| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Given area find angle/side |
| Difficulty | Moderate -0.8 This is a straightforward two-part question applying standard formulas. Part (i) uses the area formula (Area = ½ab sin C) with direct substitution to find one side. Part (ii) applies the cosine rule with all necessary values known. Both parts are routine applications requiring no problem-solving insight, making this easier than average for A-level. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 3\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2} \times 8 \times AB \times \sin 30 = 20\) | M1 | Equate correct attempt at area of triangle to 20. Must use correct formula including \(\frac{1}{2}\). Allow if subsequently evaluated in radian mode (gives \(-3.95AB = 20\)). If using \(\frac{1}{2} \times b \times h\) then must be valid use of trig to find \(h\) |
| \(AB = 10\) | A1 | Must be exactly 10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(BC^2 = 8^2 + 10^2 - 2 \times 8 \times 10 \times \cos 30\) | M1 | Attempt to use correct cosine rule, using their \(AB\). Allow M1 if not square rooted. Allow if subsequently evaluated in radian mode (gives 11.8). Allow if correct formula seen but then evaluated incorrectly (using \((8^2 + 10^2 - 2 \times 8 \times 10) \times \cos 30\) gives 1.86). Allow any equiv method as long as valid use of trig |
| \(BC = 5.04\) | A1 | If \(> 3\)sf, allow answer rounding to 5.043 with no errors seen |
**Question 1:**
$x = 3$ | A1 | cao
etc.
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# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2} \times 8 \times AB \times \sin 30 = 20$ | M1 | Equate correct attempt at area of triangle to 20. Must use correct formula including $\frac{1}{2}$. Allow if subsequently evaluated in radian mode (gives $-3.95AB = 20$). If using $\frac{1}{2} \times b \times h$ then must be valid use of trig to find $h$ |
| $AB = 10$ | A1 | Must be exactly 10 |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $BC^2 = 8^2 + 10^2 - 2 \times 8 \times 10 \times \cos 30$ | M1 | Attempt to use correct cosine rule, using their $AB$. Allow M1 if not square rooted. Allow if subsequently evaluated in radian mode (gives 11.8). Allow if correct formula seen but then evaluated incorrectly (using $(8^2 + 10^2 - 2 \times 8 \times 10) \times \cos 30$ gives 1.86). Allow any equiv method as long as valid use of trig |
| $BC = 5.04$ | A1 | If $> 3$sf, allow answer rounding to 5.043 with no errors seen |
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\includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_250_611_356_721}
The diagram shows triangle $A B C$, with $A C = 8 \mathrm {~cm}$ and angle $C A B = 30 ^ { \circ }$.\\
(i) Given that the area of the triangle is $20 \mathrm {~cm} ^ { 2 }$, find the length of $A B$.\\
(ii) Find the length of $B C$, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{OCR C2 2016 Q1 [4]}}