| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Calculate intersection coordinates algebraically |
| Difficulty | Standard +0.8 This requires solving tan x = cos x algebraically, which involves converting to sin x/cos x = cos x, leading to sin x = cos²x = 1 - sin²x, then solving a quadratic in sin x. Students must identify valid solutions in the given domain and calculate exact coordinates. This goes beyond routine trig equation solving and requires algebraic manipulation across multiple steps with careful domain consideration. |
| 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 | Marks | Guidance |
| \(\tan x = \cos x \rightarrow \sin x = \cos^2 x\) | M1 | Use \(\tan = \sin/\cos\) and multiply by \(\cos\) |
| \(\sin x = 1 - \sin^2 x\) | M1 | Use \(\cos^2 x = 1 - \sin^2 x\) |
| \(\sin x = 0.6180\). Allow \(\frac{-1+\sqrt{5}}{2}\) | M1 | Attempt soln of quadratic in \(\sin x\). Ignore solution \(-1.618\). Allow \(x = 0.618\) |
| \(x\)-coord of \(A = \sin^{-1}0.618 = 0.666\) cao | A1 | Must be radians. Accept \(0.212\pi\) |
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER: \(x\)-coord of \(B\) is \(\pi - \text{their } 0.666\) | (M1 | Expect \(2.475(3)\). Must be radians throughout |
| \(y\)-coord of \(B\) is \(\tan(\text{their } 2.475)\) or \(\cos(\text{their } 2.475)\) | M1 | |
| \(x = 2.48,\ y = -0.786\) or \(-0.787\) cao | A1 | Accept \(x = 0.788\pi\) |
| OR: \(y\)-coord of \(B\) is \(-(\cos \text{ or } \tan(\text{their } 0.666))\) | (M1 | |
| \(x\)-coord of \(B\) is \(\cos^{-1}(\text{their } y)\) or \(\pi + \tan^{-1}(\text{their } y)\) | M1 | |
| \(x = 2.48,\ y = -0.786\) or \(-0.787\) | A1 | Accept \(x = 0.788\pi\) |
| Total: | 3 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\tan x = \cos x \rightarrow \sin x = \cos^2 x$ | M1 | Use $\tan = \sin/\cos$ and multiply by $\cos$ |
| $\sin x = 1 - \sin^2 x$ | M1 | Use $\cos^2 x = 1 - \sin^2 x$ |
| $\sin x = 0.6180$. Allow $\frac{-1+\sqrt{5}}{2}$ | M1 | Attempt soln of quadratic in $\sin x$. Ignore solution $-1.618$. Allow $x = 0.618$ |
| $x$-coord of $A = \sin^{-1}0.618 = 0.666$ cao | A1 | Must be radians. Accept $0.212\pi$ |
| **Total:** | **4** | |
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER:** $x$-coord of $B$ is $\pi - \text{their } 0.666$ | (M1 | Expect $2.475(3)$. Must be radians throughout |
| $y$-coord of $B$ is $\tan(\text{their } 2.475)$ or $\cos(\text{their } 2.475)$ | M1 | |
| $x = 2.48,\ y = -0.786$ or $-0.787$ cao | A1 | Accept $x = 0.788\pi$ |
| **OR:** $y$-coord of $B$ is $-(\cos \text{ or } \tan(\text{their } 0.666))$ | (M1 | |
| $x$-coord of $B$ is $\cos^{-1}(\text{their } y)$ or $\pi + \tan^{-1}(\text{their } y)$ | M1 | |
| $x = 2.48,\ y = -0.786$ or $-0.787$ | A1 | Accept $x = 0.788\pi$ |
| **Total:** | **3** | |5\\
\includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-08_526_499_258_824}
The diagram shows the graphs of $y = \tan x$ and $y = \cos x$ for $0 \leqslant x \leqslant \pi$. The graphs intersect at points $A$ and $B$.\\
(i) Find by calculation the $x$-coordinate of $A$.\\
(ii) Find by calculation the coordinates of $B$.\\
\hfill \mbox{\textit{CAIE P1 2017 Q5 [7]}}