| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Convert equation to quadratic form |
| Difficulty | Standard +0.3 This is a standard trigonometric equation requiring conversion to quadratic form using tan x = sin x/cos x and sin²x + cos²x = 1. Part (i) involves routine algebraic manipulation and solving a quadratic, while part (ii) applies the same method with a simple substitution (2x for x). The techniques are well-practiced at A-level with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(3\sin^2 x-\cos^2 x+\cos x=0\) | M1 | Multiply by \(\cos x\) |
| Use \(s^2=1-c^2\) and simplify to 3-term quad | M1 | Expect \(4c^2-c-3=0\) |
| \(\cos x=-3/4\) and \(1\) | A1 | |
| \(x=2.42\) (allow \(0.77\pi\)) or \(0\) (extra in range max 1) | A1A1 [5] | SC1 for \(0.723\) (or \(0.23\pi\)), \(\pi\) following \(4c^2+c-3=0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2x=2\pi-\textit{their}\ 2.42\) or \(360-138.6\) | B1\(\checkmark\) | Expect \(2x=3.86\) |
| \(x=1.21\ (0.385\pi),\ 1.93\ (0.614/5\pi),\ 0,\ \pi\ (3.14)\) (extra max 1) | B1B1 [3] | Any 2 correct B1. Remaining 2 correct B1. SCB1 for all \(69.3°\), \(110.7°\), \(0°\), \(180°\). SCB1 for \(.361\), \(\pi/2\), \(2.78\) after \(4c^2+c-3=0\) |
## Question 8(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $3\sin^2 x-\cos^2 x+\cos x=0$ | M1 | Multiply by $\cos x$ |
| Use $s^2=1-c^2$ and simplify to 3-term quad | M1 | Expect $4c^2-c-3=0$ |
| $\cos x=-3/4$ and $1$ | A1 | |
| $x=2.42$ (allow $0.77\pi$) or $0$ (extra in range max 1) | A1A1 [5] | SC1 for $0.723$ (or $0.23\pi$), $\pi$ following $4c^2+c-3=0$ |
## Question 8(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2x=2\pi-\textit{their}\ 2.42$ or $360-138.6$ | B1$\checkmark$ | Expect $2x=3.86$ |
| $x=1.21\ (0.385\pi),\ 1.93\ (0.614/5\pi),\ 0,\ \pi\ (3.14)$ (extra max 1) | B1B1 [3] | Any 2 correct B1. Remaining 2 correct B1. SCB1 for all $69.3°$, $110.7°$, $0°$, $180°$. SCB1 for $.361$, $\pi/2$, $2.78$ after $4c^2+c-3=0$ |
---8 (i) Show that $3 \sin x \tan x - \cos x + 1 = 0$ can be written as a quadratic equation in $\cos x$ and hence solve the equation $3 \sin x \tan x - \cos x + 1 = 0$ for $0 \leqslant x \leqslant \pi$.\\
(ii) Find the solutions to the equation $3 \sin 2 x \tan 2 x - \cos 2 x + 1 = 0$ for $0 \leqslant x \leqslant \pi$.
\hfill \mbox{\textit{CAIE P1 2016 Q8 [8]}}