| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve double/multiple angle equation |
| Difficulty | Moderate -0.3 This is a straightforward C2 trigonometry question with two standard parts: (i) requires using inverse sine and doubling the range for the half-angle, and (ii) is a routine tan x = 3 conversion. Both are textbook exercises requiring only standard techniques with no problem-solving insight, making it slightly easier than average. |
| Spec | 1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}x = 53.1°, 126.9°\) | B1 | Obtain \(106°\), or better. Allow answers in range \([106.2, 106.3]\). Must be in degrees, so 1.85 rad is B0 |
| \(x = 106°, 254°\) | M1 | Attempt correct solution method to find second angle. Could be \(2(180° - \text{their } 53.1°)\) or \((360° - \text{their } 106°)\) |
| A1 | Obtain \(254°\), or better. Allow answers in range \([253.7°, 254°]\). A0 if extra incorrect solutions in range. SR If no working shown then allow B1 for \(106°\) and B2 for \(254°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\tan x = 3\) | B1 | State \(\tan x = 3\). Allow B1 for correct equation even if no attempt to solve. Give BOD on notation eg \(\frac{\sin}{\cos}(x)\) as long as correct equation is seen or implied |
| \(x = 71.6°, 252°\) | M1 | Attempt to solve \(\tan x = k\). Not dep on B1. Could be implied by a correct solution |
| A1 | Obtain \(71.6°\) and \(252°\), or better. A0 if extra incorrect solutions in range. Alt method: B1 Obtain \(10\sin^2 x = 9\) or \(10\cos^2 x = 1\); M1 Attempt to solve \(\sin^2 x = k\) or \(\cos^2 x = k\); A1 Obtain \(71.6°\) and \(252°\) with no extra incorrect solutions |
# Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}x = 53.1°, 126.9°$ | B1 | Obtain $106°$, or better. Allow answers in range $[106.2, 106.3]$. Must be in degrees, so 1.85 rad is B0 |
| $x = 106°, 254°$ | M1 | Attempt correct solution method to find second angle. Could be $2(180° - \text{their } 53.1°)$ or $(360° - \text{their } 106°)$ |
| | A1 | Obtain $254°$, or better. Allow answers in range $[253.7°, 254°]$. A0 if extra incorrect solutions in range. **SR** If no working shown then allow B1 for $106°$ and B2 for $254°$ |
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# Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\tan x = 3$ | B1 | State $\tan x = 3$. Allow B1 for correct equation even if no attempt to solve. Give BOD on notation eg $\frac{\sin}{\cos}(x)$ as long as correct equation is seen or implied |
| $x = 71.6°, 252°$ | M1 | Attempt to solve $\tan x = k$. Not dep on B1. Could be implied by a correct solution |
| | A1 | Obtain $71.6°$ and $252°$, or better. A0 if extra incorrect solutions in range. **Alt method: B1** Obtain $10\sin^2 x = 9$ or $10\cos^2 x = 1$; **M1** Attempt to solve $\sin^2 x = k$ or $\cos^2 x = k$; **A1** Obtain $71.6°$ and $252°$ with no extra incorrect solutions |
---2 Solve each of the following equations, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.\\
(i) $\sin \frac { 1 } { 2 } x = 0.8$\\
(ii) $\sin x = 3 \cos x$
\hfill \mbox{\textit{OCR C2 2013 Q2 [6]}}