AQA FP1 (Further Pure Mathematics 1) 2012 June

1 The quadratic equation $$5 x ^ { 2 } - 7 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }\).
3. Find a quadratic equation, with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$ (5 marks)

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

1. Find the gradient of the line passing through the point \(( - 2,14 )\) and the point on the curve for which \(x = - 2 + h\). Give your answer in the form $$p + q h + r h ^ { 2 } + h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
2. Show how the answer to part (a) can be used to find the gradient of the curve at the point ( \(- 2,14\) ). State the value of this gradient.

3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right)$$
2. Hence find the complex number \(z\) such that $$\mathrm { i } ( z + 7 ) + 3 \left( z ^ { * } - \mathrm { i } \right) = 0$$

4 Find the general solution, in degrees, of the equation $$\sin \left( 70 ^ { \circ } - \frac { 2 } { 3 } x \right) = \cos 20 ^ { \circ }$$

5 The curve \(C\) has equation \(y = \frac { x } { ( x + 1 ) ( x - 2 ) }\).
The line \(L\) has equation \(y = - \frac { 1 } { 2 }\).

1. Write down the equations of the asymptotes of \(C\).
2. The line \(L\) intersects the curve \(C\) at two points. Find the \(x\)-coordinates of these two points.
3. Sketch \(C\) and \(L\) on the same axes.
   (You are given that the curve \(C\) has no stationary points.)
4. Solve the inequality $$\frac { x } { ( x + 1 ) ( x - 2 ) } \leqslant - \frac { 1 } { 2 }$$

6

1. Using surd forms, find the matrix of a rotation about the origin through \(135 ^ { \circ }\) anticlockwise.
2. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { r r } - 1 & - 1
   1 & - 1 \end{array} \right]\).
   1. Given that \(\mathbf { M }\) represents an enlargement followed by a rotation, find the scale factor of the enlargement and the angle of the rotation.
   2. The matrix \(\mathbf { M } ^ { 2 }\) also represents an enlargement followed by a rotation. State the scale factor of the enlargement and the angle of the rotation.
   3. Show that \(\mathbf { M } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   4. Deduce that \(\mathbf { M } ^ { 2012 } = - 2 ^ { n } \mathbf { I }\) for some positive integer \(n\).

7 The equation $$24 x ^ { 3 } + 36 x ^ { 2 } + 18 x - 5 = 0$$ has one real root, \(\alpha\).

1. Show that \(\alpha\) lies in the interval \(0.1 < x < 0.2\).
2. Starting from the interval \(0.1 < x < 0.2\), use interval bisection twice to obtain an interval of width 0.025 within which \(\alpha\) must lie.
3. Taking \(x _ { 1 } = 0.2\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to four decimal places.
   (4 marks)

8 The diagram shows the ellipse \(E\) with equation $$\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$$ and the straight line \(L\) with equation $$y = x + 4$$ \includegraphics[max width=\textwidth, alt={}, center]{9f8cd5ed-f5cf-4cf6-8c92-9fd0819238ca-5_675_1120_708_468}

1. Write down the coordinates of the points where the ellipse \(E\) intersects the coordinate axes.
2. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { c } p
   0 \end{array} \right]\), where \(p\) is a constant. Write down the equation of the translated ellipse.
3. Show that, if the translated ellipse intersects the line \(L\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$9 x ^ { 2 } - ( 8 p - 40 ) x + \left( 4 p ^ { 2 } + 60 \right) = 0$$
4. Given that the line \(L\) is a tangent to the translated ellipse, find the coordinates of the two possible points of contact.
   (No credit will be given for solutions based on differentiation.)