OCR MEI C2 (Core Mathematics 2) 2006 January

1 Given that \(140 ^ { \circ } = k \pi\) radians, find the exact value of \(k\).

2 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).

3 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).

4 \begin{figure}[h] \end{figure} Fig. 4 shows a curve which passes through the points shown in the following table. Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve, the lines \(x = 1\) and \(x = 4\), and the \(x\)-axis. State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of the area of this region.

5

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x + 9\). Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show that the curve has a stationary point of inflection when \(x = 3\).

7 In Fig. 7, A and B are points on the circumference of a circle with centre O . Angle \(\mathrm { AOB } = 1.2\) radians. The arc length AB is 6 cm . \begin{figure}[h] \end{figure}

1. Calculate the radius of the circle.
2. Calculate the length of the chord AB .

8 Find \(\int \left( x ^ { \frac { 1 } { 2 } } + \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x\).

9 \begin{figure}[h] \end{figure} The graph of \(\log _ { 10 } y\) against \(x\) is a straight line as shown in Fig. 9 .

1. Find the equation for \(\log _ { 10 } y\) in terms of \(x\).
2. Find the equation for \(y\) in terms of \(x\). Section B (36 marks)

10 The equation of a curve is \(y = 7 + 6 x - x ^ { 2 }\).

1. Use calculus to find the coordinates of the turning point on this curve. Find also the coordinates of the points of intersection of this curve with the axes, and sketch the curve.
2. Find \(\int _ { 1 } ^ { 5 } \left( 7 + 6 x - x ^ { 2 } \right) \mathrm { d } x\), showing your working.
3. The curve and the line \(y = 12\) intersect at ( 1,12 ) and ( 5,12 ). Using your answer to part (ii), find the area of the finite region between the curve and the line \(y = 12\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-5_643_1034_331_513} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
3. Find the equation of the tangent to the curve at the point \(( - 1,7 )\). Find also the coordinates of the point where this tangent crosses the curve again.