AQA FP1 (Further Pure Mathematics 1) 2008 June

1 The equation $$x ^ { 2 } + x + 5 = 0$$ 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. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = - \frac { 9 } { 5 }\).
4. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).

2 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 $$3 \mathrm { i } z + 2 z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
2. Find the complex number \(z\) such that $$3 \mathrm { i } z + 2 z ^ { * } = 7 + 8 \mathrm { i }$$

3 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:

1. \(\int _ { 9 } ^ { \infty } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
2. \(\int _ { 9 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).

4 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are related by an equation of the form $$y = a x + \frac { b } { x + 2 }$$ where \(a\) and \(b\) are constants.

1. The variables \(X\) and \(Y\) are defined by \(X = x ( x + 2 ) , Y = y ( x + 2 )\). Show that \(Y = a X + b\).
2. The following approximate values of \(x\) and \(y\) have been found:

   \(x\)	1	2	3	4
   \(y\)	0.40	1.43	2.40	3.35

   1. Complete the table in Figure 1, showing values of \(X\) and \(Y\).
   2. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
   3. Estimate the values of \(a\) and \(b\).

5

1. Find, in radians, the general solution of the equation $$\cos \left( \frac { x } { 2 } + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answer in terms of \(\pi\).
2. Hence find the smallest positive value of \(x\) which satisfies this equation.

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 0 & 2
2 & 0 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { r r } 2 & 0
0 & - 2 \end{array} \right]$$

1. Calculate the matrix \(\mathbf { A B }\).
2. Show that \(\mathbf { A } ^ { 2 }\) is of the form \(k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
3. Show that \(( \mathbf { A B } ) ^ { 2 } \neq \mathbf { A } ^ { 2 } \mathbf { B } ^ { 2 }\).

7 A curve \(C\) has equation $$y = 7 + \frac { 1 } { x + 1 }$$

1. Define the translation which transforms the curve with equation \(y = \frac { 1 } { x }\) onto the curve \(C\).
2. 1. Write down the equations of the two asymptotes of \(C\).
   2. Find the coordinates of the points where the curve \(C\) intersects the coordinate axes.
3. Sketch the curve \(C\) and its two asymptotes.

8 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-04_866_883_1318_550}

1. Find the matrix of the stretch which maps \(T _ { 1 }\) to \(T _ { 2 }\).
2. The triangle \(T _ { 2 }\) is reflected in the line \(y = x\) to give a third triangle, \(T _ { 3 }\). On Figure 3, draw the triangle \(T _ { 3 }\).
3. Find the matrix of the transformation which maps \(T _ { 1 }\) to \(T _ { 3 }\).

9 The diagram shows the parabola \(y ^ { 2 } = 4 x\) and the point \(A\) with coordinates \(( 3,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-05_732_657_370_689}

1. Find an equation of the straight line having gradient \(m\) and passing through the point \(A ( 3,4 )\).
2. Show that, if this straight line intersects the parabola, then the \(y\)-coordinates of the points of intersection satisfy the equation $$m y ^ { 2 } - 4 y + ( 16 - 12 m ) = 0$$
3. By considering the discriminant of the equation in part (b), find the equations of the two tangents to the parabola which pass through \(A\).
   (No credit will be given for solutions based on differentiation.)
4. Find the coordinates of the points at which these tangents touch the parabola.