AQA C1 (Core Mathematics 1) 2005 June

1 The point \(A\) has coordinates \(( 6,5 )\) and the point \(B\) has coordinates \(( 2 , - 1 )\).

1. Find the coordinates of the midpoint of \(A B\).
2. Show that \(A B\) has length \(k \sqrt { 13 }\), where \(k\) is an integer.
3. 1. Find the gradient of the line \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x - 2 y = 8\).
4. The line \(A B\) intersects the line with equation \(2 x + y = 10\) at the point \(C\). Find the coordinates of \(C\).

2

1. Express \(x ^ { 2 } - 6 x + 16\) in the form \(( x - p ) ^ { 2 } + q\).
2. A curve has equation \(y = x ^ { 2 } - 6 x + 16\). Using your answer from part (a), or otherwise:
   1. find the coordinates of the vertex (minimum point) of the curve;
   2. sketch the curve, indicating the value where the curve crosses the \(y\)-axis;
   3. state the equation of the line of symmetry of the curve.
3. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 6 x + 16\).

3 A circle has centre \(C ( 2 , - 1 )\) and radius 5 . The point \(P\) has coordinates \(( 6,2 )\).

1. Write down an equation of the circle.
2. Verify that the point \(P\) lies on the circle.
3. Find the gradient of the line \(C P\).
4. 1. Find the gradient of a line which is perpendicular to \(C P\).
   2. Hence find an equation for the tangent to the circle at the point \(P\).

4 The curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{3729de55-7139-4f41-8584-640f173c0e09-3_444_588_411_717} The curve touches the \(x\)-axis at the point \(A ( 1,0 )\) and cuts the \(x\)-axis at the point \(B\).

1. 1. Use the factor theorem to show that \(x - 3\) is a factor of $$\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$$
   2. Hence find the coordinates of \(B\).
2. The point \(M\), shown on the diagram, is a minimum point of the curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence determine the \(x\)-coordinate of \(M\).
3. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
4. 1. Find \(\int \left( x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3 \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the coordinate axes.

5 Express each of the following in the form \(m + n \sqrt { 3 }\), where \(m\) and \(n\) are integers:

1. \(( \sqrt { 3 } + 1 ) ^ { 2 }\);
2. \(\frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).

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

1. Show that \(\mathrm { p } ( x )\) can be written in the form \(x ^ { 3 } + a x ^ { 2 } + b x - 6\), where \(a\) and \(b\) are constants whose values are to be found.
2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
   (2 marks)
3. Prove that the equation \(( x - 2 ) \left( x ^ { 2 } + x + 3 \right) = 0\) has only one real root and state its value.
   (3 marks)

7 Solve each of the following inequalities:

1. \(3 ( x - 1 ) > 3 - 5 ( x + 6 )\);
2. \(\quad x ^ { 2 } - x - 6 < 0\).

8 A line has equation \(y = m x - 1\), where \(m\) is a constant.
A curve has equation \(y = x ^ { 2 } - 5 x + 3\).

1. Show that the \(x\)-coordinate of any point of intersection of the line and the curve satisfies the equation $$x ^ { 2 } - ( 5 + m ) x + 4 = 0$$
2. Find the values of \(m\) for which the equation \(x ^ { 2 } - ( 5 + m ) x + 4 = 0\) has equal roots.
   (4 marks)
3. Describe geometrically the situation when \(m\) takes either of the values found in part (b).
   (1 mark)