Edexcel C2 (Core Mathematics 2) 2007 January

1. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$ Find

1. \(\mathrm { f } ^ { \prime \prime } ( x )\),
2. \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).

2. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 - 2 x ) ^ { 5 }\). Give each term in its simplest form.
(b) If \(x\) is small, so that \(x ^ { 2 }\) and higher powers can be ignored, show that $$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x$$ DU

3. The line joining the points \(( - 1,4 )\) and \(( 3,6 )\) is a diameter of the circle \(C\). Find an equation for \(C\).

4. Solve the equation $$5 ^ { x } = 17$$ giving your answer to 3 significant figures.

5. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise \(\mathrm { f } ( x )\) completely.
3. Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0$$

6. Find all the solutions, in the interval \(0 \leqslant x < 2 \pi\), of the equation $$2 \cos ^ { 2 } x + 1 = 5 \sin x$$ giving each solution in terms of \(\pi\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 5 )$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between \(x = 0\) and \(x = 2\) and is bounded by \(C\), the \(x\)-axis and the line \(x = 2\).
(9)

1. A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, \(\pounds C\), is given by

$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$

1. Find the value of \(v\) for which \(C\) is a minimum.
2. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
3. Calculate the minimum total cost of the journey.

9. \begin{figure}[h] \end{figure} Figure 2 shows a plan of a patio. The patio \(P Q R S\) is in the shape of a sector of a circle with centre \(Q\) and radius 6 m . Given that the length of the straight line \(P R\) is \(6 \sqrt { } 3 \mathrm {~m}\),

1. find the exact size of angle \(P Q R\) in radians.
2. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
3. Find the exact area of the triangle \(P Q R\).
4. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment \(P R S\).
5. Find, in \(m\) to 1 decimal place, the perimeter of the patio \(P Q R S\).

1. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
   1. Prove that the sum of the first \(n\) terms of this series is given by
   $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
2. Find $$\sum _ { k = 1 } ^ { 10 } 100 \left( 2 ^ { k } \right)$$
3. Find the sum to infinity of the geometric series $$\frac { 5 } { 6 } + \frac { 5 } { 18 } + \frac { 5 } { 54 } + \ldots$$
4. State the condition for an infinite geometric series with common ratio \(r\) to be convergent.