Edexcel C1 (Core Mathematics 1) 2012 January

Given that \(y = x ^ { 4 } + 6 x ^ { \frac { 1 } { 2 } }\), find in their simplest form

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\)

1. Simplify $$\sqrt { } 32 + \sqrt { } 18$$ giving your answer in the form \(a \sqrt { } 2\), where \(a\) is an integer.
2. Simplify $$\frac { \sqrt { } 32 + \sqrt { } 18 } { 3 + \sqrt { } 2 }$$ giving your answer in the form \(b \sqrt { } 2 + c\), where \(b\) and \(c\) are integers.

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

1. \(4 x - 5 > 15 - x\)
2. \(x ( x - 4 ) > 12\)

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

1. Write down an expression for \(x _ { 2 }\) in terms of \(a\).
2. Show that \(x _ { 3 } = a ^ { 2 } + 5 a + 5\) Given that \(x _ { 3 } = 41\)
3. find the possible values of \(a\).

5. The curve \(C\) has equation \(y = x ( 5 - x )\) and the line \(L\) has equation \(2 y = 5 x + 4\)

1. Use algebra to show that \(C\) and \(L\) do not intersect.
2. In the space on page 11, sketch \(C\) and \(L\) on the same diagram, showing the coordinates of the points at which \(C\) and \(L\) meet the axes.

6. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(2 x - 3 y + 12 = 0\)

1. Find the gradient of \(l _ { 1 }\). The line \(l _ { 1 }\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 1. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(B\).
2. Find an equation of \(l _ { 2 }\). The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(C\).
3. Find the area of triangle \(A B C\).

1. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,10 )\). Given that

$$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x + 5$$ find the value of \(\mathrm { f } ( 1 )\).

8. The curve \(C _ { 1 }\) has equation $$y = x ^ { 2 } ( x + 2 )$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Sketch \(C _ { 1 }\), showing the coordinates of the points where \(C _ { 1 }\) meets the \(x\)-axis.
3. Find the gradient of \(C _ { 1 }\) at each point where \(C _ { 1 }\) meets the \(x\)-axis. The curve \(C _ { 2 }\) has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where \(k\) is a constant and \(k > 2\)
4. Sketch \(C _ { 2 }\), showing the coordinates of the points where \(C _ { 2 }\) meets the \(x\) and \(y\) axes.

1. A company offers two salary schemes for a 10 -year period, Year 1 to Year 10 inclusive.

Scheme 1: Salary in Year 1 is \(\pounds P\).
Salary increases by \(\pounds ( 2 T )\) each year, forming an arithmetic sequence.
Scheme 2: Salary in Year 1 is \(\pounds ( P + 1800 )\).
Salary increases by \(\pounds T\) each year, forming an arithmetic sequence.

1. Show that the total earned under Salary Scheme 1 for the 10-year period is $$\pounds ( 10 P + 90 T )$$ For the 10-year period, the total earned is the same for both salary schemes.
2. Find the value of \(T\). For this value of \(T\), the salary in Year 10 under Salary Scheme 2 is \(\pounds 29850\)
3. Find the value of \(P\).

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$y = 2 - \frac { 1 } { x } , \quad x \neq 0$$ The curve crosses the \(x\)-axis at the point \(A\).

1. Find the coordinates of \(A\).
2. Show that the equation of the normal to \(C\) at \(A\) can be written as $$2 x + 8 y - 1 = 0$$ The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 2 .
3. Find the coordinates of \(B\).