OCR MEI C1 (Core Mathematics 1)

\includegraphics{figure_1} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \((-5, 0)\), \((-2, 0)\), \((1.5, 0)\) and \((0, -30)\).

1. Use the intersections with both axes to express the equation of the curve in a factorised form. [2]
2. Hence show that the equation of the curve may be written as \(y = 2x^3 + 11x^2 - x - 30\). [2]
3. Draw the line \(y = 5x + 10\) accurately on the graph. The curve and this line intersect at \((-2, 0)\); find graphically the \(x\)-coordinates of the other points of intersection. [3]
4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2x^2 + 7x - 20 = 0.$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. [5]

\includegraphics{figure_2} Fig. 12 shows the graph of \(y = \frac{1}{x-2}\).

1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = 2x + 3\). [3]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x-2}\). [4]

\includegraphics{figure_3} Fig. 12 shows the graph of \(y = \frac{1}{x-3}\).

1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{1}{x-3}\) and \(y = x^2 - 4x + 1\). [5]
2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
3. For the curve \(y = 4x^2 + 24x + 31\),
   1. write down the equation of the line of symmetry, [1]
   2. write down the minimum \(y\)-value on the curve. [1]

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

Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]