OCR C1 (Core Mathematics 1) 2010 January

1 Express \(x ^ { 2 } - 12 x + 1\) in the form \(( x - p ) ^ { 2 } + q\).

2
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-2_330_681_390_731} The graph of \(y = \mathrm { f } ( x )\) for \(- 2 \leqslant x \leqslant 4\) is shown above.

1. Sketch the graph of \(y = 2 \mathrm { f } ( x )\) for \(- 2 \leqslant x \leqslant 4\) on the axes provided.
2. Describe the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { f } ( x - 1 )\).

3 Find the equation of the normal to the curve \(y = x ^ { 3 } - 4 x ^ { 2 } + 7\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

4 Solve the equations

1. \(3 ^ { m } = 81\),
2. \(\left( 36 p ^ { 4 } \right) ^ { \frac { 1 } { 2 } } = 24\),
3. \(5 ^ { n } \times 5 ^ { n + 4 } = 25\).

5 Solve the equation \(x - 8 \sqrt { x } + 13 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.

6
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-2_394_846_1868_648} The diagram shows part of the curve \(y = x ^ { 2 } + 5\). The point \(A\) has coordinates ( 1,6 ). The point \(B\) has coordinates ( \(a , a ^ { 2 } + 5\) ), where \(a\) is a constant greater than 1 . The point \(C\) is on the curve between \(A\) and \(B\).

1. Find by differentiation the value of the gradient of the curve at the point \(A\).
2. The line segment joining the points \(A\) and \(B\) has gradient 2.3. Find the value of \(a\).
3. State a possible value for the gradient of the line segment joining the points \(A\) and \(C\).

7 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Each diagram shows a quadratic curve. State which diagram corresponds to the curve
   (a) \(y = ( 3 - x ) ^ { 2 }\),
   (b) \(y = x ^ { 2 } + 9\),
   (c) \(y = ( 3 - x ) ( x + 3 )\).
2. Give the equation of the curve which does not correspond to any of the equations in part (i).

8 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 4 = 0\).

1. Find the centre and radius of the circle.
2. Find the coordinates of the points where the circle meets the line with equation \(y = 3 x + 4\).

9 Given that \(\mathrm { f } ( x ) = \frac { 1 } { x } - \sqrt { x } + 3\),

1. find \(\mathrm { f } ^ { \prime } ( x )\),
2. find \(\mathrm { f } ^ { \prime \prime } ( 4 )\).

10 The quadratic equation \(k x ^ { 2 } - 30 x + 25 k = 0\) has equal roots. Find the possible values of \(k\).

11 A lawn is to be made in the shape shown below. The units are metres.
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-4_412_698_486_726}

1. The perimeter of the lawn is \(P \mathrm {~m}\). Find \(P\) in terms of \(x\).
2. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the lawn is given by \(A = 9 x ^ { 2 } + 6 x\). The perimeter of the lawn must be at least 39 m and the area of the lawn must be less than \(99 \mathrm {~m} ^ { 2 }\).
3. By writing down and solving appropriate inequalities, determine the set of possible values of \(x\).