| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.8 This is a multi-part trigonometric identity and equation question requiring compound angle formulas, double angle identities, and careful algebraic manipulation. Part (i) demands proving a non-obvious identity involving cos²(θ+45°) and double angles. Part (ii) requires applying the proven identity with substitution (θ/3), then solving a quadratic in sin(θ/3). Part (iii) involves finding the range of k for exactly two solutions, requiring understanding of the sine function's behavior and critical thinking about boundary conditions. While the techniques are C3-standard, the multi-step reasoning and the need to connect all three parts elevates this above routine exercises. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.05j 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 |
|---|---|---|
| State \(\cos \theta \cos 45 - \sin \theta \sin 45\) | B1 | or equiv including use of decimal approximation for \(\frac{1}{\sqrt{2}}\) |
| Use correct identity for \(\sin 2\theta\) or \(\cos 2\theta\) | B1 | must be used; not earned for just a separate statement |
| Attempt complete simplification of left-hand side | M1 | with relevant identities but allowing sign errors, and showing two terms involving \(\sin \theta \cos \theta\) |
| Obtain \(\sin^2 \theta\) | A1 | AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Use identity to produce equation of form \(\sin \frac{1}{2}\theta = c\) | M1 | condoning single value of constant \(c\) here (including values outside the range \(-1\) to 1); M0 for \(\sin \theta = c\) unless value(s) is) subsequently doubled |
| Obtain 70.5 or 70.6 | A1 | or greater accuracy 70.528… |
| Obtain \(-70.5\) or \(-70.6\) | A1√ | or greater accuracy \(-70.528…\); following first answer; and no other answer between \(-90\) and 90; answer(s) only : 0/3 |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(6\sin^2 \frac{1}{2}\theta = k\) | B1 | |
| Attempt to relate \(k\) to at least \(6\sin^2 30°\) | M1 | |
| Obtain \(0 < k < \frac{3}{2}\) | A1 | condone use of \(\le\) |
## (i)
State $\cos \theta \cos 45 - \sin \theta \sin 45$ | B1 | or equiv including use of decimal approximation for $\frac{1}{\sqrt{2}}$
Use correct identity for $\sin 2\theta$ or $\cos 2\theta$ | B1 | must be used; not earned for just a separate statement
Attempt complete simplification of left-hand side | M1 | with relevant identities but allowing sign errors, and showing two terms involving $\sin \theta \cos \theta$
Obtain $\sin^2 \theta$ | A1 | AG; necessary detail needed
**Total: [4]**
## (ii)
Use identity to produce equation of form $\sin \frac{1}{2}\theta = c$ | M1 | condoning single value of constant $c$ here (including values outside the range $-1$ to 1); M0 for $\sin \theta = c$ unless value(s) is) subsequently doubled
Obtain 70.5 or 70.6 | A1 | or greater accuracy 70.528…
Obtain $-70.5$ or $-70.6$ | A1√ | or greater accuracy $-70.528…$; following first answer; and no other answer between $-90$ and 90; answer(s) only : 0/3
**Total: [3]**
## (iii)
State or imply $6\sin^2 \frac{1}{2}\theta = k$ | B1 |
Attempt to relate $k$ to at least $6\sin^2 30°$ | M1 |
Obtain $0 < k < \frac{3}{2}$ | A1 | condone use of $\le$
**Total: [3]**
---\begin{enumerate}[label=(\roman*)]
\item Prove that
$$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
\item Hence solve the equation
$$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$
for $-90° < \theta < 90°$. [3]
\item It is given that there are two values of $\theta$, where $-90° < \theta < 90°$, satisfying the equation
$$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \frac{2}{3}\theta - \sin \frac{2}{3}\theta) = k,$$
where $k$ is a constant. Find the set of possible values of $k$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2013 Q9 [10]}}