Edexcel C1 (Core Mathematics 1) 2010 June

1. Write

$$\sqrt { } ( 75 ) - \sqrt { } ( 27 )$$ in the form \(k \sqrt { } x\), where \(k\) and \(x\) are integers.

2. Find $$\int \left( 8 x ^ { 3 } + 6 x ^ { \frac { 1 } { 2 } } - 5 \right) d x$$ giving each term in its simplest form.
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3. Find the set of values of \(x\) for which

1. \(3 ( x - 2 ) < 8 - 2 x\)
2. \(( 2 x - 7 ) ( 1 + x ) < 0\)
3. both \(3 ( x - 2 ) < 8 - 2 x\) and \(( 2 x - 7 ) ( 1 + x ) < 0\)

4. (a) Show that \(x ^ { 2 } + 6 x + 11\) can be written as $$( x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers to be found.
(b) In the space at the top of page 7 , sketch the curve with equation \(y = x ^ { 2 } + 6 x + 11\), showing clearly any intersections with the coordinate axes.
(c) Find the value of the discriminant of \(x ^ { 2 } + 6 x + 11\)

1. A sequence of positive numbers is defined by

$$\begin{aligned} a _ { n + 1 } & = \sqrt { } \left( a _ { n } ^ { 2 } + 3 \right) , \quad n \geqslant 1 ,
a _ { 1 } & = 2 \end{aligned}$$

1. Find \(a _ { 2 }\) and \(a _ { 3 }\), leaving your answers in surd form.
2. Show that \(a _ { 5 } = 4\)

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum point \(A\) at \(( - 2,3 )\) and a minimum point \(B\) at \(( 3 , - 5 )\). On separate diagrams sketch the curve with equation

1. \(y = \mathrm { f } ( x + 3 )\)
2. \(y = 2 \mathrm { f } ( x )\) On each diagram show clearly the coordinates of the maximum and minimum points.
   The graph of \(y = \mathrm { f } ( x ) + a\) has a minimum at (3, 0), where \(a\) is a constant.
3. Write down the value of \(a\).

1. Given that

$$y = 8 x ^ { 3 } - 4 \sqrt { } x + \frac { 3 x ^ { 2 } + 2 } { x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(6)

8. (a) Find an equation of the line joining \(A ( 7,4 )\) and \(B ( 2,0 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(b) Find the length of \(A B\), leaving your answer in surd form. The point \(C\) has coordinates ( \(2 , t\) ), where \(t > 0\), and \(A C = A B\).
(c) Find the value of \(t\).
(d) Find the area of triangle \(A B C\).
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1. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays \(\pounds a\) for their first day, \(\pounds ( a + d )\) for their second day, \(\pounds ( a + 2 d )\) for their third day, and so on, thus increasing the daily payment by \(\pounds d\) for each extra day they work.

A picker who works for all 30 days will earn \(\pounds 40.75\) on the final day.

1. Use this information to form an equation in \(a\) and \(d\). A picker who works for all 30 days will earn a total of \(\pounds 1005\)
2. Show that \(15 ( a + 40.75 ) = 1005\)
3. Hence find the value of \(a\) and the value of \(d\).

10. (a) On the axes below sketch the graphs of

1. \(y = x ( 4 - x )\)
2. \(y = x ^ { 2 } ( 7 - x )\)
   showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\)-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 4 \right) = 0\) The point \(A\) lies on both of the curves and the \(x\) and \(y\) coordinates of \(A\) are both positive.
   (c) Find the exact coordinates of \(A\), leaving your answer in the form ( \(p + q \sqrt { } 3 , r + s \sqrt { } 3\) ), where \(p , q , r\) and \(s\) are integers.
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1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\), where

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x - \frac { 5 } { \sqrt { } x } - 2$$ Given that the point \(P ( 4,5 )\) lies on \(C\), find

1. \(\mathrm { f } ( x )\),
2. an equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.