AQA FP1 (Further Pure Mathematics 1) 2009 June

1 The equation $$2 x ^ { 2 } + x - 8 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(4 \alpha ^ { 2 }\) and \(4 \beta ^ { 2 }\). Give your answer in the form \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are integers.

2 A curve has equation $$y = x ^ { 2 } - 6 x + 5$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates 2 and \(2 + h\) respectively.

1. Find, in terms of \(h\), the gradient of the line \(A B\), giving your answer in its simplest form.
2. Explain how the result of part (a) can be used to find the gradient of the curve at \(A\). State the value of this gradient.

3 The complex number \(z\) is defined by $$z = x + 2 \mathrm { i }$$ where \(x\) is real.

1. Find, in terms of \(x\), the real and imaginary parts of:
   1. \(z ^ { 2 }\);
   2. \(z ^ { 2 } + 2 z ^ { * }\).
2. Show that there is exactly one value of \(x\) for which \(z ^ { 2 } + 2 z ^ { * }\) is real.

4 The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants.

1. Given that \(Y = \log _ { 10 } y\), show that \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
2. The diagram shows the linear graph which has equation \(Y = m x + c\).
   \includegraphics[max width=\textwidth, alt={}, center]{932d4c7e-6514-4543-b1d1-753fca5a08fd-5_744_720_833_699} Use this graph to calculate:
   1. an approximate value of \(y\) when \(x = 2.3\), giving your answer to one decimal place;
   2. an approximate value of \(x\) when \(y = 80\), giving your answer to one decimal place.
      (You are not required to find the values of \(m\) and \(c\).)

5

1. Find the general solution of the equation $$\cos ( 3 x - \pi ) = \frac { 1 } { 2 }$$ giving your answer in terms of \(\pi\).
2. From your general solution, find all the solutions of the equation which lie between \(10 \pi\) and \(11 \pi\).

6 An ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Sketch the ellipse \(E\), showing the coordinates of the points of intersection of the ellipse with the coordinate axes.
2. The ellipse \(E\) is stretched with scale factor 2 parallel to the \(y\)-axis. Find and simplify the equation of the curve after the stretch.
3. The original ellipse, \(E\), is translated by the vector \(\left[ \begin{array} { l } a
   b \end{array} \right]\). The equation of the translated ellipse is $$4 x ^ { 2 } + 3 y ^ { 2 } - 8 x + 6 y = 5$$ Find the values of \(a\) and \(b\).

7

1. Using surd forms where appropriate, find the matrix which represents:
   1. a rotation about the origin through \(30 ^ { \circ }\) anticlockwise;
   2. a reflection in the line \(y = \frac { 1 } { \sqrt { 3 } } x\).
2. The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { c c } 1 & \sqrt { 3 }
   \sqrt { 3 } & - 1 \end{array} \right]$$ represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.
3. The transformation represented by \(\mathbf { A }\) is followed by the transformation represented by \(\mathbf { B }\), where $$\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 3 } & - 1
   1 & \sqrt { 3 } \end{array} \right]$$ Find the matrix of the combined transformation and give a full geometrical description of this combined transformation.

8 A curve has equation $$y = \frac { x ^ { 2 } } { ( x - 1 ) ( x - 5 ) }$$

1. Write down the equations of the three asymptotes to the curve.
2. Show that the curve has no point of intersection with the line \(y = - 1\).
3. 1. Show that, if the curve intersects the line \(y = k\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$( k - 1 ) x ^ { 2 } - 6 k x + 5 k = 0$$
   2. Show that, if this equation has equal roots, then $$k ( 4 k + 5 ) = 0$$
4. Hence find the coordinates of the two stationary points on the curve.