OCR MEI C1 (Core Mathematics 1) 2012 January

1 Find the equation of the line which is perpendicular to the line \(y = 5 x + 2\) and which passes through the point \(( 1,6 )\). Give your answer in the form \(y = a x + b\).

2

1. Evaluate \(9 ^ { - \frac { 1 } { 2 } }\).
2. Simplify \(\frac { \left( 4 x ^ { 4 } \right) ^ { 3 } y ^ { 2 } } { 2 x ^ { 2 } y ^ { 5 } }\).

3 Expand and simplify \(( n + 2 ) ^ { 3 } - n ^ { 3 }\).

4

1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).

5 Solve the following inequality. $$\frac { 2 x + 1 } { 5 } < \frac { 3 x + 4 } { 6 }$$

6 Rearrange the following equation to make \(h\) the subject. $$4 h + 5 = 9 a - h a ^ { 2 }$$

7 \begin{figure}[h] \end{figure} Fig. 7 shows the graph of \(y = \mathrm { g } ( x )\). Draw the graphs of the following.

1. \(y = \mathrm { g } ( x ) + 3\)
2. \(y = \mathrm { g } ( x + 2 )\)

8 Express \(5 x ^ { 2 } + 15 x + 12\) in the form \(a ( x + b ) ^ { 2 } + c\).
Hence state the minimum value of \(y\) on the curve \(y = 5 x ^ { 2 } + 15 x + 12\).

9 Complete each of the following by putting the best connecting symbol ( \(\Leftrightarrow , \Leftarrow\) or ⇒ ) in the box. Explain your choice, giving full reasons.

1. \(n ^ { 3 } + 1\) is an odd integer □ \(n\) is an even integer
2. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\) Section B (36 marks)

10 Point A has coordinates (4, 7) and point B has coordinates (2, 1).

1. Find the equation of the line through A and B .
2. Point C has coordinates ( \(- 1,2\) ). Show that angle \(\mathrm { ABC } = 90 ^ { \circ }\) and calculate the area of triangle ABC .
3. Find the coordinates of D , the midpoint of AC . Explain also how you can tell, without having to work it out, that \(\mathrm { A } , \mathrm { B }\) and C are all the same distance from D.

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).

1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).

12 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0 .$$
4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}