OCR C2 (Core Mathematics 2) 2009 January

1 Find

1. \(\int \left( x ^ { 3 } + 8 x - 5 \right) \mathrm { d } x\),
2. \(\int 12 \sqrt { x } \mathrm {~d} x\).

2
\includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-2_311_521_651_810} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 7 cm . The angle \(A O B\) is \(140 ^ { \circ }\).

1. Express \(140 ^ { \circ }\) in radians, giving your answer in an exact form as simply as possible.
2. Find the perimeter of the segment shaded in the diagram, giving your answer correct to 3 significant figures.

3 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 24 - \frac { 2 } { 3 } n$$

1. Write down the exact values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(k\) such that \(u _ { k } = 0\).
3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-3_570_853_269_644} The diagram shows the curve \(y = x ^ { 4 } + 3\) and the line \(y = 19\) which intersect at \(( - 2,19 )\) and \(( 2,19 )\). Use integration to find the exact area of the shaded region enclosed by the curve and the line.

5
\includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-3_623_355_1123_897} Some walkers see a tower, \(T\), in the distance and want to know how far away it is. They take a bearing from a point \(A\) and then walk for 50 m in a straight line before taking another bearing from a point \(B\). They find that angle \(T A B\) is \(70 ^ { \circ }\) and angle \(T B A\) is \(107 ^ { \circ }\) (see diagram).

1. Find the distance of the tower from \(A\).
2. They continue walking in the same direction for another 100 m to a point \(C\), so that \(A C\) is 150 m . What is the distance of the tower from \(C\) ?
3. Find the shortest distance of the walkers from the tower as they walk from \(A\) to \(C\).

6 A geometric progression has first term 20 and common ratio 0.9.

1. Find the sum to infinity.
2. Find the sum of the first 30 terms.
3. Use logarithms to find the smallest value of \(p\) such that the \(p\) th term is less than 0.4 .

7 In the binomial expansion of \(( k + a x ) ^ { 4 }\) the coefficient of \(x ^ { 2 }\) is 24 .

1. Given that \(a\) and \(k\) are both positive, show that \(a k = 2\).
2. Given also that the coefficient of \(x\) in the expansion is 128 , find the values of \(a\) and \(k\).
3. Hence find the coefficient of \(x ^ { 3 }\) in the expansion.

8

1. Given that \(\log _ { a } x = p\) and \(\log _ { a } y = q\), express the following in terms of \(p\) and \(q\).
   1. \(\log _ { a } ( x y )\)
   2. \(\log _ { a } \left( \frac { a ^ { 2 } x ^ { 3 } } { y } \right)\)
2. 1. Express \(\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x\) as a single logarithm.
   2. Hence solve the equation \(\log _ { 10 } \left( x ^ { 2 } - 10 \right) - \log _ { 10 } x = 2 \log _ { 10 } 3\).

9

1. The polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 3 x + 3$$ Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\), and hence find the other two roots.
2. Hence solve the equation $$\tan ^ { 3 } x - \tan ^ { 2 } x - 3 \tan x + 3 = 0$$ for \(0 \leqslant x \leqslant 2 \pi\). Give each solution for \(x\) in an exact form.