OCR MEI C1 (Core Mathematics 1)

1 Find the equation of the line which passes through \(( 1,3 )\) and ( 4,9 ).

2 Find the range of values of \(x\) for which \(x ^ { 2 } - 5 x + 6 \leq 0\).

3 Write \(( \sqrt { 3 } - \sqrt { 2 } ) ^ { 2 }\) in the form \(a + b \sqrt { 6 }\) where \(a\) and \(b\) are integers to be determined.

4
\includegraphics[max width=\textwidth, alt={}, center]{3b927f8b-ddf8-481d-a1ce-3b90bb1435f0-2_437_807_953_579} The graph shows a function \(y = \mathrm { f } ( x )\).
On separate graphs, sketch the graphs of the following functions:

1. \(\quad y = \mathrm { f } ( x ) + 1\),
2. \(y = \mathrm { f } ( x + 1 )\).

5 Make \(u\) the subject of the formula $$\frac { 1 } { v } - \frac { 1 } { u } = \frac { 1 } { f }$$

6 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 2 x - 8 = 0\).
Find the centre and radius of the circle.

7 Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 4 x + 4\).
Hence solve the equation \(x ^ { 3 } - x ^ { 2 } - 4 x + 4 = 0\).

8 Find the points where the line \(y = 2 x - 3\) cuts the curve \(y = x ^ { 2 } - 4 x + 5\).

9

1. Simplify \(\frac { 2 ^ { 6 } } { 8 ^ { 2 \frac { 1 } { 2 } } \times 2 ^ { - \frac { 1 } { 2 } } }\)
2. Solve the equation \(x ^ { - \frac { 1 } { 3 } } = 8\).

10 \begin{figure}[h] \end{figure} In Fig.10, A has coordinates \(( 1,1 )\) and C has coordinates \(( 3,5 )\). M is the mid-point of AC . The line \(l\) is perpendicular to AC.

1. Find the coordinates of M . Hence find the equation of \(l\).
2. The point B has coordinates \(( - 2,5 )\). Show that B lies on the line \(l\).
   Find the coordinates of the point D such that ABCD is a rhombus.
3. Find the lengths MC and MB . Hence calculate the area of the rhombus ABCD .

11

1. Multiply out \(( x - p ) ( x - q )\).
2. You are given that \(p = 2 + \sqrt { 3 }\) and \(q = 2 - \sqrt { 3 }\) are the roots of a quadratic equation. Find \(p + q\) and \(p q\) and hence find the quadratic equation with roots \(x = p\) and \(x = q\).
3. Solve the quadratic equation \(x ^ { 2 } + 5 x - 7 = 0\) giving the roots exactly.
4. Show that \(x = 1\) is the only root of the equation \(x ^ { 3 } + 2 x - 3 = 0\).
5. A quadratic equation \(x ^ { 2 } + r x + s = 0\), where \(r\) and \(s\) are integers, has two roots. One root is \(x = 3 + \sqrt { 5 }\). Without finding \(r\) or \(s\), write down the other root.

12

1. Expand \(( 1 + 2 x ) ^ { 6 }\), simplifying all the terms.
2. Hence find an expression for \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { 6 } + ( 1 - 2 x ) ^ { 6 }\) in its simplest form.
3. Substituting \(x = 0.01\) into the first two terms of \(\mathrm { f } ( x )\) gives an approximate value, z for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\). Find \(z\). By considering the value of the third term, comment on the accuracy of \(z\) as an approximation for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\).