| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Convert sin/cos ratio to tan |
| Difficulty | Standard +0.3 Part (i) is a straightforward R-formula or division approach for a linear trigonometric equation with a triple angle, requiring careful attention to the restricted domain. Part (ii) involves standard manipulation using the Pythagorean identity to form a quadratic in cos x. Both are routine C2-level techniques with no novel insight required, slightly above average due to the triple angle consideration and domain restrictions. |
| 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 | Mark | Guidance |
| \(5\sin3\theta - 7\cos3\theta = 0 \Rightarrow \tan3\theta = \frac{7}{5}\) | M1A1 | M1: Reaches \(\tan...=k\) where \(k\neq0\); A1: \(\tan...=\frac{7}{5}\) |
| \(3\theta = 0.95054...\) | dM1 | \(3\theta = \tan^{-1}\left(\frac{7}{5}\right)\). Must be \(3\theta\). Dependent on first method mark |
| \(\theta = 0.317\) or \(\theta = 1.36\) | A1 | Awrt 0.317 or awrt 1.36 |
| \(\theta = 0.317\) and \(\theta = 1.36\) only | A1 | Both values, awrt |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(9\cos^2 x + 5\cos x = 3(1-\cos^2 x)\) | M1 | Uses \(\sin^2 x = \pm 1 \pm \cos^2 x\) |
| \(12\cos^2 x + 5\cos x - 3 = 0\) | A1 | Correct 3-term quadratic |
| \((3\cos x - 1)(4\cos x + 3) = 0 \Rightarrow (\cos x) = ...\) | dM1 | Solves 3TQ in \(\cos x\) to at least one value. Dependent on first method mark |
| \(\cos x = \frac{1}{3}, -\frac{3}{4}\) | A1 | Correct values for \(\cos x\) |
| \(x = 70.5,\ 289.5,\ 138.6,\ 221.4\) | A1A1 | A1: Any 2 correct solutions (awrt); A1: All 4 answers (awrt) |
## Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $5\sin3\theta - 7\cos3\theta = 0 \Rightarrow \tan3\theta = \frac{7}{5}$ | M1A1 | M1: Reaches $\tan...=k$ where $k\neq0$; A1: $\tan...=\frac{7}{5}$ |
| $3\theta = 0.95054...$ | dM1 | $3\theta = \tan^{-1}\left(\frac{7}{5}\right)$. Must be $3\theta$. Dependent on first method mark |
| $\theta = 0.317$ or $\theta = 1.36$ | A1 | Awrt 0.317 or awrt 1.36 |
| $\theta = 0.317$ **and** $\theta = 1.36$ only | A1 | Both values, awrt |
## Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $9\cos^2 x + 5\cos x = 3(1-\cos^2 x)$ | M1 | Uses $\sin^2 x = \pm 1 \pm \cos^2 x$ |
| $12\cos^2 x + 5\cos x - 3 = 0$ | A1 | Correct 3-term quadratic |
| $(3\cos x - 1)(4\cos x + 3) = 0 \Rightarrow (\cos x) = ...$ | dM1 | Solves 3TQ in $\cos x$ to at least one value. Dependent on first method mark |
| $\cos x = \frac{1}{3}, -\frac{3}{4}$ | A1 | Correct values for $\cos x$ |
| $x = 70.5,\ 289.5,\ 138.6,\ 221.4$ | A1A1 | A1: Any 2 correct solutions (awrt); A1: All 4 answers (awrt) |5. (In this question, solutions based entirely on graphical or numerical methods are not acceptable.)\\
(i) Solve, for $0 < \theta < \frac { \pi } { 2 }$
$$5 \sin 3 \theta - 7 \cos 3 \theta = 0$$
Give each solution, in radians, to 3 significant figures.\\
(ii) Solve, for $0 < x < 360 ^ { \circ }$
$$9 \cos ^ { 2 } x + 5 \cos x = 3 \sin ^ { 2 } x$$
Give each solution, in degrees, to one decimal place.
\hfill \mbox{\textit{Edexcel C12 2018 Q5 [11]}}