Edexcel FP1 (Further Pure Mathematics 1) 2011 January

1. $$z = 5 - 3 \mathrm { i } , \quad w = 2 + 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z ^ { 2 }\),
2. \(\frac { z } { w }\).

2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0
5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1
5 & 2 \end{array} \right)$$

1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0
   0 & 1 \end{array} \right)$$
2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
3. write down \(\mathbf { C } ^ { 100 }\).
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3. $$f ( x ) = 5 x ^ { 2 } - 4 x ^ { \frac { 3 } { 2 } } - 6 , \quad x \geqslant 0$$ 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. Differentiate \(\mathrm { f } ( x )\) to find \(\mathrm { f } ^ { \prime } ( x )\).
3. 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.

4. Given that \(2 - 4 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0 ,$$ where \(p\) and \(q\) are real constants,

1. write down the other root of the equation,
2. find the value of \(p\) and the value of \(q\).

5. (a) Use the results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), to prove that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 5 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 2 ) ( n + 7 )$$ for all positive integers \(n\).
(b) Hence, or otherwise, find the value of $$\sum _ { r = 20 } ^ { 50 } r ( r + 1 ) ( r + 5 )$$

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 36 x\). The point \(S\) is the focus of \(C\).

1. Find the coordinates of \(S\).
2. Write down the equation of the directrix of \(C\). Figure 1 shows the point \(P\) which lies on \(C\), where \(y > 0\), and the point \(Q\) which lies on the directrix of \(C\). The line segment \(Q P\) is parallel to the \(x\)-axis. Given that the distance \(P S\) is 25 ,
3. write down the distance \(Q P\),
4. find the coordinates of \(P\),
5. find the area of the trapezium \(O S P Q\).

7. $$z = - 24 - 7 i$$

1. Show \(z\) on an Argand diagram.
2. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. It is given that $$w = a + b \mathrm { i } , \quad a \in \mathbb { R } , b \in \mathbb { R }$$ Given also that \(| w | = 4\) and \(\arg w = \frac { 5 \pi } { 6 }\),
3. find the values of \(a\) and \(b\),
4. find the value of \(| z w |\).

8. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 2
- 1 & 3 \end{array} \right)$$

1. Find \(\operatorname { det } \mathbf { A }\).
2. Find \(\mathbf { A } ^ { - 1 }\). The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\). Given that the area of triangle \(S\) is 72 square units,
3. find the area of triangle \(R\). The triangle \(S\) has vertices at the points \(( 0,4 ) , ( 8,16 )\) and \(( 12,4 )\).
4. Find the coordinates of the vertices of \(R\).

9. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , u _ { 4 } , \ldots\) is defined by $$u _ { n + 1 } = 4 u _ { n } + 2 , \quad u _ { 1 } = 2$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = \frac { 2 } { 3 } \left( 4 ^ { n } - 1 \right)$$

10. The point \(P \left( 6 t , \frac { 6 } { t } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\).

1. 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 \(( - 9,12 )\).
2. Find the coordinates of \(A\) and \(B\).