OCR MEI C1 (Core Mathematics 1) 2010 June

1 Find the equation of the line which is parallel to \(y = 3 x + 1\) and which passes through the point with coordinates \(( 4,5 )\).

2

1. Simplify \(\left( 5 a ^ { 2 } b \right) ^ { 3 } \times 2 b ^ { 4 }\).
2. Evaluate \(\left( \frac { 1 } { 16 } \right) ^ { - 1 }\).
3. Evaluate \(( 16 ) ^ { \frac { 3 } { 2 } }\).

4 Solve the following inequalities.

1. \(2 ( 1 - x ) > 6 x + 5\)
2. \(( 2 x - 1 ) ( x + 4 ) < 0\)

5

1. Express \(\sqrt { 48 } + \sqrt { 27 }\) in the form \(a \sqrt { 3 }\).
2. Simplify \(\frac { 5 \sqrt { 2 } } { 3 - \sqrt { 2 } }\). Give your answer in the form \(\frac { b + c \sqrt { 2 } } { d }\).

6 You are given that

• the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
• when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .

Find the values of \(k\) and \(m\).

7 Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 4 }\), simplifying the coefficients.

8 Express \(5 x ^ { 2 } + 20 x + 6\) in the form \(a ( x + b ) ^ { 2 } + c\).

9 Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x ^ { 2 } = 25$$

10

1. Solve, by factorising, the equation \(2 x ^ { 2 } - x - 3 = 0\).
2. Sketch the graph of \(y = 2 x ^ { 2 } - x - 3\).
3. Show that the equation \(x ^ { 2 } - 5 x + 10 = 0\) has no real roots.
4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2 x ^ { 2 } - x - 3\) and \(y = x ^ { 2 } - 5 x + 10\). Give your answer in the form \(a \pm \sqrt { b }\).

11 \begin{figure}[h] \end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).

1. Find the equation of the line through A and B .
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.

12 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).

1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$