AQA FP1 (Further Pure Mathematics 1) 2008 January

1 It is given that \(z _ { 1 } = 2 + \mathrm { i }\) and that \(z _ { 1 } { } ^ { * }\) is the complex conjugate of \(z _ { 1 }\).
Find the real numbers \(x\) and \(y\) such that $$x + 3 \mathrm { i } y = z _ { 1 } + 4 \mathrm { i } z _ { 1 } *$$

2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ^ { x }$$ Starting at the point \(( 1,4 )\) on the curve, use a step-by-step method with a step length of 0.01 to estimate the value of \(y\) at \(x = 1.02\). Give your answer to six significant figures.

3 Find the general solution of the equation $$\tan 4 \left( x - \frac { \pi } { 8 } \right) = 1$$ giving your answer in terms of \(\pi\).

4

1. Find $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 6 r \right)$$ expressing your answer in the form $$k n ( n + 1 ) ( n + p ) ( n + q )$$ where \(k\) is a fraction and \(p\) and \(q\) are integers.
2. It is given that $$S = \sum _ { r = 1 } ^ { 1000 } \left( r ^ { 3 } - 6 r \right)$$ Without calculating the value of \(S\), show that \(S\) is a multiple of 2008 .

5 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$$ and its asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-03_531_1013_616_516}

1. Write down the equations of the two asymptotes.
2. The points on the hyperbola for which \(x = 4\) are denoted by \(A\) and \(B\). Find, in surd form, the \(y\)-coordinates of \(A\) and \(B\).
3. The hyperbola and its asymptotes are translated by two units in the positive \(y\) direction. Write down:
   1. the \(y\)-coordinates of the image points of \(A\) and \(B\) under this translation;
   2. the equations of the hyperbola and the asymptotes after the translation.

6 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \sqrt { 3 } & 3
3 & - \sqrt { 3 } \end{array} \right]$$

1. 1. Show that $$\mathbf { M } ^ { 2 } = p \mathbf { I }$$ where \(p\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Show that the matrix \(\mathbf { M }\) can be written in the form $$q \left[ \begin{array} { c c } \cos 60 ^ { \circ } & \sin 60 ^ { \circ }
      \sin 60 ^ { \circ } & - \cos 60 ^ { \circ } \end{array} \right]$$ where \(q\) is a real number. Give the value of \(q\) in surd form.
2. The matrix \(\mathbf { M }\) represents a combination of an enlargement and a reflection. Find:
   1. the scale factor of the enlargement;
   2. the equation of the mirror line of the reflection.
3. Describe fully the geometrical transformation represented by \(\mathbf { M } ^ { 4 }\).

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows the curve $$y = x ^ { 3 } - x + 1$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates - 1 and \(- 1 + h\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-05_978_1184_676_411}

1. 1. Show that the \(y\)-coordinate of the point \(B\) is $$1 + 2 h - 3 h ^ { 2 } + h ^ { 3 }$$
   2. Find the gradient of the chord \(A B\) in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
   3. Explain how your answer to part (a)(ii) can be used to find the gradient of the tangent to the curve at \(A\). State the value of this gradient.
2. The equation \(x ^ { 3 } - x + 1 = 0\) has one real root, \(\alpha\).
   1. Taking \(x _ { 1 } = - 1\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\).
   2. On Figure 1, draw a straight line to illustrate the Newton-Raphson method as used in part (b)(i). Show the points \(\left( x _ { 2 } , 0 \right)\) and \(( \alpha , 0 )\) on your diagram.

8

1. 1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
   2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$

9 A curve \(C\) has equation $$y = \frac { 2 } { x ( x - 4 ) }$$

1. Write down the equations of the three asymptotes of \(C\).
2. The curve \(C\) has one stationary point. By considering an appropriate quadratic equation, find the coordinates of this stationary point.
   (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).