AQA C1 (Core Mathematics 1) 2010 January

1 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 13 x - 12\).

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.

2 The triangle \(A B C\) has vertices \(A ( 1,3 ) , B ( 3,7 )\) and \(C ( - 1,9 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence show that angle \(A B C\) is a right angle.
2. 1. Find the coordinates of \(M\), the mid-point of \(A C\).
   2. Show that the lengths of \(A B\) and \(B C\) are equal.
   3. Hence find an equation of the line of symmetry of the triangle \(A B C\).

3 The depth of water, \(y\) metres, in a tank after time \(t\) hours is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - 2 t ^ { 2 } + 4 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether it is a maximum value or a minimum value.
3. 1. Find the rate of change of the depth of water, in metres per hour, when \(t = 1\).
   2. Hence determine, with a reason, whether the depth of water is increasing or decreasing when \(t = 1\).

4

1. Show that \(\frac { \sqrt { 50 } + \sqrt { 18 } } { \sqrt { 8 } }\) is an integer and find its value.
   (3 marks)
2. Express \(\frac { 2 \sqrt { 7 } - 1 } { 2 \sqrt { 7 } + 5 }\) in the form \(m + n \sqrt { 7 }\), where \(m\) and \(n\) are integers.
   (4 marks)

5

1. Express \(( x - 5 ) ( x - 3 ) + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   (3 marks)
2. 1. Sketch the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
   2. Write down an equation of the tangent to the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\) at its vertex.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\).

6 The curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{2f7a8e95-4994-4732-a9a4-306c7b6cad92-3_444_819_1434_609} The curve crosses the \(x\)-axis at the origin \(O\), and the point \(A ( 2 , - 6 )\) lies on the curve.

1. 1. Find the gradient of the curve with equation \(y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }\) at the point \(A\).
   2. Hence find the equation of the normal to the curve at the point \(A\), giving your answer in the form \(x + p y + q = 0\), where \(p\) and \(q\) are integers.
2. 1. Find the value of \(\int _ { 0 } ^ { 2 } \left( 12 x ^ { 2 } - 19 x - 2 x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 4 x + 12 y + 15 = 0\).

1. Find:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. Explain why the circle lies entirely below the \(x\)-axis.
3. The point \(P\) with coordinates \(( 5 , k )\) lies outside the circle.
   1. Show that \(P C ^ { 2 } = k ^ { 2 } + 12 k + 45\).
   2. Hence show that \(k ^ { 2 } + 12 k + 20 > 0\).
   3. Find the possible values of \(k\).