| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve shifted trig equation |
| Difficulty | Standard +0.3 Part (i) is a straightforward single-step transformation (isolate sin, apply arcsin, account for phase shift) with restricted range. Part (ii) requires recognizing to use the Pythagorean identity to convert to a quadratic in cos x, then solving—a standard C2 technique but slightly more involved than routine exercises. Overall slightly easier than average A-level questions due to being standard textbook methods with no novel insight required. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sin(2\theta - 30) = -0.6\) or \(2\theta - 30 = -36.9\) or implied by 216.9 | B1 | Must contain no errors; may be implied by \(-36.9\) |
| \(2\theta - 30 = 216.87 \ (= 180 + 36.9)\) | M1 | Uses \(180 - \arcsin(-0.6)\) i.e. \(180 + 36.9\) (3rd quadrant) |
| \(\theta = \frac{216.87 + 30}{2} = 123.4\) or \(123.5\) | A1 | Answers which round to 123.4 or 123.5, must be in degrees |
| \(2\theta - 30 = 360 - 36.9\) or \(323.1\) | M1 | Uses \(360 + \arcsin(-0.6)\) i.e. \(360 - 36.9\) (4th quadrant) |
| \(\theta = \frac{323.1 + 30}{2} = 176.6\) | A1 | Answers which round to 176.6, must be in degrees |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(9\cos^2 x - 11\cos x + 3(1 - \cos^2 x) = 0\) or \(6\cos^2 x - 11\cos x + 3(\sin^2 x + \cos^2 x) = 0\) | M1 | Use of \(\sin^2 x = (1 - \cos^2 x)\) or \((\sin^2 x + \cos^2 x) = 1\) |
| \(6\cos^2 x - 11\cos x + 3 = 0\) \(\{\text{as } (\sin^2 x + \cos^2 x) = 1\}\) | A1 | Correct three term quadratic in any equivalent form |
| \((3\cos x - 1)(2\cos x - 3) = 0\) implies \(\cos x =\) | M1 | Uses standard method to solve quadratic |
| \(\cos x = \frac{1}{3}, \left(\frac{3}{2}\right)\) | A1 | A1 for \(\frac{1}{3}\) with \(\frac{3}{2}\) ignored but A0 if \(\frac{3}{2}\) is incorrect |
| \(x = 70.5\) | B1 | 70.5 or answers which round to this value |
| \(x = 360 - \text{"70.5"}\) | M1 | \(360 - \arccos(\text{their } 1/3)\) |
| \(x = 289.5\) | A1 cao | Second answer |
# Question 9:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sin(2\theta - 30) = -0.6$ or $2\theta - 30 = -36.9$ or implied by 216.9 | B1 | Must contain no errors; may be implied by $-36.9$ |
| $2\theta - 30 = 216.87 \ (= 180 + 36.9)$ | M1 | Uses $180 - \arcsin(-0.6)$ i.e. $180 + 36.9$ (3rd quadrant) |
| $\theta = \frac{216.87 + 30}{2} = 123.4$ or $123.5$ | A1 | Answers which round to 123.4 or 123.5, must be in degrees |
| $2\theta - 30 = 360 - 36.9$ or $323.1$ | M1 | Uses $360 + \arcsin(-0.6)$ i.e. $360 - 36.9$ (4th quadrant) |
| $\theta = \frac{323.1 + 30}{2} = 176.6$ | A1 | Answers which round to 176.6, must be in degrees |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $9\cos^2 x - 11\cos x + 3(1 - \cos^2 x) = 0$ or $6\cos^2 x - 11\cos x + 3(\sin^2 x + \cos^2 x) = 0$ | M1 | Use of $\sin^2 x = (1 - \cos^2 x)$ or $(\sin^2 x + \cos^2 x) = 1$ |
| $6\cos^2 x - 11\cos x + 3 = 0$ $\{\text{as } (\sin^2 x + \cos^2 x) = 1\}$ | A1 | Correct three term quadratic in any equivalent form |
| $(3\cos x - 1)(2\cos x - 3) = 0$ implies $\cos x =$ | M1 | Uses standard method to solve quadratic |
| $\cos x = \frac{1}{3}, \left(\frac{3}{2}\right)$ | A1 | A1 for $\frac{1}{3}$ with $\frac{3}{2}$ ignored but A0 if $\frac{3}{2}$ is incorrect |
| $x = 70.5$ | B1 | 70.5 or answers which round to this value |
| $x = 360 - \text{"70.5"}$ | M1 | $360 - \arccos(\text{their } 1/3)$ |
| $x = 289.5$ | A1 cao | Second answer |\begin{enumerate}
\item (i) Solve, for $0 \leqslant \theta < 180 ^ { \circ }$
\end{enumerate}
$$\sin \left( 2 \theta - 30 ^ { \circ } \right) + 1 = 0.4$$
giving your answers to 1 decimal place.\\
(ii) Find all the values of $x$, in the interval $0 \leqslant x < 360 ^ { \circ }$, for which
$$9 \cos ^ { 2 } x - 11 \cos x + 3 \sin ^ { 2 } x = 0$$
giving your answers to 1 decimal place.
You must show clearly how you obtained your answers.
\hfill \mbox{\textit{Edexcel C2 2013 Q9 [12]}}