Edexcel C1 (Core Mathematics 1)

1. (a) Write down the value of \(8 ^ { \frac { 1 } { 3 } }\).
   (b) Find the value of \(8 ^ { - \frac { 2 } { 3 } }\).
   
2. Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
   (a) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
   (b) find \(\int y \mathrm {~d} x\).
   

2. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1$$

1. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Write down the value of \(u _ { 20 }\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the origin \(O\) and through the point \(( 6,0 )\). The maximum point on the curve is \(( 3,5 )\). On separate diagrams, sketch the curve with equation

1. \(y = 3 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( x + 2 )\). On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.

5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$

8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) passes through the origin \(O\) and has gradient - 2 . The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
2. Calculate the coordinates of \(P\). Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
3. calculate the exact area of \(\triangle O C P\). \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882} \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}

9. An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.