AQA FP1 (Further Pure Mathematics 1) 2013 January

1 A curve passes through the point (1,3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { 1 + x ^ { 3 } }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 1.2\). Give your answer to four decimal places.

2

1. Solve the equation \(w ^ { 2 } + 6 w + 34 = 0\), giving your answers in the form \(p + q \mathrm { i }\), where \(p\) and \(q\) are integers.
2. It is given that \(z = \mathrm { i } ( 1 + \mathrm { i } ) ( 2 + \mathrm { i } )\).
   1. Express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are integers.
   2. Find integers \(m\) and \(n\) such that \(z + m z ^ { * } = n \mathrm { i }\).

3

1. Find the general solution of the equation $$\sin \left( 2 x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).
2. Use your general solution to find the exact value of the greatest solution of this equation which is less than \(6 \pi\).

4 Show that the improper integral \(\int _ { 25 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\) has a finite value and find that value.

5 The roots of the quadratic equation $$x ^ { 2 } + 2 x - 5 = 0$$ are \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Calculate the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
3. Find a quadratic equation which has roots \(\alpha ^ { 3 } \beta + 1\) and \(\alpha \beta ^ { 3 } + 1\).

6

1. The matrix \(\mathbf { X }\) is defined by \(\left[ \begin{array} { l l } 1 & 2
   3 & 0 \end{array} \right]\).
   1. Given that \(\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2
      3 & 6 \end{array} \right]\), find the value of \(m\).
   2. Show that \(\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }\), where \(n\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. It is given that \(\mathbf { A } = \left[ \begin{array} { r r } 1 & 0
   0 & - 1 \end{array} \right]\).
   1. Describe the geometrical transformation represented by \(\mathbf { A }\).
   2. The matrix \(\mathbf { B }\) represents an anticlockwise rotation through \(45 ^ { \circ }\) about the origin. Show that \(\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1
      1 & 1 \end{array} \right]\), where \(k\) is a surd.
   3. Find the image of the point \(P ( - 1,2 )\) under an anticlockwise rotation through \(45 ^ { \circ }\) about the origin, followed by the transformation represented by \(\mathbf { A }\).
      \(7 \quad\) The variables \(y\) and \(x\) are related by an equation of the form $$y = a x ^ { n }$$ where \(a\) and \(n\) are constants. Let \(Y = \log _ { 10 } y\) and \(X = \log _ { 10 } x\).

1. Show that there is a linear relationship between \(Y\) and \(X\).
2. The graph of \(Y\) against \(X\) is shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-4_748_858_849_593} Find the value of \(n\) and the value of \(a\).

8

1. Show that $$\sum _ { r = 1 } ^ { n } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right) = n ( n + p ) ( n + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
2. Hence find the value of $$\sum _ { r = 11 } ^ { 20 } 2 r \left( 2 r ^ { 2 } - 3 r - 1 \right)$$ (2 marks)

9 An ellipse is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-5_453_633_365_699} The ellipse intersects the \(x\)-axis at the points \(A\) and \(B\). The equation of the ellipse is $$\frac { ( x - 4 ) ^ { 2 } } { 4 } + y ^ { 2 } = 1$$

1. Find the \(x\)-coordinates of \(A\) and \(B\).
2. The line \(y = m x ( m > 0 )\) is a tangent to the ellipse, with point of contact \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$\left( 1 + 4 m ^ { 2 } \right) x ^ { 2 } - 8 x + 12 = 0$$
   2. Hence find the exact value of \(m\).
   3. Find the coordinates of \(P\).