AQA C1 (Core Mathematics 1) 2014 June

1 The point \(A\) has coordinates \(( - 1,2 )\) and the point \(B\) has coordinates \(( 3 , - 5 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
2. The midpoint of \(A B\) is \(M\).
   1. Find the coordinates of \(M\).
   2. Find an equation of the line which passes through \(M\) and which is perpendicular to \(A B\). [3 marks]
3. The point \(C\) has coordinates \(( k , 2 k + 3 )\). Given that the distance from \(A\) to \(C\) is \(\sqrt { 13 }\), find the two possible values of the constant \(k\).
   [0pt] [4 marks]

2 A rectangle has length \(( 9 + 5 \sqrt { 3 } ) \mathrm { cm }\) and area \(( 15 + 7 \sqrt { 3 } ) \mathrm { cm } ^ { 2 }\).
Find the width of the rectangle, giving your answer in the form \(( m + n \sqrt { 3 } ) \mathrm { cm }\), where \(m\) and \(n\) are integers.
[0pt] [4 marks]

3 A curve has equation \(y = 2 x ^ { 5 } + 5 x ^ { 4 } - 1\).

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. The point on the curve where \(x = - 1\) is \(P\).
   1. Determine whether \(y\) is increasing or decreasing at \(P\), giving a reason for your answer.
   2. Find an equation of the tangent to the curve at \(P\).
3. The point \(Q ( - 2,15 )\) also lies on the curve. Verify that \(Q\) is a maximum point of the curve.
   [0pt] [4 marks]

4

1. 1. Express \(16 - 6 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\) where \(p\) and \(q\) are integers.
   2. Hence write down the maximum value of \(16 - 6 x - x ^ { 2 }\).
2. 1. Factorise \(16 - 6 x - x ^ { 2 }\).
   2. Sketch the curve with equation \(y = 16 - 6 x - x ^ { 2 }\), stating the values of \(x\) where the curve crosses the \(x\)-axis and the value of the \(y\)-intercept.
      [0pt] [3 marks]

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

1. Given that \(x + 3\) is a factor of \(\mathrm { p } ( x )\), show that $$3 c - d = 8$$
2. The remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\) is 65 . Obtain a further equation in \(c\) and \(d\).
3. Use the equations from parts (a) and (b) to find the value of \(c\) and the value of \(d\). [3 marks]

6 The diagram shows a curve and a line which intersect at the points \(A , B\) and \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation \(y = x ^ { 3 } - x ^ { 2 } - 5 x + 7\) and the straight line has equation \(y = x + 7\). The point \(B\) has coordinates ( 0,7 ).

1. 1. Show that the \(x\)-coordinates of the points \(A\) and \(C\) satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
   2. Find the coordinates of the points \(A\) and \(C\).
2. Find \(\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x\).
3. Find the area of the shaded region \(R\) bounded by the curve and the line segment \(A B\).
   [0pt] [4 marks]
   \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0\). The point \(A ( 3 , - 2 )\) lies on the circle.

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. 1. Write down the coordinates of \(C\).
   2. Show that the circle has radius \(n \sqrt { 5 }\), where \(n\) is an integer.
3. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(x + p y = q\), where \(p\) and \(q\) are integers.
4. The point \(B\) lies on the tangent to the circle at \(A\) and the length of \(B C\) is 6. Find the length of \(A B\).
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{f2124c89-79de-4758-b7b8-ff273345b9dd-8_1421_1709_1286_153}

8 Solve the following inequalities:

1. \(\quad 3 ( 1 - 2 x ) - 5 ( 3 x + 2 ) > 0\)
2. \(\quad 6 x ^ { 2 } \leqslant x + 12\)
   [0pt] [4 marks]