Edexcel C2 (Core Mathematics 2)

1. A circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0 .$$

1. Find the coordinates of the centre of \(C\).
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.

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1. A circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0 .$$

1. Find the coordinates of the centre of \(C\).
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.
   3. Given that \(2 \sin 2 \theta = \cos 2 \theta\),
3. show that \(\tan 2 \theta = 0.5\).
4. Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2 \theta ^ { \circ } = \cos 2 \theta ^ { \circ }\).
   4. \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c\), where \(c\) is a constant. Given that \(\mathrm { f } ( 4 ) = 0\),
5. find the value of \(c\),
6. factorise \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
   (c Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(\mathrm { f } ( x ) = 0\).
   5. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13c2bf9f-f87a-420c-8cdc-9deb688112ae-3_538_618_283_749}
   \end{figure} Figure 1 shows the sector \(O A B\) of a circle of radius \(r \mathrm {~cm}\). The area of the sector is \(15 \mathrm {~cm} ^ { 2 }\) and \(\angle A O B = 1.5\) radians.
7. Prove that \(r = 2 \sqrt { } 5\).
8. Find, in cm , the perimeter of the sector \(O A B\). The segment \(R\), shaded in Fig 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
9. Calculate, to 3 decimal places, the area of \(R\).
   6. The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find
10. the common ratio of the series,
11. the first term of the series,
12. the sum to infinity of the series.
13. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series.
    7. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180 .$$
14. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis
15. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
16. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\)

8. (i) Solve, for \(0 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$\sin \left( 2 x + 50 ^ { \circ } \right) = 0.6$$ giving your answers to 1 decimal place.
(ii) In the triangle \(A B C , A C = 18 \mathrm {~cm} , \angle A B C = 60 ^ { \circ }\) and \(\sin A = \frac { 1 } { 3 }\).
(a Use the sine rule to show that \(B C = 4 \sqrt { } 3\).
(b) Find the exact value of \(\cos A\).

9. \begin{figure}[h] \end{figure} Figure 2 shows the line with equation \(y = 9 - x\) and the curve with equation \(y = x ^ { 2 } - 2 x + 3\). 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 \(B\). The shaded region \(R\) is bounded by the line and the curve.
2. Calculate the area of \(R\).