Edexcel C2 (Core Mathematics 2)

A circle has the equation \(x^2 + y^2 - 6y - 7 = 0\).

1. Find the coordinates of the centre of the circle. [2]
2. Find the radius of the circle. [2]

\includegraphics{figure_1} Figure 1 shows the sector \(OAB\) of a circle, centre \(O\), in which \(\angle AOB = 2.5\) radians. Given that the perimeter of the sector is 36 cm,

1. find the length \(OA\), [2]
2. find the area of the shaded segment. [3]

\includegraphics{figure_2} Figure 2 shows the curves with equations \(y = 7 - 2x - 3x^2\) and \(y = \frac{2}{x}\). The two curves intersect at the points \(P\), \(Q\) and \(R\).

1. Show that the \(x\)-coordinates of \(P\), \(Q\) and \(R\) satisfy the equation $$3x^3 + 2x^2 - 7x + 2 = 0.$$ [2] Given that \(P\) has coordinates \((-2, -1)\),
2. find the coordinates of \(Q\) and \(R\). [6]

1. Expand \((1 + x)^4\) in ascending powers of \(x\). [2]
2. Using your expansion, express each of the following in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.
   1. \((1 + \sqrt{2})^4\)
   2. \((1 - \sqrt{2})^8\) [7]

1. Describe fully a single transformation that maps the graph of \(y = 3^x\) onto the graph of \(y = (\frac{1}{3})^x\). [1]
2. Sketch on the same diagram the curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\), showing the coordinates of any points where each curve crosses the coordinate axes. [3]

The curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\) intersect at the point \(P\).

1. Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt{2}\). [5]

A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),

1. find the values of \(a\) and \(b\), [6]
2. find the coordinates of the other stationary point of the curve. [3]

\includegraphics{figure_3} Figure 3 shows part of the curve \(y = \text{f}(x)\) where $$\text{f}(x) = \frac{1 - 8x^3}{x^2}, \quad x \neq 0.$$

1. Solve the equation \(\text{f}(x) = 0\). [3]
2. Find \(\int \text{f}(x) \, dx\). [3]
3. Find the area of the shaded region bounded by the curve \(y = \text{f}(x)\), the \(x\)-axis and the line \(x = 2\). [3]

1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos(2x - \frac{\pi}{6}) = \frac{1}{2}.$$ [7]

The second and fifth terms of a geometric series are \(-48\) and \(6\) respectively.

1. Find the first term and the common ratio of the series. [5]
2. Find the sum to infinity of the series. [2]
3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2^{6-n}\). [5]