AQA C1 (Core Mathematics 1) 2009 January

1 The points \(A\) and \(B\) have coordinates \(( 1,6 )\) and \(( 5 , - 2 )\) respectively. The mid-point of \(A B\) is \(M\).

1. Find the coordinates of \(M\).
2. Find the gradient of \(A B\), giving your answer in its simplest form.
3. A straight line passes through \(M\) and is perpendicular to \(A B\).
   1. Show that this line has equation \(x - 2 y + 1 = 0\).
   2. Given that this line passes through the point \(( k , k + 5 )\), find the value of the constant \(k\).

2

1. Factorise \(2 x ^ { 2 } - 5 x + 3\).
2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - 5 x + 3 < 0\).

3

1. Express \(\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
2. Express \(\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }\) in the form \(k \sqrt { 5 }\), where \(k\) is an integer.

4

1. 1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
2. A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
   1. Write down the coordinates of the minimum point of the curve.
   2. Sketch the curve, showing the value of the intercept on the \(y\)-axis.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 2 x + 5\).

5 A model car moves so that its distance, \(x\) centimetres, from a fixed point \(O\) after time \(t\) seconds is given by $$x = \frac { 1 } { 2 } t ^ { 4 } - 20 t ^ { 2 } + 66 t , \quad 0 \leqslant t \leqslant 4$$

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

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x - 10\).
   1. Use the Factor Theorem to show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are constants.
2. The curve \(C\) with equation \(y = x ^ { 3 } + x - 10\), sketched below, crosses the \(x\)-axis at the point \(Q ( 2,0 )\).
   \includegraphics[max width=\textwidth, alt={}, center]{22c93dd5-d96a-4e31-8507-9c802e386231-3_444_547_1781_756}
   1. Find the gradient of the curve \(C\) at the point \(Q\).
   2. Hence find an equation of the tangent to the curve \(C\) at the point \(Q\).
   3. Find \(\int \left( x ^ { 3 } + x - 10 \right) \mathrm { d } x\).
   4. Hence find the area of the shaded region bounded by the curve \(C\) and the coordinate axes.

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0\).

1. 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 \(D\) has coordinates (7, -2).
   1. Verify that the point \(D\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(D\), giving your answer in the form \(m x + n y = p\), where \(m , n\) and \(p\) are integers.
4. 1. A line has equation \(y = k x\). Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$
   2. Find the values of \(k\) for which the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (d)(ii).