OCR MEI C2 (Core Mathematics 2) 2009 June

1 Use an isosceles right-angled triangle to show that \(\cos 45 ^ { \circ } = \frac { 1 } { \sqrt { 2 } }\).

2 Find \(\int _ { 1 } ^ { 2 } \left( 12 x ^ { 5 } + 5 \right) \mathrm { d } x\).

3

1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.

4 A sector of a circle of radius 18.0 cm has arc length 43.2 cm .

1. Find in radians the angle of the sector.
2. Find this angle in degrees, giving your answer to the nearest degree.

5

1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).

6 Use calculus to find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 6 x ^ { 2 } - 15 x\). Hence find the set of values of \(x\) for which \(x ^ { 3 } - 6 x ^ { 2 } - 15 x\) is an increasing function.

7 Show that the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) may be written in the form $$4 \sin ^ { 2 } \theta - \sin \theta = 0$$ Hence solve the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8 The gradient of a curve is \(3 \sqrt { x } - 5\). The curve passes through the point ( 4,6 ). Find the equation of the curve.

9 Simplify

1. \(10 - 3 \log _ { a } a\),
2. \(\frac { \log _ { 10 } a ^ { 5 } + \log _ { 10 } \sqrt { a } } { \log _ { 10 } a }\). Section B (36 marks)

1. On the insert, complete the table and plot \(h\) against \(\log _ { 10 } t\), drawing by eye a line of best fit.
2. Use your graph to find an equation for \(h\) in terms of \(\log _ { 10 } t\) for this model.
3. Find the height of the tree at age 100 years, as predicted by this model.
4. Find the age of the tree when it reaches a height of 29 m , according to this model.
5. Comment on the suitability of the model when the tree is very young.

11

1. In a 'Make Ten' quiz game, contestants get \(\pounds 10\) for answering the first question correctly, then a further \(\pounds 20\) for the second question, then a further \(\pounds 30\) for the third, and so on, until they get a question wrong and are out of the game.
   (A) Haroon answers six questions correctly. Show that he receives a total of \(\pounds 210\).
   (B) State, in a simple form, a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant who receives \(\pounds 10350\) from this game.
2. In a 'Double Your Money' quiz game, contestants get \(\pounds 5\) for answering the first question correctly, then a further \(\pounds 10\) for the second question, then a further \(\pounds 20\) for the third, and so on doubling the amount for each question until they get a question wrong and are out of the game.
   (A) Gary received \(\pounds 75\) from the game. How many questions did he get right?
   (B) Bethan answered 9 questions correctly. How much did she receive from the game?
   (C) State a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant in this game who receives \(\pounds 2621435\).

12

1. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
2. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
3. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
4. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
5. This tangent crosses the \(x\)-axis at the point P . The curve crosses the positive \(x\)-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.