AQA FP1 (Further Pure Mathematics 1) 2009 January

1 A curve passes through the point \(( 0,1 )\) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + x ^ { 2 } }$$ Starting at the point \(( 0,1 )\), use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 0.4\). Give your answer to five decimal places.

2 The complex number \(2 + 3 \mathrm { i }\) is a root of the quadratic equation $$x ^ { 2 } + b x + c = 0$$ where \(b\) and \(c\) are real numbers.

1. Write down the other root of this equation.
2. Find the values of \(b\) and \(c\).

3 Find the general solution of the equation $$\tan \left( \frac { \pi } { 2 } - 3 x \right) = \sqrt { 3 }$$

4 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right)$$

1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that \(S _ { n } = n ^ { 3 }\).
2. Hence show that \(\sum _ { r = n + 1 } ^ { 2 n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = k n ^ { 3 }\) for some integer \(k\).

5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } k & k
k & - k \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } - k & k
k & k \end{array} \right]$$ where \(k\) is a constant.

1. Find, in terms of \(k\) :
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { A } ^ { 2 }\).
2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
3. It is now given that \(k = 1\).
   1. Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
   2. The matrix \(\mathbf { A }\) represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.

6 A curve has equation $$y = \frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the equations of the three asymptotes of this curve.
   2. State the coordinates of the points at which the curve intersects the \(x\)-axis.
   3. Sketch the curve.
      (You are given that the curve has no stationary points.)
2. Hence, or otherwise, solve the inequality $$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$

7 The points \(P ( a , c )\) and \(Q ( b , d )\) lie on the curve with equation \(y = \mathrm { f } ( x )\). The straight line \(P Q\) intersects the \(x\)-axis at the point \(R ( r , 0 )\). The curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis at the point \(S ( \beta , 0 )\).
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1. Show that $$r = a + c \left( \frac { b - a } { c - d } \right)$$
2. Given that $$a = 2 , b = 3 \text { and } \mathrm { f } ( x ) = 20 x - x ^ { 4 }$$
   1. find the value of \(r\);
   2. show that \(\beta - r \approx 0.18\).

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

1. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 3 } { 4 } } \mathrm {~d} x\);
2. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\);
3. \(\quad \int _ { 1 } ^ { \infty } \left( x ^ { - \frac { 3 } { 4 } } - x ^ { - \frac { 5 } { 4 } } \right) \mathrm { d } x\).

9 A hyperbola \(H\) has equation $$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$

1. Find the equations of the two asymptotes of \(H\), giving each answer in the form \(y = m x\).
2. Draw a sketch of the two asymptotes of \(H\), using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola \(H\).
3. 1. Show that, if the line \(y = x + c\) intersects \(H\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
   2. Hence show that the line \(y = x + c\) intersects \(H\) in two distinct points, whatever the value of \(c\).
   3. Find, in terms of \(c\), the \(y\)-coordinates of these two points.