| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Given sin/cos/tan, find other expressions |
| Difficulty | Standard +0.3 This is a straightforward application of double angle formulae with algebraic manipulation. Part (a) requires expanding sin 2α, factoring, and solving; part (b) uses cos 2β = 2cos²β - 1 to form a quadratic. Both are standard C3 exercises requiring routine technique rather than insight, making them slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a) State \(14\sin \alpha \cos \alpha = 3\sin \alpha\) | B1 | or unsimplified equiv such as \(7(2\sin \alpha \cos \alpha) = 3\sin \alpha\) |
| Attempt to find value of \(\cos \alpha\) | M1 | by valid process; may be implied |
| Obtain \(\frac{3}{14}\) | A1 | 3 marks: exact answer required; ignore subsequent work to find angle |
| (b) Attempt use of identity for \(\cos 2\beta\) | M1 | of form \(\pm 2\cos^2 \beta \pm 1\); initial use of \(\cos^2 \beta - \sin^2 \beta\) needs attempt to express \(\sin^2 \beta\) in terms of \(\cos^2 \beta\) to earn M1 |
| Obtain \(6\cos^2 \beta + 19\cos \beta + 10\) | A1 | or unsimplified equiv or equiv involving \(\sec \beta\) |
| Attempt solution of 3-term quadratic eqn | M1 | for \(\cos \beta\) (or after adjustment) for \(\sec \beta\) |
| Use \(\sec \beta = \frac{1}{\cos \beta}\) at some stage | M1 | or equiv |
| Obtain \(-\frac{3}{2}\) | A1 | 5 marks: or equiv; and (finally) no other answer |
**(a)** State $14\sin \alpha \cos \alpha = 3\sin \alpha$ | B1 | or unsimplified equiv such as $7(2\sin \alpha \cos \alpha) = 3\sin \alpha$
Attempt to find value of $\cos \alpha$ | M1 | by valid process; may be implied
Obtain $\frac{3}{14}$ | A1 | 3 marks: exact answer required; ignore subsequent work to find angle
**(b)** Attempt use of identity for $\cos 2\beta$ | M1 | of form $\pm 2\cos^2 \beta \pm 1$; initial use of $\cos^2 \beta - \sin^2 \beta$ needs attempt to express $\sin^2 \beta$ in terms of $\cos^2 \beta$ to earn M1
Obtain $6\cos^2 \beta + 19\cos \beta + 10$ | A1 | or unsimplified equiv or equiv involving $\sec \beta$
Attempt solution of 3-term quadratic eqn | M1 | for $\cos \beta$ (or after adjustment) for $\sec \beta$
Use $\sec \beta = \frac{1}{\cos \beta}$ at some stage | M1 | or equiv
Obtain $-\frac{3}{2}$ | A1 | 5 marks: or equiv; and (finally) no other answer
---3
\begin{enumerate}[label=(\alph*)]
\item Given that $7 \sin 2 \alpha = 3 \sin \alpha$, where $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, find the exact value of $\cos \alpha$.
\item Given that $3 \cos 2 \beta + 19 \cos \beta + 13 = 0$, where $90 ^ { \circ } < \beta < 180 ^ { \circ }$, find the exact value of $\sec \beta$.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2011 Q3 [8]}}