Edexcel C1 (Core Mathematics 1)

1. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7 .
   (b) Hence, or otherwise, evaluate \(\sum _ { r = 1 } ^ { 142 } ( 7 r + 2 )\).
2. Solve the simultaneous equations

$$\begin{gathered} x - 3 y + 1 = 0
x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{gathered}$$

1. The first three terms of an arithmetic series are \(p , 5 p - 8\), and \(3 p + 8\) respectively.
   1. Show that \(p = 4\).
   2. Find the value of the 40th term of this series.
   3. \(\mathrm { f } ( x ) = x ^ { 2 } - k x + 9\), where \(k\) is a constant.
   4. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions.
   Given that \(k = 4\),
2. express \(\mathrm { f } ( x )\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found,

5. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } }$$

1. Use integration to find \(y\) in terms of \(x\).
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).

6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$

7. \begin{figure}[h] \end{figure} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 1.

1. Find the coordinates of the centre of the circle.
2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.