Edexcel C1 (Core Mathematics 1) 2009 June

2. Given that \(32 \sqrt { } 2 = 2 ^ { a }\), find the value of \(a\).

3. Given that \(y = 2 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } , x \neq 0\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\), simplifying each term.

4. Find the set of values of \(x\) for which

1. \(4 x - 3 > 7 - x\)
2. \(2 x ^ { 2 } - 5 x - 12 < 0\)
3. both \(4 x - 3 > 7 - x\) and \(2 x ^ { 2 } - 5 x - 12 < 0\)

5. A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and finished in 1990 (Year 40). The numbers of houses built each year form an arithmetic sequence with first term \(a\) and common difference \(d\). Given that 2400 new houses were built in 1960 and 600 new houses were built in 1990, find

1. the value of \(d\),
2. the value of \(a\),
3. the total number of houses built in Oldtown over the 40-year period.

6. The equation \(x ^ { 2 } + 3 p x + p = 0\), where \(p\) is a non-zero constant, has equal roots. Find the value of \(p\).

7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 2 a _ { n } - 7 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
2. Show that \(a _ { 3 } = 4 k - 21\). Given that \(\sum _ { r = 1 } ^ { 4 } a _ { r } = 43\),
3. find the value of \(k\).

8. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) have coordinates \(( 6,7 )\) and \(( 8,2 )\) respectively.
The line \(l\) passes through the point \(A\) and is perpendicular to the line \(A B\), as shown in Figure 1.

1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. Given that \(l\) intersects the \(y\)-axis at the point \(C\), find
2. the coordinates of \(C\),
3. the area of \(\triangle O C B\), where \(O\) is the origin.

9. $$f ( x ) = \frac { ( 3 - 4 \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = 9 x ^ { - \frac { 1 } { 2 } } + A x ^ { \frac { 1 } { 2 } } + B\), where \(A\) and \(B\) are constants to be found.
2. Find \(\mathrm { f } ^ { \prime } ( x )\).
3. Evaluate \(\mathrm { f } ^ { \prime } ( 9 )\).

10. (a) Factorise completely \(x ^ { 3 } - 6 x ^ { 2 } + 9 x\)
(b) Sketch the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. Using your answer to part (b), or otherwise,
(c) sketch, on a separate diagram, the curve with equation $$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$ showing the coordinates of the points at which the curve meets the \(x\)-axis.

11. The curve \(C\) has equation $$y = x ^ { 3 } - 2 x ^ { 2 } - x + 9 , \quad x > 0$$ The point \(P\) has coordinates (2, 7).

1. Show that \(P\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The point \(Q\) also lies on \(C\).
   Given that the tangent to \(C\) at \(Q\) is perpendicular to the tangent to \(C\) at \(P\),
3. show that the \(x\)-coordinate of \(Q\) is \(\frac { 1 } { 3 } ( 2 + \sqrt { 6 } )\).