Edexcel C1 (Core Mathematics 1) 2013 June

Given \(y = x ^ { 3 } + 4 x + 1\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 3\)

Express \(\frac { 15 } { \sqrt { 3 } } - \sqrt { 27 }\) in the form \(k \sqrt { } 3\), where \(k\) is an integer.

Find $$\int \left( 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x$$ giving each term in its simplest form.

4. The line \(L _ { 1 }\) has equation \(4 x + 2 y - 3 = 0\)

1. Find the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) is perpendicular to \(L _ { 1 }\) and passes through the point \(( 2,5 )\).
2. Find the equation of \(L _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

5. Solve

1. \(2 ^ { y } = 8\)
2. \(2 ^ { x } \times 4 ^ { x + 1 } = 8\)

6. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } \ldots\) is defined by $$\begin{gathered} x _ { 1 } = 1
x _ { n + 1 } = \left( x _ { n } \right) ^ { 2 } - k x _ { n } , \quad n \geqslant 1 \end{gathered}$$ where \(k\) is a constant, \(k \neq 0\)

1. Find an expression for \(x _ { 2 }\) in terms of \(k\).
2. Show that \(x _ { 3 } = 1 - 3 k + 2 k ^ { 2 }\) Given also that \(x _ { 3 } = 1\),
3. calculate the value of \(k\).
4. Hence find the value of \(\sum _ { n = 1 } ^ { 100 } x _ { n }\)

7. Each year, Abbie pays into a savings scheme. In the first year she pays in \(\pounds 500\). Her payments then increase by \(\pounds 200\) each year so that she pays \(\pounds 700\) in the second year, \(\pounds 900\) in the third year and so on.

1. Find out how much Abbie pays into the savings scheme in the tenth year. Abbie pays into the scheme for \(n\) years until she has paid in a total of \(\pounds 67200\).
2. Show that \(n ^ { 2 } + 4 n - 24 \times 28 = 0\)
3. Hence find the number of years that Abbie pays into the savings scheme.

1. A rectangular room has a width of \(x \mathrm {~m}\).

The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m ,

1. show that \(x > 2.8\) Given also that the area of the room is less than \(21 \mathrm {~m} ^ { 2 }\),
2. 1. write down an inequality, in terms of \(x\), for the area of the room.
   2. Solve this inequality.
3. Hence find the range of possible values for \(x\).

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 2,0 )\).
The curve \(C\) has a maximum at the point ( 0,4 ).

1. The equation of the curve \(C\) can be written in the form $$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$ where \(a\), \(b\) and \(c\) are integers.
   Calculate the values of \(a , b\) and \(c\).
2. Sketch the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) in the space provided on page 24 Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.

10. A curve has equation \(y = \mathrm { f } ( x )\). The point \(P\) with coordinates \(( 9,0 )\) lies on the curve. Given that $$\mathrm { f } ^ { \prime } ( x ) = \frac { x + 9 } { \sqrt { } x } , \quad x > 0$$

1. find \(\mathrm { f } ( x )\).
2. Find the \(x\)-coordinates of the two points on \(y = \mathrm { f } ( x )\) where the gradient of the curve is equal to 10

11. \begin{figure}[h] \end{figure} The line \(y = x + 2\) meets the curve \(x ^ { 2 } + 4 y ^ { 2 } - 2 x = 35\) at the points \(A\) and \(B\) as shown in Figure 2.

1. Find the coordinates of \(A\) and the coordinates of \(B\).
2. Find the distance \(A B\) in the form \(r \sqrt { 2 }\) where \(r\) is a rational number.