AQA C1 (Core Mathematics 1) 2006 June

1 The point \(A\) has coordinates \(( 1,7 )\) and the point \(B\) has coordinates \(( 5,1 )\).

1. 1. Find the gradient of the line \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x + 2 y = 17\).
2. The line \(A B\) intersects the line with equation \(x - 4 y = 8\) at the point \(C\). Find the coordinates of \(C\).
3. Find an equation of the line through \(A\) which is perpendicular to \(A B\).

2

1. Express \(x ^ { 2 } + 8 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
2. Hence, or otherwise, show that the equation \(x ^ { 2 } + 8 x + 19 = 0\) has no real solutions.
3. Sketch the graph of \(y = x ^ { 2 } + 8 x + 19\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 8 x + 19\).

3 A curve has equation \(y = 7 - 2 x ^ { 5 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find an equation for the tangent to the curve at the point where \(x = 1\).
3. Determine whether \(y\) is increasing or decreasing when \(x = - 2\).

4

1. Express \(( 4 \sqrt { 5 } - 1 ) ( \sqrt { 5 } + 3 )\) in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers.
2. Show that \(\frac { \sqrt { 75 } - \sqrt { 27 } } { \sqrt { 3 } }\) is an integer and find its value.

5 The curve with equation \(y = x ^ { 3 } - 10 x ^ { 2 } + 28 x\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{f2c95d73-d3fe-48f7-af07-84f12bb06727-3_483_899_402_568} The curve crosses the \(x\)-axis at the origin \(O\) and the point \(A ( 3,21 )\) lies on the curve.

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence show that the curve has a stationary point when \(x = 2\) and find the \(x\)-coordinate of the other stationary point.
2. 1. Find \(\int \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x\).
   2. Hence show that \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x = 56 \frac { 1 } { 4 }\).
   3. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 3 x\).

1. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
3. 1. Use the Remainder Theorem to find the remainder, \(r\), when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
   2. Using algebraic division, or otherwise, express \(\mathrm { p } ( x )\) in the form $$( x - 2 ) \left( x ^ { 2 } + a x + b \right) + r$$ where \(a , b\) and \(r\) are constants.

7 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0\).

1. Find:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle in the form \(p \sqrt { 2 }\), where \(p\) is an integer.
2. A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
3. A line has equation \(y = 2 k - x\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
   2. Find the values of \(k\) for which the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (c)(ii).