OCR C2 (Core Mathematics 2) 2007 January

1 In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms.

2 The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 8 cm . The angle \(A O B\) is \(46 ^ { \circ }\).

1. Express \(46 ^ { \circ }\) in radians, correct to 3 significant figures.
2. Find the length of the arc \(A B\).
3. Find the area of the sector \(O A B\).

3

1. Find \(\int ( 4 x - 5 ) \mathrm { d } x\).
2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x - 5\). The curve passes through the point (3,7). Find the equation of the curve.

4 In a triangle \(A B C , A B = 5 \sqrt { 2 } \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and angle \(B = 60 ^ { \circ }\).

1. Find the exact area of the triangle, giving your answer as simply as possible.
2. Find the length of \(A C\), correct to 3 significant figures.

5

1. 1. Express \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x\) as a single logarithm.
   2. Hence solve the equation \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x = 2\).
2. Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int _ { 3 } ^ { 9 } \log _ { 10 } x \mathrm {~d} x ,$$ giving your answer correct to 3 significant figures.

6

1. Find and simplify the first four terms in the expansion of \(( 1 + 4 x ) ^ { 7 }\) in ascending powers of \(x\).
2. In the expansion of $$( 3 + a x ) ( 1 + 4 x ) ^ { 7 }$$ the coefficient of \(x ^ { 2 }\) is 1001 . Find the value of \(a\).
3. (a) Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), indicating the coordinates of any points where the curve meets the axes.
   (b) Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
4. Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(- 180 ^ { \circ }\) and \(180 ^ { \circ }\).

8 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 7 x + 33\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
2. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form as simply as possible. On its first trip between Malby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses \(2 \%\) more coal than the previous trip.

1. Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures.
2. There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02 ^ { N } \leqslant 1.52$$
3. Hence, by using logarithms, find the greatest number of trips possible.

10
\includegraphics[max width=\textwidth, alt={}, center]{dd199f4d-8cf3-4b1e-92aa-d54e9e94da57-4_693_931_269_607} The diagram shows the graph of \(y = 1 - 3 x ^ { - \frac { 1 } { 2 } }\).

1. Verify that the curve intersects the \(x\)-axis at \(( 9,0 )\).
2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\) ). Given that the area of the shaded region is 4 square units, find the value of \(a\).