| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Convert sin/cos ratio to tan |
| Difficulty | Moderate -0.8 Part (i) is a straightforward manipulation dividing by cos θ to get tan θ = 1/3, then using inverse tan. Part (ii) requires recognizing to divide by cos²2x to get tan²2x = 1/3, then solving tan 2x = ±1/√3 within the extended range for 2x. Both parts are routine textbook exercises requiring standard techniques with no novel problem-solving, making this easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\tan\theta = 1/3\) | M1 | |
| \(\theta = 18.4°\) only | A1 [2] | Ignore solns. outside range \(0\to180\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\tan 2x = (\pm)1/\sqrt{3}\) — Must be sq. root soi | M1 | \(\sin 2x = (\pm)1/2\) or \(\cos 2x = (\pm)\sqrt{3}/2\); using \(c^2+s^2=1\). Not \(\tan x = (\pm)\frac{1}{\sqrt{3}}\) etc. |
| \((x) = 15\) | A1 | ft for \((90 \pm \text{ their } 15)\) or \((180 - \text{their } 15)\) |
| \((x) =\) any correct second value \((75, 105, 165)\) | A1\(\checkmark\) | All four correct. Extra solns in range \(-1\) |
| \((x) =\) cao | A1 [4] |
## Question 4(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan\theta = 1/3$ | M1 | |
| $\theta = 18.4°$ only | A1 [2] | Ignore solns. outside range $0\to180$ |
## Question 4(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\tan 2x = (\pm)1/\sqrt{3}$ — Must be sq. root soi | M1 | $\sin 2x = (\pm)1/2$ **or** $\cos 2x = (\pm)\sqrt{3}/2$; using $c^2+s^2=1$. Not $\tan x = (\pm)\frac{1}{\sqrt{3}}$ etc. |
| $(x) = 15$ | A1 | ft for $(90 \pm \text{ their } 15)$ or $(180 - \text{their } 15)$ |
| $(x) =$ any correct second value $(75, 105, 165)$ | A1$\checkmark$ | All four correct. Extra solns in range $-1$ |
| $(x) =$ cao | A1 [4] | |
---4 (i) Express the equation $3 \sin \theta = \cos \theta$ in the form $\tan \theta = k$ and solve the equation for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.\\
(ii) Solve the equation $3 \sin ^ { 2 } 2 x = \cos ^ { 2 } 2 x$ for $0 ^ { \circ } < x < 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2015 Q4 [6]}}