OCR MEI C1 (Core Mathematics 1)

1

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.

2 Answer part (i) of this question on the insert provided.
The insert shows the graph of \(y = \frac { 1 } { x }\).

1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.

3 Factorise and hence simplify \(\frac { 3 x ^ { 2 } - 7 x + 4 } { x ^ { 2 } - 1 }\).

1. Prove that 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all even positive integers \(n\).
2. Determine whether 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all positive integers \(n\).
3. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
4. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
5. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
6. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.

6

1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).