AQA C1 (Core Mathematics 1) 2006 January

1

1. Simplify \(( \sqrt { 5 } + 2 ) ( \sqrt { 5 } - 2 )\).
2. Express \(\sqrt { 8 } + \sqrt { 18 }\) in the form \(n \sqrt { 2 }\), where \(n\) is an integer.

2 The point \(A\) has coordinates \(( 1,1 )\) and the point \(B\) has coordinates \(( 5 , k )\). The line \(A B\) has equation \(3 x + 4 y = 7\).

1. 1. Show that \(k = - 2\).
   2. Hence find the coordinates of the mid-point of \(A B\).
2. Find the gradient of \(A B\).
3. The line \(A C\) is perpendicular to the line \(A B\).
   1. Find the gradient of \(A C\).
   2. Hence find an equation of the line \(A C\).
   3. Given that the point \(C\) lies on the \(x\)-axis, find its \(x\)-coordinate.

3

1. 1. Express \(x ^ { 2 } - 4 x + 9\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence, or otherwise, state the coordinates of the minimum point of the curve with equation \(y = x ^ { 2 } - 4 x + 9\).
2. The line \(L\) has equation \(y + 2 x = 12\) and the curve \(C\) has equation \(y = x ^ { 2 } - 4 x + 9\).
   1. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation $$x ^ { 2 } - 2 x - 3 = 0$$
   2. Hence find the coordinates of the points of intersection of \(L\) and \(C\).

4 The quadratic equation \(x ^ { 2 } + ( m + 4 ) x + ( 4 m + 1 ) = 0\), where \(m\) is a constant, has equal roots.

1. Show that \(m ^ { 2 } - 8 m + 12 = 0\).
2. Hence find the possible values of \(m\).

5 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x + 6 y = 11\).

1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. The point \(O\) has coordinates \(( 0,0 )\).
   1. Find the length of CO .
   2. Hence determine whether the point \(O\) lies inside or outside the circle, giving a reason for your answer.

6 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$$

1. 1. Using the factor theorem, show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Hence express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Sketch the curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 10 x + 8\), showing the coordinates of the points where the curve cuts the axes.
   (You are not required to calculate the coordinates of the stationary points.)

7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$

1. Find:
   1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
   (2 marks)
3. 1. Verify that \(V\) has a stationary value when \(t = 1\).
      (2 marks)
   2. Determine whether this is a maximum or minimum value.
      (2 marks)

8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\).
\includegraphics[max width=\textwidth, alt={}, center]{81f6fc30-982b-47b5-bab3-076cc0cc6563-5_479_816_406_596} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).

1. Find the area of the rectangle \(A B C D\).
2. 1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
   3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).