| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Equation with non-equation preliminary part (sketch/proof/identity) |
| Difficulty | Moderate -0.8 This is a straightforward C2 trigonometry question testing basic graph sketching, calculator use for inverse trig, and the standard tan x = 2 manipulation. All parts are routine textbook exercises requiring only direct application of standard techniques with no problem-solving insight needed, making it easier than average. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i)(a) Correct shape of \(\cos x\) graph | B1, B1 (2 marks) | Correct shape of cosecant graph; \((90°, 0)\), \((270°, 0)\) and \((0, 2)\) stated or implied |
| (i)(b) \(\cos x = 0.4\) \(x = 66.4°, 294°\) | M1, A1, A1∇ (3 marks) | Divide by 2, and attempt to solve for \(x\); Correct answer of \(66.4°\) / 1.16 rads; Second correct answer only, in degrees, following their \(x\) |
| (ii) \(\tan x = 2\) \(x = 63.4°, -117°\) | M1, A1, A1∇ (3 marks) | Use of \(\tan x = \frac{\sin x}{\cos x}\) (or square and use \(\sin^2 x + \cos^2 x = 1\)); Correct answer of \(63.4°\) / 1.56 rads; Second correct answer only, in degrees, following their \(x\) |
**(i)(a)** Correct shape of $\cos x$ graph | B1, B1 (2 marks) | Correct shape of cosecant graph; $(90°, 0)$, $(270°, 0)$ and $(0, 2)$ stated or implied
**(i)(b)** $\cos x = 0.4$ $x = 66.4°, 294°$ | M1, A1, A1∇ (3 marks) | Divide by 2, and attempt to solve for $x$; Correct answer of $66.4°$ / 1.16 rads; Second correct answer only, in degrees, following their $x$
**(ii)** $\tan x = 2$ $x = 63.4°, -117°$ | M1, A1, A1∇ (3 marks) | Use of $\tan x = \frac{\sin x}{\cos x}$ (or square and use $\sin^2 x + \cos^2 x = 1$); Correct answer of $63.4°$ / 1.56 rads; Second correct answer only, in degrees, following their $x$
**Total: 8 marks**\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = 2 \cos x$ for values of $x$ such that $0° \leq x \leq 360°$, indicating the coordinates of any points where the curve meets the axes. [2]
\item Solve the equation $2 \cos x = 0.8$, giving all values of $x$ between $0°$ and $360°$. [3]
\end{enumerate}
\item Solve the equation $2 \cos x = \sin x$, giving all values of $x$ between $-180°$ and $180°$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2007 Q7 [8]}}