OCR MEI C1 (Core Mathematics 1)

1 You are given that \(a = \frac { 3 } { 2 } , b = \frac { 9 \sqrt { 17 } } { 4 }\) and \(c = \frac { 9 + \sqrt { 17 } } { 4 }\). Show that \(a + b + c = a b c\).
\(2 \quad\) (i) Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
(ii) Factorise \(x ^ { 2 } - 4 \quad x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.

3 Simplify \(\left( m ^ { 2 } + 1 \right) ^ { 2 } - \left( m ^ { 2 } - 1 \right) ^ { 2 }\), showing your method.
Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer.
\includegraphics[max width=\textwidth, alt={}, center]{6978a576-e86a-48d4-93a4-268845ea6699-1_414_590_1328_772}

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

5

1. Write \(x ^ { 2 } - 7 x + 6\) in the form \(( x - a ) ^ { 2 } + b\).
2. State the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 7 x + 6\).
3. Find the coordinates of the points where the graph of \(y = x ^ { 2 } - 7 x + 6\) crosses the axes and sketch the graph.
4. Show that the graphs of \(y = x ^ { 2 } - 7 x + 6\) and \(y = x ^ { 2 } - 3 x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6978a576-e86a-48d4-93a4-268845ea6699-3_643_864_390_622} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
5. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
6. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
7. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
8. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point.

7

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