OCR MEI C2 (Core Mathematics 2) 2015 June

1

1. Differentiate \(12 \sqrt [ 3 ] { x }\).
2. Integrate \(\frac { 6 } { x ^ { 3 } }\).

2 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).

3 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression.

4 A sector of a circle has angle 1.5 radians and area \(27 \mathrm {~cm} ^ { 2 }\). Find the perimeter of the sector.

5 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.

6

1. On the same axes, sketch the curves \(y = 3 ^ { x }\) and \(y = 3 ^ { 2 x }\), identifying clearly which is which.
2. Given that \(3 ^ { 2 x } = 729\), find in either order the values of \(3 ^ { x }\) and \(x\).

7 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).

8 Fig. 8 shows the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\). It is a straight line passing through the points \(( 2,8 )\) and \(( 0,2 )\). \begin{figure}[h] \end{figure} Find the equation relating \(\log _ { 10 } y\) and \(\log _ { 10 } x\) and hence find the equation relating \(y\) and \(x\).

9 \begin{figure}[h] \end{figure}

1. Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing's cross-section. Calculate angle A and the area of the cross-section.
2. Jean then modifies her design for the wing. Fig. 9.2 shows the new cross-section, with 1 unit for each of \(x\) and \(y\) representing one centimetre. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_431_1682_970_194} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} Here are some of the coordinates that Jean used to draw the new cross-section.

   Upper surface		Lower surface
   \(x\)	\(y\)	\(x\)	\(y\)
   0	0	0	0
   4	1.45	4	-0.85
   8	1.56	8	-0.76
   12	1.27	12	-0.55
   16	1.04	16	-0.30
   20	0	20	0

   Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section.

10 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).

1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve.
3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\). Write down the coordinates of the minimum point of the transformed curve.

11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.

1. How many of Jill's descendants would there be in generation 8 ?
2. How many of Jill's descendants would there be altogether in the first 15 generations?
3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}