Edexcel C1 (Core Mathematics 1)

1. (a) Given that \(8 = 2 ^ { k }\), write down the value of \(k\).
   (b) Given that \(4 ^ { x } = 8 ^ { 2 - x }\), find the value of \(x\).
2. Given that \(( 2 + \sqrt { } 7 ) ( 4 - \sqrt { 7 } ) = a + b \sqrt { } 7\), where a and \(b\) are integers,
   (a) find the value of a and the value of \(b\).

Given that \(\frac { 2 + \sqrt { 7 } } { 4 + \sqrt { 7 } } = c + d \sqrt { 7 }\) where \(c\) and \(d\) are rational numbers,
(b) find the value of \(c\) and the value of \(d\).

3. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } } .$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find \(\int y \mathrm {~d} x\).

4. (a) By completing the square, find in terms of \(k\) the roots of the equation $$x ^ { 2 } + 2 k x - 7 = 0 .$$ (b) Prove that, for all values of \(k\), the roots of \(x ^ { 2 } + 2 k x - 7 = 0\) are real and different.
(c) Given that \(k = \sqrt { } 2\), find the exact roots of the equation.

5. The straight line \(l _ { 1 }\) has equation \(4 y + x = 0\). The straight line \(l _ { 2 }\) has equation \(y = 2 x - 3\).

1. On the same axes, sketch the graphs of \(l _ { 1 }\) and \(l _ { 2 }\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).
2. Calculate, as exact fractions, the coordinates of \(A\).
3. Find an equation of the line through \(A\) which is perpendicular to \(l _ { 1 }\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays \(\pounds 500\). Her payments then increase by \(\pounds 50\) each year, so that she pays \(\pounds 550\) in the second year, \(\pounds 600\) in the third year, and so on.

1. Find the amount that Anne will pay in the 40th year.
2. Find the total amount that Anne will pay in over the 40 years. Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in \(\pounds 890\) and his payments then increase by \(\pounds d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
3. find the value of \(d\).

7. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) have coordinates \(( 2 , - 3 )\) and \(( 8,5 )\) respectively, and \(A B\) is a chord of a circle with centre \(C\), as shown in Fig. 1.

1. Find the gradient of \(A B\). The point \(M\) is the mid-point of \(A B\).
2. Find an equation for the line through \(C\) and \(M\).
   (5) Given that the \(x\)-coordinate of \(C\) is 4 ,
3. find the \(y\)-coordinate of \(C\),
   (2)
4. show that the radius of the circle is \(\frac { 5 \sqrt { } 17 } { 4 }\).
   (4)

8. \begin{figure}[h] \end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2 x cm by x cm and height \(h \mathrm {~cm}\), as shown in Fig. 4. Given that the capacity of a carton has to be \(1030 \mathrm {~cm} ^ { 3 }\),

1. express \(h\) in terms of \(x\),
2. show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$