AQA FP1 (Further Pure Mathematics 1) 2010 January

1 The quadratic equation $$3 x ^ { 2 } - 6 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 6\).
3. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\).

2 The complex number \(z\) is defined by $$z = 1 + \mathrm { i }$$

1. Find the value of \(z ^ { 2 }\), giving your answer in its simplest form.
2. Hence show that \(z ^ { 8 } = 16\).
3. Show that \(\left( z ^ { * } \right) ^ { 2 } = - z ^ { 2 }\).

3 Find the general solution of the equation $$\sin \left( 4 x + \frac { \pi } { 4 } \right) = 1$$

4 It is given that $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 4
3 & 1 \end{array} \right]$$ and that \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. Show that \(( \mathbf { A } - \mathbf { I } ) ^ { 2 } = k \mathbf { I }\) for some integer \(k\).
2. Given further that $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 3
   p & 1 \end{array} \right]$$ find the integer \(p\) such that $$( \mathbf { A } - \mathbf { B } ) ^ { 2 } = ( \mathbf { A } - \mathbf { I } ) ^ { 2 }$$

5

1. Explain why \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\) is an improper integral.
2. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
   1. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\);
   2. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\).

6 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a rectangle \(R _ { 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-4_652_1136_470_429}

1. The rectangle \(R _ { 1 }\) is mapped onto a second rectangle, \(R _ { 2 }\), by a transformation with matrix \(\left[ \begin{array} { l l } 3 & 0
   0 & 2 \end{array} \right]\).
   1. Calculate the coordinates of the vertices of the rectangle \(R _ { 2 }\).
   2. On Figure 1, draw the rectangle \(R _ { 2 }\).
2. The rectangle \(R _ { 2 }\) is rotated through \(90 ^ { \circ }\) clockwise about the origin to give a third rectangle, \(R _ { 3 }\).
   1. On Figure 1, draw the rectangle \(R _ { 3 }\).
   2. Write down the matrix of the rotation which maps \(R _ { 2 }\) onto \(R _ { 3 }\).
3. Find the matrix of the transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).

7 A curve \(C\) has equation \(y = \frac { 1 } { ( x - 2 ) ^ { 2 } }\).

1. 1. Write down the equations of the asymptotes of the curve \(C\).
   2. Sketch the curve \(C\).
2. The line \(y = x - 3\) intersects the curve \(C\) at a point which has \(x\)-coordinate \(\alpha\).
   1. Show that \(\alpha\) lies within the interval \(3 < x < 4\).
   2. Starting from the interval \(3 < x < 4\), use interval bisection twice to obtain an interval of width 0.25 within which \(\alpha\) must lie.

8

1. Show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r$$ can be expressed in the form $$k n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(k\) is a rational number and \(a , b\) and \(c\) are integers.
2. Show that there is exactly one positive integer \(n\) for which $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r = 8 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

9 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ and its asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-6_798_939_612_555} The constants \(a\) and \(b\) are positive integers.
The point \(A\) on the hyperbola has coordinates ( 2,0 ).
The equations of the asymptotes are \(y = 2 x\) and \(y = - 2 x\).

1. Show that \(a = 2\) and \(b = 4\).
2. The point \(P\) has coordinates ( 1,0 ). A straight line passes through \(P\) and has gradient \(m\). Show that, if this line intersects the hyperbola, the \(x\)-coordinates of the points of intersection satisfy the equation $$\left( m ^ { 2 } - 4 \right) x ^ { 2 } - 2 m ^ { 2 } x + \left( m ^ { 2 } + 16 \right) = 0$$
3. Show that this equation has equal roots if \(3 m ^ { 2 } = 16\).
4. There are two tangents to the hyperbola which pass through \(P\). Find the coordinates of the points at which these tangents touch the hyperbola.
   (No credit will be given for solutions based on differentiation.)