Edexcel C2 (Core Mathematics 2)

1. The point \(A\) has coordinates \(( 2,5 )\) and the point \(B\) has coordinates \(( - 2,8 )\). Find, in cartesian form, an equation of the circle with diameter \(A B\).

\begin{figure}[h] \end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.

1. Write down the value of \(a\) and the value of \(b\).
2. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
3. Find, to 3 significant figures, the length of \(P T\).

3. \(f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9\), where \(p\) is a constant.

1. Given that \(\mathrm { f } ( n )\) has a remainder of 3 when it is divided by ( \(n + 2\) ), prove that \(p = 6\).
2. Show that \(\mathrm { f } ( n )\) can be written in the form \(( n + 2 ) ( n + q ) ( n + r ) + 3\), where \(q\) and \(r\) are integers to be found.
3. Hence show that \(\mathrm { f } ( n )\) is divisible by 3 for all positive integer values of \(n\).

4. (a) Sketch, for \(0 \leq x \leq 360 ^ { \circ }\), the graph of \(y = \sin \left( x + 30 ^ { \circ } \right)\).
(b) Write down the coordinates of the points at which the graph meets the axes.
(c) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(\sin \left( x + 30 ^ { \circ } \right) = - \frac { 1 } { 2 }\).

5. The expansion of \(( 2 - p x ) ^ { 6 }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\), is $$64 + A x + 135 x ^ { 2 }$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\).

6. Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which $$2 \cos ^ { 2 } \theta - \cos \theta - 1 = \sin ^ { 2 } \theta$$ Give your answers to 1 decimal place where appropriate.
(8)

7. \section*{Figure 2} Fig. 2 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.

1. Prove that \(r = 2 \sqrt { 5 }\).
2. 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 \(A B\).
3. Calculate, to 3 decimal places, the area of \(R\).

8. \begin{figure}[h] \end{figure} Fig. 3 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 \(R\).
   [0pt] [P1 June 2003 Question 7]

9. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find the coordinates of each of the stationary points,
3. determine the nature of each stationary point. The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
4. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.