OCR C2 (Core Mathematics 2) 2008 January

1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 11 cm . The angle \(A O B\) is 0.7 radians. Find the area of the segment shaded in the diagram.

2 Use the trapezium rule, with 3 strips each of width 2, to estimate the value of $$\int _ { 1 } ^ { 7 } \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$$

3 Express each of the following as a single logarithm:

1. \(\log _ { a } 2 + \log _ { a } 3\),
2. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\).

4
\includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-2_515_713_1567_715} In the diagram, angle \(B D C = 50 ^ { \circ }\) and angle \(B C D = 62 ^ { \circ }\). It is given that \(A B = 10 \mathrm {~cm} , A D = 20 \mathrm {~cm}\) and \(B C = 16 \mathrm {~cm}\).

1. Find the length of \(B D\).
2. Find angle \(B A D\).

5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \sqrt { x }\). The curve passes through the point (4,50). Find the equation of the curve.

6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence it is.
3. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).

7
\includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-3_579_557_858_794} The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\).

1. Explain why \(\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 3 x \right) \mathrm { d } x\) does not give the total area of the regions shaded in the diagram.
2. Use integration to find the exact total area of the shaded regions.

8 The first term of a geometric progression is 10 and the common ratio is 0.8.

1. Find the fourth term.
2. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
3. The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as $$0.8 ^ { N } < 0.0002 ,$$ and use logarithms to find the smallest possible value of \(N\).

9

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_376_764_276_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the curve \(y = 2 \sin x\) for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). State the coordinates of the maximum and minimum points on this part of the curve.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_371_766_959_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Fig. 2 shows the curve \(y = 2 \sin x\) and the line \(y = k\). The smallest positive solution of the equation \(2 \sin x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\), and in the range \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\),
   (a) another solution of the equation \(2 \sin x = k\),
   (b) one solution of the equation \(2 \sin x = - k\).
3. Find the \(x\)-coordinates of the points where the curve \(y = 2 \sin x\) intersects the curve \(y = 2 - 3 \cos ^ { 2 } x\), for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

10

1. Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
2. Hence show that \(( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }\) can be written as $$320 x ^ { 3 } + k x$$ where the value of the constant \(k\) is to be stated.
3. Verify that \(x = 2\) is a root of the equation $$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$ and find the other possible values of \(x\).