AQA C1 (Core Mathematics 1) 2012 January

1 The point \(A\) has coordinates (6, -4) and the point \(B\) has coordinates (-2, 7).

1. Given that the point \(O\) has coordinates \(( 0,0 )\), show that the length of \(O A\) is less than the length of \(O B\).
2. 1. Find the gradient of \(A B\).
   2. Find an equation of the line \(A B\) in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
3. The point \(C\) has coordinates \(( k , 0 )\). The line \(A C\) is perpendicular to the line \(A B\). Find the value of the constant \(k\).

2

1. Factorise \(x ^ { 2 } - 4 x - 12\).
2. Sketch the graph with equation \(y = x ^ { 2 } - 4 x - 12\), stating the values where the curve crosses the coordinate axes.
3. 1. Express \(x ^ { 2 } - 4 x - 12\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are positive integers.
   2. Hence find the minimum value of \(x ^ { 2 } - 4 x - 12\).
4. The curve with equation \(y = x ^ { 2 } - 4 x - 12\) is translated by the vector \(\left[ \begin{array} { r } - 3
   2 \end{array} \right]\). Find an equation of the new curve. You need not simplify your answer.

3

1. 1. Simplify \(( 3 \sqrt { 2 } ) ^ { 2 }\).
   2. Show that \(( 3 \sqrt { 2 } - 1 ) ^ { 2 } + ( 3 + \sqrt { 2 } ) ^ { 2 }\) is an integer and find its value.
2. Express \(\frac { 4 \sqrt { 5 } - 7 \sqrt { 2 } } { 2 \sqrt { 5 } + \sqrt { 2 } }\) in the form \(m - \sqrt { n }\), where \(m\) and \(n\) are integers.

4 The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}

1. Given that \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\), find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find an equation of the tangent to the curve at the point \(A ( - 1,0 )\).
3. Verify that the point \(B\), where \(x = 1\), is a minimum point of the curve.
4. The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\).
   \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
   1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve between \(A\) and \(B\) and the line segments \(A O\) and \(O B\).

5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x - 12\), where \(c\) and \(d\) are constants.

1. When \(\mathrm { p } ( x )\) is divided by \(x + 2\), the remainder is - 150 . Show that \(2 c - d + 65 = 0\).
2. Given that \(x - 3\) is a factor of \(\mathrm { p } ( x )\), find another equation involving \(c\) and \(d\).
3. By solving these two equations, find the value of \(c\) and the value of \(d\).

6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.

1. The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
2. The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
3. Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
4. Hence determine the possible values of the width of the garden.
   \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\).
5. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
6. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
7. Sketch the circle.
8. A line has equation \(y = k x + 6\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
   2. The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
   3. Hence find the values of \(k\) for which the line is a tangent to the circle.