Edexcel M2 (Mechanics 2) 2014 January

2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving each term in its simplest form.
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4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at \(( 0,3 )\) and has a minimum at \(P ( 4,2 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( x + 4 )\),
2. \(y = 2 \mathrm { f } ( x )\). On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.

6. \begin{figure}[h] \end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 7\)
The line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.

1. 1. State the gradient of \(l _ { 1 }\)
   2. Write down the coordinates of the point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B ( 1,5 )\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle A B C = 90 ^ { \circ }\),
2. find an equation of \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The rectangle \(A B C D\), shown shaded in Figure 2, has vertices at the points \(A , B , C\) and \(D\).
3. Find the exact area of rectangle \(A B C D\).