Edexcel C2 (Core Mathematics 2)

1. \begin{figure}[h] \end{figure} Figure 1 shows the sector \(O A B\) of a circle of radius 9.2 cm and centre \(O\).
Given that the area of the sector is \(37.4 \mathrm {~cm} ^ { 2 }\), find to 3 significant figures

1. the size of \(\angle A O B\) in radians,
2. the perimeter of the sector.

2. The first three terms of a geometric series are ( \(p - 1\) ), 2 and ( \(2 p + 5\) ) respectively, where \(p\) is a constant. Find the two possible values of \(p\).

3. Find the area of the finite region enclosed by the curve \(y = 5 x - x ^ { 2 }\) and the \(x\)-axis.

4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta ,$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).

5. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6 ,$$ and that \(x = - 3\) is a solution to the equation \(\mathrm { f } ( x ) = 0\),

1. find the value of the constant \(p\),
2. show that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) there is a remainder of 50 ,
3. find the other solutions to the equation \(\mathrm { f } ( x ) = 0\), giving your answers to 2 decimal places.

6. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0 .$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\).
3. Sketch C. Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).

7. Given that for small values of \(x\) $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 } ,$$ where \(n\) is an integer and \(n > 1\),

1. show that \(n = 16\) and find the value of \(a\),
2. use your value of \(a\) and a suitable value of \(x\) to estimate the value of (0.9985) \({ } ^ { 16 }\), giving your answer to 5 decimal places.

8. (a) Given that $$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x ,$$ show that $$y = 2 x + 1 .$$ (b) Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x
& 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$

9. \begin{figure}[h] \end{figure} Figure 2 shows a tray made from sheet metal.
The horizontal base is a rectangle measuring \(8 x \mathrm {~cm}\) by \(y \mathrm {~cm}\) and the two vertical sides are trapezia of height \(x \mathrm {~cm}\) with parallel edges of length \(8 x \mathrm {~cm}\) and \(10 x \mathrm {~cm}\). The remaining two sides are rectangles inclined at \(45 ^ { \circ }\) to the horizontal. Given that the capacity of the tray is \(900 \mathrm {~cm} ^ { 3 }\),

1. find an expression for \(y\) in terms of \(x\),
2. show that the area of metal used to make the tray, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 18 x ^ { 2 } + \frac { 200 ( 4 + \sqrt { 2 } ) } { x } ,$$
3. find to 3 significant figures, the value of \(x\) for which \(A\) is stationary,
4. find the minimum value of \(A\) and show that it is a minimum.