| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Triangle with circular sector |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard C2 content: cosine rule to find an angle, then sector area and arc length formulas. Part (a) is given as 'show that', making it easier. The remaining parts require direct application of memorized formulas with minimal problem-solving. Slightly above average difficulty only due to the multi-step nature and need to distinguish major/minor arcs. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| \(9^2 = 4^2 + 6^2 - 2\times4\times6\cos\alpha \Rightarrow \cos\alpha = \frac{4^2+6^2-9^2}{2\times4\times6} = -\frac{29}{48} = -0.604...\) | M1 | Correct use of cosine rule leading to a value for \(\cos\alpha\) |
| \(\alpha = 2.22\) | A1 | 2.22 must be seen here |
| Answer | Marks | Guidance |
|---|---|---|
| \(XY^2 = 4^2+6^2-2\times4\times6\cos2.22 \Rightarrow XY^2 = 81.01..., XY = 9.00...\) | M1 A1 | Correct use of cosine rule leading to a value for \(XY^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\pi - 2.22 (= 4.06366...)\) | B1 | \(2\pi - 2.22\) or awrt 4.06 |
| \(\frac{1}{2}\times4^2\times"4.06"\) | M1 | Correct method for major sector area |
| \(32.5\) | A1 | Awrt 32.5 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\pi\times4^2\) | B1 | Correct expression for circle area |
| \(\pi\times4^2 - \frac{1}{2}\times4^2\times2.22 = 32.5\) | M1 | Correct method for circle minus minor sector area |
| \(= 32.5\) | A1 | Awrt 32.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Area of triangle \(= \frac{1}{2}\times4\times6\times\sin2.22 (= 9.56)\) | B1 | Correct expression for area of triangle XYZ |
| Area required \(= "9.56" + "32.5"\) | M1 | Their triangle XYZ + part (b) answer or correct attempt at major sector |
| \(= 42.1\text{ cm}^2\) or \(42.0\text{ cm}^2\) | A1 | Awrt 42.1 or 42.0 (or just 42) |
| Answer | Marks | Guidance |
|---|---|---|
| Arc length \(= 4\times4.06 (= 16.24)\) or \(8\pi - 4\times2.22\) | M1A1ft | M1: \(4\times their(2\pi-2.22)\) or circumference minus minor arc; A1: Correct ft expression |
| Perimeter \(= ZY + WY +\) Arc Length | M1 | \(9 + 2 +\) Any Arc |
| Perimeter \(= 27.2\) or \(27.3\) | A1 | Awrt 27.2 or awrt 27.3 |
## Question 7:
### Part (a):
| $9^2 = 4^2 + 6^2 - 2\times4\times6\cos\alpha \Rightarrow \cos\alpha = \frac{4^2+6^2-9^2}{2\times4\times6} = -\frac{29}{48} = -0.604...$ | M1 | Correct use of cosine rule leading to a value for $\cos\alpha$ |
|---|---|---|
| $\alpha = 2.22$ | A1 | 2.22 must be seen here |
### Part (a) Way 2:
| $XY^2 = 4^2+6^2-2\times4\times6\cos2.22 \Rightarrow XY^2 = 81.01..., XY = 9.00...$ | M1 A1 | Correct use of cosine rule leading to a value for $XY^2$ |
|---|---|---|
### Part (b):
| $2\pi - 2.22 (= 4.06366...)$ | B1 | $2\pi - 2.22$ or awrt 4.06 |
|---|---|---|
| $\frac{1}{2}\times4^2\times"4.06"$ | M1 | Correct method for major sector area |
| $32.5$ | A1 | Awrt 32.5 |
### Part (b) Way 2:
| $\pi\times4^2$ | B1 | Correct expression for circle area |
|---|---|---|
| $\pi\times4^2 - \frac{1}{2}\times4^2\times2.22 = 32.5$ | M1 | Correct method for circle minus minor sector area |
| $= 32.5$ | A1 | Awrt 32.5 |
### Part (c):
| Area of triangle $= \frac{1}{2}\times4\times6\times\sin2.22 (= 9.56)$ | B1 | Correct expression for area of triangle XYZ |
|---|---|---|
| Area required $= "9.56" + "32.5"$ | M1 | Their triangle XYZ + part (b) answer or correct attempt at major sector |
| $= 42.1\text{ cm}^2$ or $42.0\text{ cm}^2$ | A1 | Awrt 42.1 or 42.0 (or just 42) |
### Part (d):
| Arc length $= 4\times4.06 (= 16.24)$ or $8\pi - 4\times2.22$ | M1A1ft | M1: $4\times their(2\pi-2.22)$ or circumference minus minor arc; A1: Correct ft expression |
|---|---|---|
| Perimeter $= ZY + WY +$ Arc Length | M1 | $9 + 2 +$ Any Arc |
| Perimeter $= 27.2$ or $27.3$ | A1 | Awrt 27.2 or awrt 27.3 |
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{6f31b6f1-33b5-4bca-9030-cf93760b454d-09_432_656_210_644}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The triangle $X Y Z$ in Figure 1 has $X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}$ and angle $Z X Y = \alpha$. The point $W$ lies on the line $X Y$.
The circular arc $Z W$, in Figure 1 is a major arc of the circle with centre $X$ and radius 4 cm .
\begin{enumerate}[label=(\alph*)]
\item Show that, to 3 significant figures, $\alpha = 2.22$ radians.
\item Find the area, in $\mathrm { cm } ^ { 2 }$, of the major sector $X Z W X$.
The region enclosed by the major arc $Z W$ of the circle and the lines $W Y$ and $Y Z$ is shown shaded in Figure 1.
Calculate
\item the area of this shaded region,
\item the perimeter $Z W Y Z$ of this shaded region.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2013 Q7 [12]}}