Edexcel FP1 (Further Pure Mathematics 1) 2013 June

1. The complex numbers \(z\) and \(w\) are given by

$$z = 8 + 3 \mathrm { i } , \quad w = - 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,

1. \(z - w\),
2. \(z w\).

2. (i) $$\mathbf { A } = \left( \begin{array} { c c } 2 k + 1 & k
- 3 & - 5 \end{array} \right) , \quad \text { where } k \text { is a constant }$$ Given that $$\mathbf { B } = \mathbf { A } + 3 \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, find

1. \(\mathbf { B }\) in terms of \(k\),
2. the value of \(k\) for which \(\mathbf { B }\) is singular.
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { r } 2
   - 3

4 \end{array} \right) , \quad \mathbf { D } = \left( \begin{array} { l l l } 2 & - 1 & 5 \end{array} \right)$$ and $$\mathbf { E } = \mathbf { C D }$$ find \(\mathbf { E }\).
3. $$f ( x ) = \frac { 1 } { 2 } x ^ { 4 } - x ^ { 3 } + x - 3$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between \(x = 2\) and \(x = 2.5\)
   [0pt]
2. Starting with the interval [2,2.5] use interval bisection twice to find an interval of width 0.125 which contains \(\alpha\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\beta\) in the interval \([ - 2 , - 1 ]\).
3. Taking - 1.5 as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\beta\). Give your answer to 2 decimal places.
   4. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 2 x + 5 \right)$$
4. Find the four roots of \(\mathrm { f } ( x ) = 0\)
5. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.

5. \begin{figure}[h] \end{figure} Figure 1 shows a rectangular hyperbola \(H\) with parametric equations $$x = 3 t , \quad y = \frac { 3 } { t } , \quad t \neq 0$$ The line \(L\) with equation \(6 y = 4 x - 15\) intersects \(H\) at the point \(P\) and at the point \(Q\) as shown in Figure 1.

1. Show that \(L\) intersects \(H\) where \(4 t ^ { 2 } - 5 t - 6 = 0\)
2. Hence, or otherwise, find the coordinates of points \(P\) and \(Q\).

6. $$\mathbf { A } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { l l } 2 & 3
1 & 4 \end{array} \right)$$ The transformation represented by \(\mathbf { B }\) followed by the transformation represented by \(\mathbf { A }\) is equivalent to the transformation represented by \(\mathbf { P }\).

1. Find the matrix \(\mathbf { P }\). Triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { P }\). Given that the area of triangle \(T ^ { \prime }\) is 24 square units,
2. find the area of triangle \(T\). Triangle \(T ^ { \prime }\) is transformed to the original triangle \(T\) by the matrix represented by \(\mathbf { Q }\).
3. Find the matrix \(\mathbf { Q }\).

7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a t ^ { 2 } , 2 a t \right)\) is a general point on \(C\).

1. Show that the equation of the tangent to \(C\) at \(P \left( a t ^ { 2 } , 2 a t \right)\) is $$t y = x + a t ^ { 2 }$$ The tangent to \(C\) at \(P\) meets the \(y\)-axis at a point \(Q\).
2. Find the coordinates of \(Q\). Given that the point \(S\) is the focus of \(C\),
3. show that \(P Q\) is perpendicular to \(S Q\).

8. (a) Prove by induction, that for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } r ( 2 r - 1 ) = \frac { 1 } { 6 } n ( n + 1 ) ( 4 n - 1 )$$ (b) Hence, show that $$\sum _ { r = n + 1 } ^ { 3 n } r ( 2 r - 1 ) = \frac { 1 } { 3 } n \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be found.

9. The complex number \(w\) is given by $$w = 10 - 5 \mathrm { i }$$

1. Find \(| w |\).
2. Find arg \(w\), giving your answer in radians to 2 decimal places. The complex numbers \(z\) and \(w\) satisfy the equation $$( 2 + i ) ( z + 3 i ) = w$$
3. Use algebra to find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where \(\lambda\) is a real constant,
4. find the value of \(\lambda\).

10. (i) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to evaluate $$\sum _ { r = 1 } ^ { 24 } \left( r ^ { 3 } - 4 r \right)$$ (ii) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 0 } ^ { n } \left( r ^ { 2 } - 2 r + 2 n + 1 \right) = \frac { 1 } { 6 } ( n + 1 ) ( n + a ) ( b n + c )$$ for all integers \(n \geqslant 0\), where \(a , b\) and \(c\) are constant integers to be found.