Edexcel C2 (Core Mathematics 2)

A circle \(C\) has equation $$x^2 + y^2 - 10x + 6y - 15 = 0.$$

1. Find the coordinates of the centre of \(C\). [2]
2. Find the radius of \(C\). [2]

Express \(\frac{y + 3}{(y + 1)(y + 2)} - \frac{y + 1}{(y + 2)(y + 3)}\) as a single fraction in its simplest form. [5]

Given that \(2 \sin 2\theta = \cos 2\theta\),

1. show that \(\tan 2\theta = 0.5\). [1]
2. Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2\theta° = \cos 2\theta°\). [5]

$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),

1. find the value of \(c\), [2]
2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]

\includegraphics{figure_1} Figure 1 shows the sector \(OAB\) of a circle of radius \(r\) cm. The area of the sector is 15 cm\(^2\) and \(\angle AOB = 1.5\) radians.

1. Prove that \(r = 2\sqrt{5}\). [3]
2. Find, in cm, the perimeter of the sector \(OAB\). [2]

The segment \(R\), shaded in Fig 1, is enclosed by the arc \(AB\) and the straight line \(AB\).

1. Calculate, to 3 decimal places, the area of \(R\). [3]

The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

1. the common ratio of the series, [2]
2. the first term of the series, [2]
3. the sum to infinity of the series. [2]
4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]

$$f(x) = 5\sin 3x°, \quad 0 \leq x \leq 180.$$

1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis [3]
2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]

1. Solve, for \(0° < x < 180°\), the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [7]
2. In the triangle \(ABC\), \(AC = 18\) cm, \(\angle ABC = 60°\) and \(\sin A = \frac{1}{3}\).
   1. Use the sine rule to show that \(BC = 4\sqrt{3}\). [4]
   2. Find the exact value of \(\cos A\). [2]

\includegraphics{figure_2} Figure 2 shows the line with equation \(y = 9 - x\) and the curve with equation \(y = x^2 - 2x + 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the line and the curve.

1. Calculate the area of \(R\). [7]