AQA FP1 (Further Pure Mathematics 1) 2005 January

1 The equation $$x ^ { 2 } - 5 x - 2 = 0$$ has roots \(\alpha\) and \(\beta\).

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

2 A curve has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Sketch the curve, showing the coordinates of the points of intersection with the coordinate axes.
2. Calculate the \(y\)-coordinates of the points of intersection of the curve with the line \(x = 1\). Give your answers in the form \(p \sqrt { 2 }\), where \(p\) is a rational number.
3. The curve is translated one unit in the positive \(x\) direction. Write down the equation of the curve after the translation.

3 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Write down, in terms of \(x\) and \(y\), an expression for \(z ^ { * }\), the complex conjugate of \(z\).
2. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$2 z - \mathrm { i } z ^ { * }$$
3. Find the complex number \(z\) such that $$2 z - \mathrm { i } z ^ { * } = 3 \mathrm { i }$$

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

1. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } \mathrm {~d} x\);
   (3 marks)
2. \(\quad \int _ { 2 } ^ { \infty } \left( 8 x ^ { - 3 } + 1 \right) \mathrm { d } x\);
3. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x\).

5

1. The transformation \(T _ { 1 }\) is defined by the matrix $$\left[ \begin{array} { l l } 0 & 1
   1 & 0 \end{array} \right]$$ Describe this transformation geometrically.
2. The transformation \(T _ { 2 }\) is an anticlockwise rotation about the origin through an angle of \(60 ^ { \circ }\). Find the matrix of the transformation \(T _ { 2 }\). Use surds in your answer where appropriate.
   (3 marks)
3. Find the matrix of the transformation obtained by carrying out \(T _ { 1 }\) followed by \(T _ { 2 }\).
   (3 marks)

6 The angle \(x\) radians satisfies the equation $$\cos \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { \sqrt { 2 } }$$

1. Find the general solution of this equation, giving the roots as exact values in terms of \(\pi\).
2. Find the number of roots of the equation which lie between 0 and \(2 \pi\).

7 [Figure 1, printed on the insert, is provided for use in this question.]
The variables \(x\) and \(y\) are known to be related by an equation of the form $$y ^ { 3 } = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants. Experimental evidence has provided the following approximate values:

1. On Figure 1, draw a linear graph connecting the variables \(X\) and \(Y\), where $$X = x ^ { 2 } \quad \text { and } \quad Y = y ^ { 3 }$$
2. From your graph, find approximate values for the constants \(a\) and \(b\).

8 [Figure 2, printed on the insert, is provided for use in this question.]
The diagram shows a part of the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 3 } - 2 x - 1$$ The point \(P\) has coordinates \(( 1 , - 2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{a77cc9c3-5ff6-4abc-931e-e811740267f2-05_606_565_717_740}

1. Taking \(x _ { 1 } = 1\) as a first approximation to a root of the equation \(\mathrm { f } ( x ) = 0\), use the NewtonRaphson method to find a second approximation, \(x _ { 2 }\), to the root.
2. On Figure 2, draw a straight line to illustrate the Newton-Raphson method as used in part (a). Mark \(x _ { 1 }\) and \(x _ { 2 }\) on Figure 2
3. By considering \(f ( 2 )\), show that the second approximation found in part (a) is not as good as the first approximation.
4. Taking \(x _ { 1 } = 1.6\) as a first approximation to the root, use the Newton-Raphson method to find a second approximation to the root. Give your answer to three decimal places.
   (2 marks)

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 2 x + 2 } { x ^ { 2 } }$$

1. Write down the equations of the two asymptotes to the curve \(y = \mathrm { f } ( x )\).
2. By considering the expression \(x ^ { 2 } + 2 x + 2\) :
   1. show that the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis;
   2. find the non-real roots of the equation \(\mathrm { f } ( x ) = 0\).
3. 1. Show that, if the equation \(\mathrm { f } ( x ) = k\) has two equal roots, then $$4 - 8 ( 1 - k ) = 0$$
   2. Deduce that the graph of \(y = \mathrm { f } ( x )\) has exactly one stationary point and find its coordinates.