AQA C1 (Core Mathematics 1) 2013 January

1 The point \(A\) has coordinates \(( - 3,2 )\) and the point \(B\) has coordinates \(( 7 , k )\).
The line \(A B\) has equation \(3 x + 5 y = 1\).

1. 1. Show that \(k = - 4\).
   2. Hence find the coordinates of the midpoint of \(A B\).
2. Find the gradient of \(A B\).
3. A line which passes through the point \(A\) is perpendicular to the line \(A B\). Find an equation of this line, giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
4. The line \(A B\), with equation \(3 x + 5 y = 1\), intersects the line \(5 x + 8 y = 4\) at the point \(C\). Find the coordinates of \(C\).

2 A bird flies from a tree. At time \(t\) seconds, the bird's height, \(y\) metres, above the horizontal ground is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - t ^ { 2 } + 5 , \quad 0 \leqslant t \leqslant 4$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
2. 1. Find the rate of change of height of the bird in metres per second when \(t = 1\).
   2. Determine, with a reason, whether the bird's height above the horizontal ground is increasing or decreasing when \(t = 1\).
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) when \(t = 2\).
   2. Given that \(y\) has a stationary value when \(t = 2\), state whether this is a maximum value or a minimum value.

3

1. 1. Express \(\sqrt { 18 }\) in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
   2. Simplify \(\frac { \sqrt { 8 } } { \sqrt { 18 } + \sqrt { 32 } }\).
2. Express \(\frac { 7 \sqrt { 2 } - \sqrt { 3 } } { 2 \sqrt { 2 } - \sqrt { 3 } }\) in the form \(m + \sqrt { n }\), where \(m\) and \(n\) are integers.

4

1. 1. Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Use the result from part (a)(i) to show that the equation \(x ^ { 2 } - 6 x + 11 = 0\) has no real solutions.
2. A curve has equation \(y = x ^ { 2 } - 6 x + 11\).
   1. Find the coordinates of the vertex of the curve.
   2. Sketch the curve, indicating the value of \(y\) where the curve crosses the \(y\)-axis.
   3. Describe the geometrical transformation that maps the curve with equation \(y = x ^ { 2 } - 6 x + 11\) onto the curve with equation \(y = x ^ { 2 }\).

5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$$

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 \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as a product of linear factors.
3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\), stating the values of \(x\) where the curve meets the \(x\)-axis.

6 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 10 x ^ { 4 } - 6 x ^ { 2 } + 5$$ The curve passes through the point \(P ( 1,4 )\).

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. Find the equation of the curve.

7 A circle with centre \(C ( - 3,2 )\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 12$$

1. Find the \(y\)-coordinates of the points where the circle crosses the \(y\)-axis.
2. Find the radius of the circle.
3. The point \(P ( 2,5 )\) lies outside the circle.
   1. Find the length of \(C P\), giving your answer in the form \(\sqrt { n }\), where \(n\) is an integer.
   2. The point \(Q\) lies on the circle so that \(P Q\) is a tangent to the circle. Find the length of \(P Q\).

8 A curve has equation \(y = 2 x ^ { 2 } - x - 1\) and a line has equation \(y = k ( 2 x - 3 )\), where \(k\) is a constant.

1. Show that the \(x\)-coordinate of any point of intersection of the curve and the line satisfies the equation $$2 x ^ { 2 } - ( 2 k + 1 ) x + 3 k - 1 = 0$$
2. The curve and the line intersect at two distinct points.
   1. Show that \(4 k ^ { 2 } - 20 k + 9 > 0\).
   2. Find the possible values of \(k\).