Edexcel C2 (Core Mathematics 2) 2012 January

1. A geometric series has first term \(a = 360\) and common ratio \(r = \frac { 7 } { 8 }\)

Giving your answers to 3 significant figures where appropriate, find

1. the 20 th term of the series,
2. the sum of the first 20 terms of the series,
3. the sum to infinity of the series.

2. A circle \(C\) has centre \(( - 1,7 )\) and passes through the point \(( 0,0 )\). Find an equation for \(C\).
(4)

3. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$$ giving each term in its simplest form.
(b) Use your expansion to estimate the value of \(( 1.025 ) ^ { 8 }\), giving your answer to 4 decimal places.

4. Given that \(y = 3 x ^ { 2 }\),

1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$

1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 3\), where \(a\) and \(b\) are constants.

Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 7 ,

1. show that \(2 a - b = 6\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4 ,
2. find the value of \(a\) and the value of \(b\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the curve with equation $$y = \frac { 16 } { x ^ { 2 } } - \frac { x } { 2 } + 1 , \quad x > 0$$ The finite region \(R\), bounded by the lines \(x = 1\), the \(x\)-axis and the curve, is shown shaded in Figure 1. The curve crosses the \(x\)-axis at the point \(( 4,0 )\).

1. Complete the table with the values of \(y\) corresponding to \(x = 2\) and 2.5

   \(x\)	1	1.5	2	2.5	3	3.5	4
   \(y\)	16.5	7.361			1.278	0.556	0

2. Use the trapezium rule with all the values in the completed table to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
3. Use integration to find the exact value for the area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 2 shows \(A B C\), a sector of a circle of radius 6 cm with centre \(A\). Given that the size of angle \(B A C\) is 0.95 radians, find

1. the length of the \(\operatorname { arc } B C\),
2. the area of the sector \(A B C\). The point \(D\) lies on the line \(A C\) and is such that \(A D = B D\). The region \(R\), shown shaded in Figure 2, is bounded by the lines \(C D , D B\) and the \(\operatorname { arc } B C\).
3. Show that the length of \(A D\) is 5.16 cm to 3 significant figures. Find
4. the perimeter of \(R\),
5. the area of \(R\), giving your answer to 2 significant figures.

8. \begin{figure}[h] \end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
3. Use calculus to find the minimum value of \(P\).
4. Find the width of each rectangle when the perimeter is a minimum. Give your answer to the nearest centimetre.

1. (i) Find the solutions of the equation \(\sin \left( 3 x - 15 ^ { \circ } \right) = \frac { 1 } { 2 }\), for which \(0 \leqslant x \leqslant 180 ^ { \circ }\) (6)
   (ii)

\begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation $$y = \sin ( a x - b ) , \text { where } a > 0,0 < b < \pi$$ The curve cuts the \(x\)-axis at the points \(P , Q\) and \(R\) as shown.
Given that the coordinates of \(P , Q\) and \(R\) are \(\left( \frac { \pi } { 10 } , 0 \right) , \left( \frac { 3 \pi } { 5 } , 0 \right)\) and \(\left( \frac { 11 \pi } { 10 } , 0 \right)\) respectively, find the values of \(a\) and \(b\).