OCR MEI C1 (Core Mathematics 1)

You are given that \(a = \frac{3}{2}\), \(b = \frac{9 - \sqrt{17}}{4}\) and \(c = \frac{9 + \sqrt{17}}{4}\). Show that \(a + b + c = abc\). [4]

1. Simplify \(3a^3b \times 4(ab)^2\). [2]
2. Factorise \(x^2 - 4\) and \(x^2 - 5x + 6\). Hence express \(\frac{x^2 - 4}{x^2 - 5x + 6}\) as a fraction in its simplest form. [3]

Simplify \((m^2 + 1)^2 - (m^2 - 1)^2\), showing your method. Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer. [4] \includegraphics{figure_3}

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]

1. Write \(x^2 - 7x + 6\) in the form \((x - a)^2 + b\). [3]
2. State the coordinates of the minimum point on the graph of \(y = x^2 - 7x + 6\). [2]
3. Find the coordinates of the points where the graph of \(y = x^2 - 7x + 6\) crosses the axes and sketch the graph. [5]
4. Show that the graphs of \(y = x^2 - 7x + 6\) and \(y = x^2 - 3x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. [3]

\includegraphics{figure_6} Fig. 11 shows a sketch of the curve with equation \(y = (x - 4)^2 - 3\).

1. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point. [2]
2. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary. [4]
3. The curve is translated by \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\). Show that the equation of the translated curve may be written as \(y = x^2 - 12x + 33\). [2]
4. Show that the line \(y = 8 - 2x\) meets the curve \(y = x^2 - 12x + 33\) at just one point, and find the coordinates of this point. [5]

1. Describe fully the transformation which maps the curve \(y = x^2\) onto the curve \(y = (x + 4)^2\). [2]
2. Sketch the graph of \(y = x^2 - 4\). [2]