Edexcel C1 (Core Mathematics 1) 2006 June

Find \(\int \left( 6 x ^ { 2 } + 2 + x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving each term in its simplest form.

Find the set of values of \(x\) for which $$x ^ { 2 } - 7 x - 18 > 0 .$$

3. On separate diagrams, sketch the graphs of

1. \(y = ( x + 3 ) ^ { 2 }\),
2. \(y = ( x + 3 ) ^ { 2 } + k\), where \(k\) is a positive constant. Show on each sketch the coordinates of each point at which the graph meets the axes.

4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} & a _ { 1 } = 3 \\ & a _ { n + 1 } = 3 a _ { n } - 5 , \quad n \geqslant 1 . \end{aligned}$$

1. Find the value of \(a _ { 2 }\) and the value of \(a _ { 3 }\).
2. Calculate the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\).

5. Differentiate with respect to \(x\)

1. \(x ^ { 4 } + 6 \sqrt { } x\),
2. \(\frac { ( x + 4 ) ^ { 2 } } { x }\).

6. (a) Expand and simplify \(( 4 + \sqrt { 3 } ) ( 4 - \sqrt { 3 } )\).
(b) Express \(\frac { 26 } { 4 + \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.

7. An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day, he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a \mathrm {~km}\) and common difference \(d \mathrm {~km}\). He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period.
Find the value of \(a\) and the value of \(d\).

8. The equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\), where \(p\) is a positive constant, has equal roots.

1. Find the value of \(p\).
2. For this value of \(p\), solve the equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\).

9. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } - 6 x \right) ( x - 2 ) + 3 x\),

1. express \(\mathrm { f } ( x )\) in the form \(x \left( a x ^ { 2 } + b x + c \right)\), where \(a\), \(b\) and \(c\) are constants.
2. Hence factorise \(\mathrm { f } ( x )\) completely.
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of each point at which the graph meets the axes.

10. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \neq 0\), passes through the point ( \(3,7 \frac { 1 } { 2 }\) ). Given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 3 } { x ^ { 2 } }\),

1. find \(\mathrm { f } ( x )\).
2. Verify that \(f ( - 2 ) = 5\).
3. Find an equation for the tangent to \(C\) at the point ( \(- 2,5\) ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. The line \(l _ { 1 }\) passes through the points \(P ( - 1,2 )\) and \(Q ( 11,8 )\).
   1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   The line \(l _ { 2 }\) passes through the point \(R ( 10,0 )\) and is perpendicular to \(l _ { 1 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(S\).
2. Calculate the coordinates of \(S\).
3. Show that the length of \(R S\) is \(3 \sqrt { 5 }\).
4. Hence, or otherwise, find the exact area of triangle \(P Q R\).