OCR C2 (Core Mathematics 2) 2005 June

1 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by $$u _ { n } = 3 n - 1 ,$$ for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\), and state what type of sequence \(S\) is.
2. Evaluate \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-2_579_895_817_625} A sector \(O A B\) of a circle of radius \(r \mathrm {~cm}\) has angle \(\theta\) radians. The length of the arc of the sector is 12 cm and the area of the sector is \(36 \mathrm {~cm} ^ { 2 }\) (see diagram).

1. Write down two equations involving \(r\) and \(\theta\).
2. Hence show that \(r = 6\), and state the value of \(\theta\).
3. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\).

3

1. Find \(\int ( 2 x + 1 ) ( x + 3 ) \mathrm { d } x\).
2. Evaluate \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).

4
\includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-3_309_1084_269_532} In the diagram, \(A B C D\) is a quadrilateral in which \(A D\) is parallel to \(B C\). It is given that \(A B = 9 , B C = 6\), \(C A = 5\) and \(C D = 15\).

1. Show that \(\cos B C A = - \frac { 1 } { 3 }\), and hence find the value of \(\sin B C A\).
2. Find the angle \(A D C\) correct to the nearest \(0.1 ^ { \circ }\).

5 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and that the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) is 16 .

1. Find the values of \(a\) and \(b\).
2. Hence verify that \(\mathrm { f } ( 2 ) = 0\), and factorise \(\mathrm { f } ( x )\) completely.

6

1. Find the binomial expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 }\), simplifying the terms.
2. Hence find \(\int \left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 } \mathrm {~d} x\).

7

1. Evaluate \(\log _ { 5 } 15 + \log _ { 5 } 20 - \log _ { 5 } 12\).
2. Given that \(y = 3 \times 10 ^ { 2 x }\), show that \(x = a \log _ { 10 } ( b y )\), where the values of the constants \(a\) and \(b\) are to be found.

8 The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9 . The amount pumped in 2001 was 100000 barrels.

1. Calculate the amount pumped in 2004. It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.
2. Calculate in which year the amount pumped will fall below 5000 barrels.
3. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production.

9

1. 1. Write down the exact values of \(\cos \frac { 1 } { 6 } \pi\) and \(\tan \frac { 1 } { 3 } \pi\) (where the angles are in radians). Hence verify that \(x = \frac { 1 } { 6 } \pi\) is a solution of the equation $$2 \cos x = \tan 2 x$$
   2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2 x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2 x$$
2. 1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.)
   2. State with a reason whether this approximation is an underestimate or an overestimate.