OCR MEI C1 (Core Mathematics 1) 2008 June

1 Solve the inequality \(3 x - 1 > 5 - x\).

2

1. Find the points of intersection of the line \(2 x + 3 y = 12\) with the axes.
2. Find also the gradient of this line.

3

1. Solve the equation \(2 x ^ { 2 } + 3 x = 0\).
2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 x - k = 0\) has no real roots.

4 Given that \(n\) is a positive integer, write down whether the following statements are always true (T), always false (F) or could be either true or false (E).

1. \(2 n + 1\) is an odd integer
2. \(3 n + 1\) is an even integer
3. \(n\) is odd \(\Rightarrow n ^ { 2 }\) is odd
4. \(n ^ { 2 }\) is odd \(\Rightarrow n ^ { 3 }\) is even

5 Make \(x\) the subject of the equation \(y = \frac { x + 3 } { x - 2 }\).

6

1. Find the value of \(\left( \frac { 1 } { 25 } \right) ^ { - \frac { 1 } { 2 } }\).
2. Simplify \(\frac { \left( 2 x ^ { 2 } y ^ { 3 } z \right) ^ { 5 } } { 4 y ^ { 2 } z }\).

7

1. Express \(\frac { 1 } { 5 + \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.
2. Expand and simplify \(( 3 - 2 \sqrt { 7 } ) ^ { 2 }\).

8 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 5 - 2 x ) ^ { 5 }\).

9 Solve the equation \(y ^ { 2 } - 7 y + 12 = 0\).
Hence solve the equation \(x ^ { 4 } - 7 x ^ { 2 } + 12 = 0\). Section B (36 marks)

10

1. Express \(x ^ { 2 } - 6 x + 2\) in the form \(( x - a ) ^ { 2 } - b\).
2. State the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 2\).
3. Sketch the graph of \(y = x ^ { 2 } - 6 x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis.
4. Solve the simultaneous equations \(y = x ^ { 2 } - 6 x + 2\) and \(y = 2 x - 14\). Hence show that the line \(y = 2 x - 14\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 2\).

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).

1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).

12

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.