sqrt{3}$ and $d$ replaced by either $2\cos(\textit{their}\,x)$ or $\pm\sqrt{3}$ |
| $-325\sqrt{3}$ | A1 | Must be exact |
| **Total** | **3** | |5 The first, third and fifth terms of an arithmetic progression are $2 \cos x , - 6 \sqrt { 3 } \sin x$ and $10 \cos x$ respectively, where $\frac { 1 } { 2 } \pi < x < \pi$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $x$.
\item Hence find the exact sum of the first 25 terms of the progression.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q5 [6]}}