AQA C1 (Core Mathematics 1) 2007 June

1 The points \(A\) and \(B\) have coordinates \(( 6 , - 1 )\) and \(( 2,5 )\) respectively.

1. 1. Show that the gradient of \(A B\) is \(- \frac { 3 } { 2 }\).
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
2. 1. Find an equation of the line which passes through \(B\) and which is perpendicular to the line \(A B\).
   2. The point \(C\) has coordinates ( \(k , 7\) ) and angle \(A B C\) is a right angle. Find the value of the constant \(k\).

2

1. Express \(\frac { \sqrt { 63 } } { 3 } + \frac { 14 } { \sqrt { 7 } }\) in the form \(n \sqrt { 7 }\), where \(n\) is an integer.
2. Express \(\frac { \sqrt { 7 } + 1 } { \sqrt { 7 } - 2 }\) in the form \(p \sqrt { 7 } + q\), where \(p\) and \(q\) are integers.

3

1. 1. Express \(x ^ { 2 } + 10 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Write down the coordinates of the vertex (minimum point) of the curve with equation \(y = x ^ { 2 } + 10 x + 19\).
   3. Write down the equation of the line of symmetry of the curve \(y = x ^ { 2 } + 10 x + 19\).
   4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 10 x + 19\).
2. Determine the coordinates of the points of intersection of the line \(y = x + 11\) and the curve \(y = x ^ { 2 } + 10 x + 19\).

4 A model helicopter takes off from a point \(O\) at time \(t = 0\) and moves vertically so that its height, \(y \mathrm {~cm}\), above \(O\) after time \(t\) seconds is given by $$y = \frac { 1 } { 4 } t ^ { 4 } - 26 t ^ { 2 } + 96 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether this stationary value is a maximum value or a minimum value.
   (4 marks)
3. Find the rate of change of \(y\) with respect to \(t\) when \(t = 1\).
4. Determine whether the height of the helicopter above \(O\) is increasing or decreasing at the instant when \(t = 3\).

5 A circle with centre \(C\) has equation \(( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25\).

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the point \(N ( 0 , - 2 )\) lies on the circle.
   2. Sketch the circle.
   3. Find an equation of the normal to the circle at the point \(N\).
3. The point \(P\) has coordinates (2, 6).
   1. Find the distance \(P C\), leaving your answer in surd form.
   2. Find the length of a tangent drawn from \(P\) to the circle.

6

1. The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5\).
   1. Use the Factor Theorem to show that \(x - 1\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) in the form \(( x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root and state its value.
2. The curve with equation \(y = x ^ { 3 } + 4 x - 5\) is sketched below.
   \includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529} The curve cuts the \(x\)-axis at the point \(A ( 1,0 )\) and the point \(B ( 2,11 )\) lies on the curve.
   1. Find \(\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(A B\).

7 The quadratic equation $$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$ where \(k\) is a constant, has real roots.

1. Show that \(2 k ^ { 2 } - 5 k + 2 \leqslant 0\).
2. 1. Factorise \(2 k ^ { 2 } - 5 k + 2\).
   2. Hence, or otherwise, solve the quadratic inequality $$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$