OCR MEI C1 (Core Mathematics 1) 2006 January

2 \begin{figure}[h] \end{figure} Fig. 2 shows graphs \(A\) and \(B\).

1. State the transformation which maps graph \(A\) onto graph \(B\).
2. The equation of graph \(A\) is \(y = \mathrm { f } ( x )\). Which one of the following is the equation of graph \(B\) ? $$\begin{array} { l l l l } y = \mathrm { f } ( x ) + 2 & y = \mathrm { f } ( x ) - 2 & y = \mathrm { f } ( x + 2 ) & y = \mathrm { f } ( x - 2 )
   y = 2 \mathrm { f } ( x ) & y = \mathrm { f } ( x + 3 ) & y = \mathrm { f } ( x - 3 ) & y = 3 \mathrm { f } ( x ) \end{array}$$

3 Find the binomial expansion of \(( 2 + x ) ^ { 4 }\), writing each term as simply as possible.

4 Solve the inequality \(\frac { 3 ( 2 x + 1 ) } { 4 } > - 6\).

5 Make \(C\) the subject of the formula \(P = \frac { C } { C + 4 }\).

6 When \(x ^ { 3 } + 3 x + k\) is divided by \(x - 1\), the remainder is 6 . Find the value of \(k\). \begin{figure}[h] \end{figure} The line AB has equation \(y = 4 x - 5\) and passes through the point \(\mathrm { B } ( 2,3 )\), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C . Find the equation of the line BC and the \(x\)-coordinate of C .

8

1. Simplify \(5 \sqrt { 8 } + 4 \sqrt { 50 }\). Express your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
2. Express \(\frac { \sqrt { 3 } } { 6 - \sqrt { 3 } }\) in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are rational.

9

1. Find the range of values of \(k\) for which the equation \(x ^ { 2 } + 5 x + k = 0\) has one or more real roots.
2. Solve the equation \(4 x ^ { 2 } + 20 x + 25 = 0\).

10 A circle has equation \(x ^ { 2 } + y ^ { 2 } = 45\).

1. State the centre and radius of this circle.
2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B . Show that the distance AB is \(\sqrt { 162 }\).

11

1. Write \(x ^ { 2 } - 7 x + 6\) in the form \(( x - a ) ^ { 2 } + b\).
2. State the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 7 x + 6\).
3. Find the coordinates of the points where the graph of \(y = x ^ { 2 } - 7 x + 6\) crosses the axes and sketch the graph.
4. Show that the graphs of \(y = x ^ { 2 } - 7 x + 6\) and \(y = x ^ { 2 } - 3 x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection.

12

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).