OCR MEI C2 (Core Mathematics 2)

1 Find all the angles in the range \(0 ^ { 0 } \leq x \leq 360 ^ { 0 }\) satisfying the equation \(\sin x + \frac { 1 } { 2 } \sqrt { 3 } = 0\).

2 Solve the equation \(3 ^ { x } = 15\), giving your answer correct to 4 decimal places.

3 The sum to infinity of a geometric series is 5 and the first term is 2 .
Find the common ratio of the series.

4 The first 3 terms of an arithmetical progression are 7, 5.9 and 4.8.
Find

1. the common difference,
2. the smallest value of \(n\) for which the sum to \(n\) terms is negative.

5 The gradient of a curve is given by the function \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - x\).
The curve passes through the point \(( 1,2 )\).
Find the equation of the curve.

6 Evaluate \(\int _ { 1 } ^ { 2 } \left( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).

7

1. Using the triangle, show that \(\sin ^ { 2 } x + \cos ^ { 2 } x = 1\).
2. Hence prove that
   \includegraphics[max width=\textwidth, alt={}]{73d1c02b-1b7b-426d-a171-c762597cfed4-2_255_501_1779_1022} \(1 + \tan ^ { 2 } x = \frac { 1 } { \cos ^ { 2 } x }\).

8 Draw two sketches of the graph of \(y = \sin x\) in the range \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\).

1. On the first sketch, draw also a sketch of \(y = \sin ( 2 x )\).
2. On the second sketch, draw also a sketch of \(y = 2 \sin x\).

9 A sector of a circle has an angle of 0.8 radians. The arc length is 5 cm . Calculate the radius of the circle and the area of the sector.

10 At 1200 the captain of a ship observes that the bearing of a lighthouse is \(340 ^ { \circ }\). His position is at A.
At 1230 he takes another bearing of the lighthouse and finds it to be \(030 ^ { \circ }\). During this time the ship moves on a constant course of \(280 ^ { \circ }\) to the point B . His plot on the chart is as shown in Fig. 11 below. \begin{figure}[h] \end{figure}

1. Write down the size of the angles LAB and LBA .
2. The captain believes that at A he is 5 km from L . Assuming that LA is exactly 5 km , show that LB is 4.61 km , correct to 2 decimal places, and find AB . Hence calculate the speed of the ship.
3. The speed of the ship is actually 10 kilometres per hour. Given that the bearings of \(340 ^ { \circ }\) and \(030 ^ { \circ }\) and the ship's course of \(280 ^ { \circ }\) are all accurate, calculate the true value of the distance LA.

1. The speed-time graph on the insert sheet provides the axes and the first two points plotted. Plot the remainder of these points and join them with a smooth curve. The area between this curve and the \(t\)-axis represents the distance travelled by the car in this time.
2. Using the trapezium rule with 6 values of \(t\) estimate the area under the curve to give the distance travelled. Illustrate on your graph the area found.
3. John's teacher suggests that the equation of the curve could be \(v = 6 t - \frac { 1 } { 2 } t ^ { 2 }\). Find, by calculus, the area between this curve and the \(t\) axis.
4. Plot this curve on your graph. Comment on whether the estimates obtained in parts (ii) and (iii) are overestimates or underestimates. 12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure}
5. Express \(y\) as a function of \(x\).
6. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
7. Prove that for the maximum value of \(A\), \(y = x\) exactly.
   \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
   Insert sheet for question 11}

12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h] \end{figure}

1. Express \(y\) as a function of \(x\).
2. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
3. Prove that for the maximum value of \(A\), \(y = x\) exactly.
   \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
   Insert sheet for question 11} 11 Speed-time graph with the first two points plotted.
   \includegraphics[max width=\textwidth, alt={}, center]{73d1c02b-1b7b-426d-a171-c762597cfed4-5_768_1772_1389_205}