OCR C1 (Core Mathematics 1) 2007 January

1 Express \(\frac { 5 } { 2 - \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.

2 Evaluate

1. \(6 ^ { 0 }\),
2. \(2 ^ { - 1 } \times 32 ^ { \frac { 4 } { 5 } }\).

3 Solve the inequalities

1. \(3 ( x - 5 ) \leqslant 24\),
2. \(5 x ^ { 2 } - 2 > 78\).

4 Solve the equation \(x ^ { \frac { 2 } { 3 } } + 3 x ^ { \frac { 1 } { 3 } } - 10 = 0\).

5
\includegraphics[max width=\textwidth, alt={}, center]{82ae6eec-3007-467c-90df-18f2adb9ccb1-2_634_926_1242_612} The graph of \(y = \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\) is shown above.

1. Sketch the graph of \(y = - \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\).
2. The point \(P ( 1,1 )\) on \(y = \mathrm { f } ( x )\) is transformed to the point \(Q\) on \(y = 3 \mathrm { f } ( x )\). State the coordinates of \(Q\).
3. Describe the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { f } ( x + 2 )\).

6

1. Express \(2 x ^ { 2 } - 24 x + 80\) in the form \(a ( x - b ) ^ { 2 } + c\).
2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } - 24 x + 80\).
3. State the equation of the tangent to the curve \(y = 2 x ^ { 2 } - 24 x + 80\) at its minimum point.

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases.

1. \(y = 5 x + 3\)
2. \(y = \frac { 2 } { x ^ { 2 } }\)
3. \(y = ( 2 x + 1 ) ( 5 x - 7 )\)

8

1. Find the coordinates of the stationary points of the curve \(y = 27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Determine, in each case, whether the stationary point is a maximum or minimum point.
3. Hence state the set of values of \(x\) for which \(27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\) is an increasing function.
   \(9 \quad A\) is the point \(( 2,7 )\) and \(B\) is the point \(( - 1 , - 2 )\).

1. Find the equation of the line through \(A\) parallel to the line \(y = 4 x - 5\), giving your answer in the form \(y = m x + c\).
2. Calculate the length of \(A B\), giving your answer in simplified surd form.
3. Find the equation of the line which passes through the mid-point of \(A B\) and which is perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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

1. Find the centre and radius of the circle.
2. The circle passes through the point \(( - 3 , k )\), where \(k < 0\). Find the value of \(k\).
3. Find the coordinates of the points where the circle meets the line with equation \(x + y = 6\).