OCR MEI C1 (Core Mathematics 1)

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3x^2 + 6x + 10\) and the line \(y = 2 - 4x\). [5]
2. Write \(3x^2 + 6x + 10\) in the form \(a(x + b)^2 + c\). [4]
3. Hence or otherwise, show that the graph of \(y = 3x^2 + 6x + 10\) 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 \(y = x^2 - 5x + 5\) for \(0 \leq x \leq 5\). [4]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac{1}{x}\) and \(y = x^2 - 5x + 5\) satisfy the equation \(x^3 - 5x^2 + 5x - 1 = 0\). [2]
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x^3 - 5x^2 + 5x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x^3 - 5x^2 + 5x - 1 = 0\) is rational. [5]

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

1. Prove that 12 is a factor of \(3n^2 + 6n\) for all even positive integers \(n\). [3]
2. Determine whether 12 is a factor of \(3n^2 + 6n\) for all positive integers \(n\). [2]

1. Write \(x^2 - 5x + 8\) in the form \((x - a)^2 + b\) and hence show that \(x^2 - 5x + 8 > 0\) for all values of \(x\). [4]
2. Sketch the graph of \(y = x^2 - 5x + 8\), showing the coordinates of the turning point. [3]
3. Find the set of values of \(x\) for which \(x^2 - 5x + 8 > 14\). [3]
4. If \(f(x) = x^2 - 5x + 8\), does the graph of \(y = f(x) - 10\) cross the \(x\)-axis? Show how you decide. [2]

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