OCR MEI C1 (Core Mathematics 1) 2012 June

1 Find the equation of the line with gradient - 2 which passes through the point \(( 3,1 )\). Give your answer in the form \(y = a x + b\). Find also the points of intersection of this line with the axes.

2 Make \(b\) the subject of the following formula. $$a = \frac { 2 } { 3 } b ^ { 2 } c$$

3

1. Evaluate \(\left( \frac { 1 } { 5 } \right) ^ { - 2 }\).
2. Evaluate \(\left( \frac { 8 } { 27 } \right) ^ { \frac { 2 } { 3 } }\).

4 Factorise and hence simplify the following expression. $$\frac { x ^ { 2 } - 9 } { x ^ { 2 } + 5 x + 6 }$$

5

1. Simplify \(\frac { 10 ( \sqrt { 6 } ) ^ { 3 } } { \sqrt { 24 } }\).
2. Simplify \(\frac { 1 } { 4 - \sqrt { 5 } } + \frac { 1 } { 4 + \sqrt { 5 } }\).

6

1. Evaluate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).

8 The function \(\mathrm { f } ( x ) = x ^ { 4 } + b x + c\) is such that \(\mathrm { f } ( 2 ) = 0\). Also, when \(\mathrm { f } ( x )\) is divided by \(x + 3\), the remainder is 85 . Find the values of \(b\) and \(c\).

9 Simplify \(( n + 3 ) ^ { 2 } - n ^ { 2 }\). Hence explain why, when \(n\) is an integer, \(( n + 3 ) ^ { 2 } - n ^ { 2 }\) is never an even number. Given also that \(( n + 3 ) ^ { 2 } - n ^ { 2 }\) is divisible by 9 , what can you say about \(n\) ? \begin{figure}[h] \end{figure} Fig. 10 is a sketch of quadrilateral ABCD with vertices \(\mathrm { A } ( 1,5 ) , \mathrm { B } ( - 1,1 ) , \mathrm { C } ( 3 , - 1 )\) and \(\mathrm { D } ( 11,5 )\).

1. Show that \(\mathrm { AB } = \mathrm { BC }\).
2. Show that the diagonals AC and BD are perpendicular.
3. Find the midpoint of AC . Show that BD bisects AC but AC does not bisect BD .

11 A cubic curve has equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis where \(x = - \frac { 1 } { 2 } , - 2\) and 5 .

1. Write down three linear factors of \(\mathrm { f } ( x )\). Hence find the equation of the curve in the form \(y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 0 } { - 8 }\). State the coordinates of the point where the translated curve intersects the \(y\)-axis.
4. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression in factorised form for \(\mathrm { g } ( x )\) and state the coordinates of the point where the curve \(y = \mathrm { g } ( x )\) intersects the \(y\)-axis. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0088b5e7-d587-419a-a13b-87527ac658c4-4_1287_1410_292_315} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\).
5. Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
6. Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
7. Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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