AQA C1 (Core Mathematics 1) 2016 June

1 The line \(A B\) has equation \(5 x + 3 y + 3 = 0\).

1. The line \(A B\) is parallel to the line with equation \(y = m x + 7\). Find the value of \(m\).
2. The line \(A B\) intersects the line with equation \(3 x - 2 y + 17 = 0\) at the point \(B\). Find the coordinates of \(B\).
3. The point with coordinates \(( 2 k + 3,4 - 3 k )\) lies on the line \(A B\). Find the value of \(k\).
   [0pt] [2 marks]

2

1. Simplify \(( 3 \sqrt { 5 } ) ^ { 2 }\).
2. Express \(\frac { ( 3 \sqrt { 5 } ) ^ { 2 } + \sqrt { 5 } } { 7 + 3 \sqrt { 5 } }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
   [0pt] [4 marks]

3

1. 1. Express \(x ^ { 2 } - 7 x + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 7 x + 2\).
2. Describe the geometrical transformation which maps the graph of \(y = x ^ { 2 } - 7 x + 2\) onto the graph of \(y = ( x - 4 ) ^ { 2 }\).
   [0pt] [3 marks]

4 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 8 x + 48\).

1. 1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as a product of three linear factors.
2. 1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
   2. Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + b x + c \right) + r\), where \(b , c\) and \(r\) are integers. [3 marks]

5 A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
4. The point \(T\) lies on the tangent to the circle at \(A\) such that \(A T = 4\). Find the length of \(C T\).

6

1. A curve has equation \(y = 8 - 4 x - 2 x ^ { 2 }\).
   1. Find the values of \(x\) where the curve crosses the \(x\)-axis, giving your answer in the form \(m \pm \sqrt { n }\), where \(m\) and \(n\) are integers.
   2. Sketch the curve, giving the value of the \(y\)-intercept.
2. A line has equation \(y = k ( x + 4 )\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the line with the curve \(y = 8 - 4 x - 2 x ^ { 2 }\) satisfy the equation $$2 x ^ { 2 } + ( k + 4 ) x + 4 ( k - 2 ) = 0$$
   2. Find the values of \(k\) for which the line is a tangent to the curve \(y = 8 - 4 x - 2 x ^ { 2 }\).
      [0pt] [3 marks]

7 The diagram shows the sketch of a curve and the tangent to the curve at \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{0d5b9235-af2b-4fd5-8fcf-b2b45e3c0a3c-14_519_817_356_614} The curve has equation \(y = 4 - x ^ { 2 } - 3 x ^ { 3 }\) and the point \(P ( - 2,24 )\) lies on the curve. The tangent at \(P\) crosses the \(x\)-axis at \(Q\).

1. 1. Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
   2. Hence find the \(x\)-coordinate of \(Q\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 4 - x ^ { 2 } - 3 x ^ { 3 } \right) \mathrm { d } x\).
   2. The point \(R ( 1,0 )\) lies on the curve. Calculate the area of the shaded region bounded by the curve and the lines \(P Q\) and \(Q R\).
      [0pt] [3 marks]

8 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).