Edexcel C2 (Core Mathematics 2)

\begin{enumerate} \item (a) Using the factor theorem, show that \(( x + 3 )\) is a factor of \(x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24\).
(b) Factorise \(x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24\) completely. \item (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
(b) Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers. \item The first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + p x ) ^ { n }\), are \(1 - 18 x + 36 p ^ { 2 } x ^ { 2 }\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\).
[0pt] [P2 January 2003 Question 2] \item A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 8 y - 75 = 0\).

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). \end{enumerate} A second circle has centre at the point \(( 15,12 )\) and radius 10.
2. Sketch both circles on a single diagram and find the coordinates of the point where they touch.
   (4)
   [0pt] [P3 June 2003 Question 3]

5. (i) Differentiate \(2 x ^ { 3 } + \sqrt { } x + \frac { x ^ { 2 } + 2 x } { x ^ { 2 } }\) with respect to \(x\)
(ii) Evaluate \(\int _ { 1 } ^ { 4 } \left( \frac { x } { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).

6. A geometric series has first term 1200. Its sum to infinity is 960 .

1. Show that the common ratio of the series is \(- \frac { 1 } { 4 }\).
2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series.
3. Write down an expression for the sum of the first \(n\) terms of the series. Given that \(n\) is odd,
4. prove that the sum of the first \(n\) terms of the series is \(960 \left( 1 + 0.25 ^ { n } \right)\).

7. On a journey, the average speed of a car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For \(v \geq 5\), the cost per kilometre, \(C\) pence, of the journey is modelled by \(C = \frac { 160 } { v } + \frac { v ^ { 2 } } { 100 }\).
Using this model,

1. show, by calculus, that there is a value of \(v\) for which \(C\) has a stationary value, and find this value of \(v\).
2. Justify that this value of \(v\) gives a minimum value of \(C\).
3. Find the minimum value of \(C\) and hence find the minimum cost of a 250 km car journey.

8. \begin{figure}[h] \end{figure} The curve \(C\), shown in Fig. 1, represents the graph of \(y = \frac { x ^ { 2 } } { 25 } , x \geq 0\).
The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates 5 and 10 respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\).
2. Find an equation of the tangent to \(C\) at \(A\). The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
3. For points \(( x , y )\) on \(C\), express \(x\) in terms of \(y\).
4. Use integration to find the area of \(R\).

9. (i) Solve, for \(0 ^ { \circ } < x < 180 ^ { \circ }\), the equation \(\sin \left( 2 x + 50 ^ { \circ } \right) = 0.6\), giving your answers to 1 d. p.
(ii) In the triangle \(A B C , A C = 18 \mathrm {~cm} , \angle A B C = 60 ^ { \circ }\) and \(\sin A = \frac { 1 } { 3 }\).

1. Use the sine rule to show that \(B C = 4 \sqrt { } 3\).
2. Find the exact value of \(\cos A\). L