Edexcel C1 (Core Mathematics 1)

1. Given that \(8 = 2^k\), write down the value of \(k\). [1]
2. Given that \(4^x = 8^{2-x}\), find the value of \(x\). [4]

Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,

1. find the value of \(a\) and the value of \(b\). [2]

Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,

1. find the value of \(c\) and the value of \(d\). [3]

\(y = 7 + 10x^{\frac{1}{3}}\).

1. Find \(\frac{dy}{dx}\). [2]
2. Find \(\int y \, dx\). [3]

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

The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).

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. [3]

The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Calculate, as exact fractions, the coordinates of \(A\). [3]
2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]

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

1. Find the amount that Anne will pay in the 40th year. [2]
2. Find the total amount that Anne will pay in over the 40 years. [2]

Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in £890 and his payments then increase by £\(d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,

1. find the value of \(d\). [4]

\includegraphics{figure_1} The points \(A\) and \(B\) have coordinates \((2, -3)\) and \((8, 5)\) respectively, and \(AB\) is a chord of a circle with centre \(C\), as shown in Fig. 1.

1. Find the gradient of \(AB\). [2]

The point \(M\) is the mid-point of \(AB\).

1. Find an equation for the line through \(C\) and \(M\). [5]

Given that the \(x\)-coordinate of \(C\) is 4,

1. find the \(y\)-coordinate of \(C\), [2]
2. show that the radius of the circle is \(\frac{5\sqrt{17}}{4}\). [4]

\includegraphics{figure_4} 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 \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]