OCR MEI C1 (Core Mathematics 1)

1 Sketch the graph of \(y = 9 - x ^ { 2 }\).

2 \begin{figure}[h] \end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).

1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).

3 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12\).

1. Show that \(x = - 2\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Divide \(\mathrm { f } ( x )\) by \(x + 6\).
3. Express \(\mathrm { f } ( x )\) in fully factorised form.
4. Sketch the graph of \(y = \mathrm { f } ( x )\).
5. Solve the equation \(\mathrm { f } ( x ) = 12\).

4

1. Sketch the graph of \(y = x ( x - 3 ) ^ { 2 }\).
2. Show that the equation \(x ( x - 3 ) ^ { 2 } = 2\) can be expressed as \(x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0\).
3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i).

5

1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.