OCR MEI C2 (Core Mathematics 2)

1 Fig. 11.1 shows a village green which is bordered by 3 straight roads \(\mathrm { AB } , \mathrm { BC }\) and CA . The road AC runs due North and the measurements shown are in metres. \begin{figure}[h] \end{figure}

1. Calculate the bearing of B from C , giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of the village green. The road \(A B\) is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-1_432_574_1377_779} \captionsetup{labelformat=empty} \caption{Not to scale}
   \end{figure} Fig. 11.2 The new road is an arc of a circle with centre O and radius 130 m .
3. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures.
   (B) Show that the area of land added to the village green is \(5300 \mathrm {~m} ^ { 2 }\) correct to 2 significant figures. Fig. 4 For triangle ABC shown in Fig. 4, calculate
4. the length of BC , Not to scale
5. the area of triangle ABC .

3

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-3_769_766_174_770} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} A boat travels from P to Q and then to R . As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045 ^ { \circ }\). R is 9.2 km from P on a bearing of \(113 ^ { \circ }\), so that angle QPR is \(68 ^ { \circ }\). Calculate the distance and bearing of R from Q .
2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-3_517_1472_1433_414} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure} BC is an arc of a circle with centre A and radius 80 cm . Angle \(\mathrm { CAB } = \frac { 2 \pi } { 3 }\) radians.
   EC is an arc of a circle with centre D and radius \(r \mathrm {~cm}\). Angle CDE is a right angle.
   1. Calculate the area of sector ABC .
   2. Show that \(r = 40 \sqrt { 3 }\) and calculate the area of triangle CDA.
   3. Hence calculate the area of cross-section of the rudder.

4 Arrowline Enterprises is considering two possible logos: \begin{figure}[h] \end{figure}

1. Fig. 10.1 shows the first logo ABCD . It is symmetrical about AC . Find the length of AB and hence find the area of this logo.
2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm . ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that \(\mathrm { ST } = 16.2 \mathrm {~cm}\) to 3 significant figures.
   Find the area and perimeter of this logo.

5

1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-5_659_867_348_716} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
   \end{figure} (A) Calculate the total length of the course for this race.
   (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC .
2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{139b1905-2035-4503-9ffb-3e6e81f78ef9-5_705_822_1624_739} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{figure}