AQA FP1 (Further Pure Mathematics 1) 2007 June

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 2 & 1
3 & 8 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } 1 & 2
3 & 4 \end{array} \right]$$ The matrix \(\mathbf { M } = \mathbf { A } - 2 \mathbf { B }\).

1. Show that \(\mathbf { M } = n \left[ \begin{array} { r r } 0 & - 1
   - 1 & 0 \end{array} \right]\), where \(n\) is a positive integer.
   (2 marks)
2. The matrix \(\mathbf { M }\) represents a combination of an enlargement of scale factor \(p\) and a reflection in a line \(L\). State the value of \(p\) and write down the equation of \(L\).
3. Show that $$\mathbf { M } ^ { 2 } = q \mathbf { I }$$ where \(q\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

2

1. Show that the equation $$x ^ { 3 } + x - 7 = 0$$ has a root between 1.6 and 1.8.
2. Use interval bisection twice, starting with the interval in part (a), to give this root to one decimal place.

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 $$z - 3 \mathbf { i } z ^ { * }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
2. Find the complex number \(z\) such that $$z - 3 \mathrm { i } z ^ { * } = 16$$

4 The quadratic equation $$2 x ^ { 2 } - x + 4 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 1 } { 4 }\).
3. Find a quadratic equation with integer coefficients such that the roots of the equation are $$\frac { 4 } { \alpha } \text { and } \frac { 4 } { \beta }$$ (3 marks)

5 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.] 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. The following approximate values of \(x\) and \(y\) have been found.

1. Complete the table in Figure 1, showing values of \(x\) and \(Y\), where \(Y = \log _ { 10 } y\). Give each value of \(Y\) to three decimal places.
2. Show that, if \(y = a b ^ { x }\), then \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
3. Draw on Figure 2 a linear graph relating \(x\) and \(Y\).
4. Hence find estimates for the values of \(a\) and \(b\).

6 Find the general solution of the equation $$\sin \left( 2 x - \frac { \pi } { 2 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).

7 A curve has equation $$y = \frac { 3 x - 1 } { x + 2 }$$

1. Write down the equations of the two asymptotes to the curve.
2. Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
3. Hence, or otherwise, solve the inequality $$0 < \frac { 3 x - 1 } { x + 2 } < 3$$

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

1. \(\quad \int _ { 0 } ^ { 1 } \left( x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } x\);
2. \(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).

9 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows the curve with equation $$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$ and the straight line with equation $$x + y = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}

1. Write down the exact coordinates of the points where the curve with equation \(\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1\) intersects the coordinate axes.
2. The curve is translated \(k\) units in the positive \(x\) direction, where \(k\) is a constant. Write down, in terms of \(k\), the equation of the curve after this translation.
3. Show that, if the line \(x + y = 2\) intersects the translated curve, the \(x\)-coordinates of the points of intersection must satisfy the equation $$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
4. Hence find the two values of \(k\) for which the line \(x + y = 2\) is a tangent to the translated curve. Give your answer in the form \(p \pm \sqrt { q }\), where \(p\) and \(q\) are integers.
5. On Figure 3, show the translated curves corresponding to these two values of \(k\). \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2 (for use in Question 5)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3 (for use in Question 9)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
   \end{figure}