OCR MEI C1 (Core Mathematics 1) 2009 January

1 State the value of each of the following.

1. \(2 ^ { - 3 }\)
2. \(9 ^ { 0 }\)

2 Find the equation of the line passing through \(( - 1 , - 9 )\) and \(( 3,11 )\). Give your answer in the form \(y = m x + c\).

3 Solve the inequality \(7 - x < 5 x - 2\).

4 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } + a x - 6\) and that \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the value of \(a\).

5

1. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( x ^ { 2 } - 3 \right) \left( x ^ { 3 } + 7 x + 1 \right)\).
2. Find the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).

6 Solve the equation \(\frac { 3 x + 1 } { 2 x } = 4\).

7

1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
2. Simplify \(\left( 4 a ^ { 3 } b ^ { 5 } \right) ^ { 2 }\).

8 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.

9 Rearrange \(y + 5 = x ( y + 2 )\) to make \(y\) the subject of the formula.

10

1. Express \(\sqrt { 75 } + \sqrt { 48 }\) in the form \(a \sqrt { 3 }\).
2. Express \(\frac { 14 } { 3 - \sqrt { 2 } }\) in the form \(b + c \sqrt { d }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c7fbeb8f-d874-4756-aa53-5471b215902f-3_773_961_354_591} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows the points A and B , which have coordinates \(( - 1,0 )\) and \(( 11,4 )\) respectively.

1. Show that the equation of the circle with AB as diameter may be written as $$( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 40$$
2. Find the coordinates of the points of intersection of this circle with the \(y\)-axis. Give your answer in the form \(a \pm \sqrt { b }\).
3. Find the equation of the tangent to the circle at B . Hence find the coordinates of the points of intersection of this tangent with the axes.

12

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.

13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).

1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.