AQA C1 (Core Mathematics 1) 2011 June

1 The line \(A B\) has equation \(7 x + 3 y = 13\).

1. Find the gradient of \(A B\).
2. The point \(C\) has coordinates \(( - 1,3 )\).
   1. Find an equation of the line which passes through the point \(C\) and which is parallel to \(A B\).
   2. The point \(\left( 1 \frac { 1 } { 2 } , - 1 \right)\) is the mid-point of \(A C\). Find the coordinates of the point \(A\).
3. The line \(A B\) intersects the line with equation \(3 x + 2 y = 12\) at the point \(B\). Find the coordinates of \(B\).

2

1. 1. Express \(\sqrt { 48 }\) in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
   2. Simplify \(\frac { \sqrt { 48 } + 2 \sqrt { 27 } } { \sqrt { 12 } }\), giving your answer as an integer.
2. Express \(\frac { 1 - 5 \sqrt { 5 } } { 3 + \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.

3 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank after time \(t\) seconds is given by $$V = \frac { t ^ { 3 } } { 4 } - 3 t + 5$$

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} t }\).
2. 1. Find the rate of change of volume, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 1\).
   2. Hence determine, with a reason, whether the volume is increasing or decreasing when \(t = 1\).
3. 1. Find the positive value of \(t\) for which \(V\) has a stationary value.
   2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\), and hence determine whether this stationary value is a maximum value or a minimum value.
      (3 marks)

4

1. Express \(x ^ { 2 } + 5 x + 7\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   (3 marks)
2. A curve has equation \(y = x ^ { 2 } + 5 x + 7\).
   1. Find the coordinates of the vertex of the curve.
   2. State the equation of the line of symmetry of the curve.
   3. Sketch the curve, stating 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 \(y = x ^ { 2 } + 5 x + 7\).

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

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
2. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
3. 1. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\) in the form \(( x + 1 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   2. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

6 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{c44c229e-44b2-4799-9c9c-bfccdd09d450-4_590_787_365_625} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and passes through the point \(B ( 1,2 )\).

1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 3 } - 2 x ^ { 2 } + 3 \right) \mathrm { d } x\).
   (5 marks)
2. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) and the line \(A B\).
   (3 marks)

7 Solve each of the following inequalities:

1. \(\quad 2 ( 4 - 3 x ) > 5 - 4 ( x + 2 )\);
2. \(\quad 2 x ^ { 2 } + 5 x \geqslant 12\).

8 A circle has centre \(C ( 3 , - 8 )\) and radius 10 .

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis.
3. The tangent to the circle at the point \(A\) has gradient \(\frac { 5 } { 2 }\). Find an equation of the line \(C A\), giving your answer in the form \(r x + s y + t = 0\), where \(r , s\) and \(t\) are integers.
4. The line with equation \(y = 2 x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.