Edexcel C2 (Core Mathematics 2) 2008 June

1. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 39 x + 20$$

1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise f ( \(x\) ) completely.

2. $$y = \sqrt { } \left( 5 ^ { x } + 2 \right)$$

1. Complete the table below, giving the values of \(y\) to 3 decimal places.

   \(x\)	0	0.5	1	1.5	2
   \(y\)			2.646	3.630

2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } \sqrt { } \left( 5 ^ { x } + 2 \right) \mathrm { d } x\).

3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 10 }\), where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 3 }\) is double the coefficient of \(x ^ { 2 }\),
(b) find the value of \(a\).

4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
(b) Solve the equation \(5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0\).

5. The circle \(C\) has centre \(( 3,1 )\) and passes through the point \(P ( 8,3 )\).

1. Find an equation for \(C\).
2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6. A geometric series has first term 5 and common ratio \(\frac { 4 } { 5 }\). Calculate

1. the 20th term of the series, to 3 decimal places,
2. the sum to infinity of the series. Given that the sum to \(k\) terms of the series is greater than 24.95,
3. show that \(k > \frac { \log 0.002 } { \log 0.8 }\),
4. find the smallest possible value of \(k\).

7. \begin{figure}[h] \end{figure} Figure 1 shows \(A B C\), a sector of a circle with centre \(A\) and radius 7 cm .
Given that the size of \(\angle B A C\) is exactly 0.8 radians, find

1. the length of the arc \(B C\),
2. the area of the sector \(A B C\). The point \(D\) is the mid-point of \(A C\). The region \(R\), shown shaded in Figure 1, is bounded by \(C D , D B\) and the arc \(B C\). Find
3. the perimeter of \(R\), giving your answer to 3 significant figures,
4. the area of \(R\), giving your answer to 3 significant figures.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = 10 + 8 x + x ^ { 2 } - x ^ { 3 }\).
The curve has a maximum turning point \(A\).

1. Using calculus, show that the \(x\)-coordinate of \(A\) is 2 . The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(y\)-axis and the line from \(O\) to \(A\), where \(O\) is the origin.
2. Using calculus, find the exact area of \(R\).

9. Solve, for \(0 \leqslant x < 360 ^ { \circ }\),

1. \(\quad \sin \left( x - 20 ^ { \circ } \right) = \frac { 1 } { \sqrt { 2 } }\)
2. \(\cos 3 x = - \frac { 1 } { 2 }\)