AQA C1 (Core Mathematics 1) 2008 June

1 The straight line \(L\) has equation \(y = 3 x - 1\) and the curve \(C\) has equation $$y = ( x + 3 ) ( x - 1 )$$

1. Sketch on the same axes the line \(L\) and the curve \(C\), showing the values of the intercepts on the \(x\)-axis and the \(y\)-axis.
2. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation \(x ^ { 2 } - x - 2 = 0\).
3. Hence find the coordinates of the points of intersection of \(L\) and \(C\).

2 It is given that \(x = \sqrt { 3 }\) and \(y = \sqrt { 12 }\).
Find, in the simplest form, the value of:

1. \(x y\);
2. \(\frac { y } { x }\);
3. \(( x + y ) ^ { 2 }\).

3 Two numbers, \(x\) and \(y\), are such that \(3 x + y = 9\), where \(x \geqslant 0\) and \(y \geqslant 0\). It is given that \(V = x y ^ { 2 }\).

1. Show that \(V = 81 x - 54 x ^ { 2 } + 9 x ^ { 3 }\).
2. 1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} x } = k \left( x ^ { 2 } - 4 x + 3 \right)\), and state the value of the integer \(k\).
   2. Hence find the two values of \(x\) for which \(\frac { \mathrm { d } V } { \mathrm {~d} x } = 0\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
4. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) for each of the two values of \(x\) found in part (b)(ii).
   2. Hence determine the value of \(x\) for which \(V\) has a maximum value.
   3. Find the maximum value of \(V\).

4

1. Express \(x ^ { 2 } - 3 x + 4\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   (2 marks)
2. Hence write down the minimum value of the expression \(x ^ { 2 } - 3 x + 4\).
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 3 x + 4\).

5 The curve with equation \(y = 16 - x ^ { 4 }\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-3_435_663_824_685} The points \(A ( - 2,0 ) , B ( 2,0 )\) and \(C ( 1,15 )\) lie on the curve.

1. Find an equation of the straight line \(A C\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 16 - x ^ { 4 } \right) \mathrm { d } x\).
   2. Hence calculate the area of the shaded region bounded by the curve and the line \(A C\).

6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
2. 1. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of linear factors.
3. 1. The curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) passes through the point \(( 0 , k )\). State the value of \(k\).
   2. Sketch the graph of \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\), indicating the values of \(x\) where the curve touches or crosses the \(x\)-axis.

7 The circle \(S\) has centre \(C ( 8,13 )\) and touches the \(x\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}

1. Write down an equation for \(S\), giving your answer in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. The point \(P\) with coordinates \(( 3,1 )\) lies on the circle.
   1. Find the gradient of the straight line passing through \(P\) and \(C\).
   2. Hence find an equation of the tangent to the circle \(S\) at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   3. The point \(Q\) also lies on the circle \(S\), and the length of \(P Q\) is 10 . Calculate the shortest distance from \(C\) to the chord \(P Q\).

8 The quadratic equation \(( k + 1 ) x ^ { 2 } + 4 k x + 9 = 0\) has real roots.

1. Show that \(4 k ^ { 2 } - 9 k - 9 \geqslant 0\).
2. Hence find the possible values of \(k\).