AQA C1 (Core Mathematics 1) 2012 June

1 Express \(\frac { 5 \sqrt { 3 } - 6 } { 2 \sqrt { 3 } + 3 }\) in the form \(m + n \sqrt { 3 }\), where \(m\) and \(n\) are integers.
(4 marks)

2 The line \(A B\) has equation \(4 x - 3 y = 7\).

1. 1. Find the gradient of \(A B\).
   2. Find an equation of the straight line that is parallel to \(A B\) and which passes through the point \(C ( 3 , - 5 )\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
2. The line \(A B\) intersects the line with equation \(3 x - 2 y = 4\) at the point \(D\). Find the coordinates of \(D\).
3. The point \(E\) with coordinates \(( k - 2,2 k - 3 )\) lies on the line \(A B\). Find the value of the constant \(k\).

3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$$

1. 1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Verify that \(\mathrm { p } ( 0 ) > \mathrm { p } ( 1 )\).
3. Sketch the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6\), indicating the values where the curve crosses the \(x\)-axis.

4 The diagram shows a solid cuboid with sides of lengths \(x \mathrm {~cm} , 3 x \mathrm {~cm}\) and \(y \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-3_349_472_376_769} The total surface area of the cuboid is \(32 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(3 x ^ { 2 } + 4 x y = 16\).
   2. Hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cuboid is given by $$V = 12 x - \frac { 9 x ^ { 3 } } { 4 }$$
2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
3. 1. Verify that a stationary value of \(V\) occurs when \(x = \frac { 4 } { 3 }\).
   2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = \frac { 4 } { 3 }\).

5

1. 1. Express \(x ^ { 2 } - 3 x + 5\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = x ^ { 2 } - 3 x + 5\).
2. The curve \(C\) with equation \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = x + 5\) intersect at the point \(A ( 0,5 )\) and at the point \(B\), as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-4_471_707_653_676}
   1. Find the coordinates of the point \(B\).
   2. Find \(\int \left( x ^ { 2 } - 3 x + 5 \right) \mathrm { d } x\).
   3. Find the area of the shaded region \(R\) bounded by the curve \(C\) and the line segment \(A B\).

6 The circle with centre \(C ( 5,8 )\) touches the \(y\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-5_485_631_370_715}

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. 1. Verify that the point \(A ( 2,12 )\) lies on the circle.
   2. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(s x + t y + u = 0\), where \(s , t\) and \(u\) are integers.
3. The points \(P\) and \(Q\) lie on the circle, and the mid-point of \(P Q\) is \(M ( 7,12 )\).
   1. Show that the length of \(C M\) is \(n \sqrt { 5 }\), where \(n\) is an integer.
   2. Hence find the area of triangle \(P C Q\).

7 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve \(C\) at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 20 x - 6 x ^ { 2 } - 16$$

1. 1. Show that \(y\) is increasing when \(3 x ^ { 2 } - 10 x + 8 < 0\).
   2. Solve the inequality \(3 x ^ { 2 } - 10 x + 8 < 0\).
2. The curve \(C\) passes through the point \(P ( 2,3 )\).
   1. Verify that the tangent to the curve at \(P\) is parallel to the \(x\)-axis.
   2. The point \(Q ( 3 , - 1 )\) also lies on the curve. The normal to the curve at \(Q\) and the tangent to the curve at \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
      (7 marks)