Edexcel F1 (Further Pure Mathematics 1) Specimen

1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by

$$z _ { 1 } = 2 + 8 \mathrm { i } \quad \text { and } \quad z _ { 2 } = 1 - \mathrm { i }$$ Find, showing your working,

1. \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
2. the value of \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right|\),
3. the value of \(\arg \frac { z _ { 1 } } { z _ { 2 } }\), giving your answer in radians to 2 decimal places.

2. $$f ( x ) = 5 x ^ { 2 } - 4 x ^ { \frac { 3 } { 2 } } - 6 , \quad x \geqslant 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 1.6,1.8 ]\)

1. Use linear interpolation once on the interval \([ 1.6,1.8 ]\) to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
2. Taking 1.7 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 8 x\).
The point \(P\) lies on \(C\), where \(y > 0\), and the point \(Q\) lies on \(C\), where \(y < 0\) The line segment \(P Q\) is parallel to the \(y\)-axis. Given that the distance \(P Q\) is 12 ,

1. write down the \(y\) coordinate of \(P\),
2. find the \(x\) coordinate of \(P\). Figure 1 shows the point \(S\) which is the focus of \(C\). The line \(l\) passes through the point \(P\) and the point \(S\).
3. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

1. The quadratic equation

$$5 x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 6 } { 5 }\)
3. Find a quadratic equation with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \text { and } \beta + \frac { 1 } { \beta }$$

1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),

$$\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3 \text { is divisible by } 4$$

6. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + k )$$ where \(k\) is a constant to be found.
(b) Hence evaluate \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\)

1. The point \(\mathrm { P } \left( 6 \mathrm { t } , \frac { 6 } { \mathrm { t } } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\) (a) Show that an equation for the tangent to \(H\) at \(P\) is

$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
(b) Find the coordinates of \(A\) and \(B\).

8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where, $$P = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$

1. Describe fully the transformation \(U\). The transformation \(V\), represented by the matrix \(\mathbf { Q }\), is a stretch scale factor 3 parallel to the \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
3. Find the matrix \(\mathbf { R }\).
   (ii) $$S = \left( \begin{array} { r r } 1 & - 3
   3 & 1 \end{array} \right)$$ Given that the matrix \(\mathbf { S }\) represents an enlargement, with a positive scale factor and centre \(( 0,0 )\), followed by a rotation with centre \(( 0,0 )\),
4. find the scale factor of the enlargement,
5. find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.