Edexcel C1 (Core Mathematics 1) 2010 January

Given that \(y = x ^ { 4 } + x ^ { \frac { 1 } { 3 } } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).

1. Expand and simplify \(( 7 + \sqrt { 5 } ) ( 3 - \sqrt { 5 } )\).
2. Express \(\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.

The line \(l _ { 1 }\) has equation \(3 x + 5 y - 2 = 0\)

1. Find the gradient of \(l _ { 1 }\). The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 3,1 )\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

4. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 35\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.

5. Solve the simultaneous equations $$\begin{array} { r } y - 3 x + 2 = 0
y ^ { 2 } - x - 6 x ^ { 2 } = 0 \end{array}$$

6. The curve \(C\) has equation $$y = \frac { ( x + 3 ) ( x - 8 ) } { x } , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
2. Find an equation of the tangent to \(C\) at the point where \(x = 2\)

7. Jill gave money to a charity over a 20 -year period, from Year 1 to Year 20 inclusive. She gave \(\pounds 150\) in Year \(1 , \pounds 160\) in Year 2, \(\pounds 170\) in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.

1. Find the amount of money she gave in Year 10.
2. Calculate the total amount of money she gave over the 20 -year period. Kevin also gave money to the charity over the same 20 -year period. He gave \(\pounds A\) in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference \(\pounds 30\). The total amount of money that Kevin gave over the 20 -year period was twice the total amount of money that Jill gave.
3. Calculate the value of \(A\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve has a maximum point \(( - 2,5 )\) and an asymptote \(y = 1\), as shown in Figure 1. On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( x ) + 2\)
2. \(y = 4 \mathrm { f } ( x )\)
3. \(y = \mathrm { f } ( \mathrm { x } + 1 )\) On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.

1. (a) Factorise completely \(x ^ { 3 } - 4 x\)
   (b) Sketch the curve \(C\) with equation

$$y = x ^ { 3 } - 4 x ,$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. The point \(A\) with \(x\)-coordinate - 1 and the point \(B\) with \(x\)-coordinate 3 lie on the curve \(C\).
(c) Find an equation of the line which passes through \(A\) and \(B\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
(d) Show that the length of \(A B\) is \(k \sqrt { } 10\), where \(k\) is a constant to be found.

10. $$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\). Given that the equation \(\mathrm { f } ( x ) = 0\) has no real roots,
2. find the set of possible values of \(k\). Given that \(k = 1\),
3. sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any point at which the graph crosses a coordinate axis.