Edexcel FP1 (Further Pure Mathematics 1) 2014 June

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

$$z _ { 1 } = p + 2 i \text { and } z _ { 2 } = 1 - 2 i$$ where \(p\) is an integer.

1. Find \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b\) i where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right| = 13\),
2. find the possible values of \(p\).

2. $$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 5 } { 2 x ^ { \frac { 3 } { 2 } } } + 2 x - 3 , \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.1,1.5].
2. Find f'(x).
3. Using \(x _ { 0 } = 1.1\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 decimal places.

3. Given that 2 and \(1 - 5 \mathrm { i }\) are roots of the equation $$x ^ { 3 } + p x ^ { 2 } + 30 x + q = 0 , \quad p , q \in \mathbb { R }$$

1. write down the third root of the equation.
2. Find the value of \(p\) and the value of \(q\).
3. Show the three roots of this equation on a single Argand diagram.

4. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } 1 & 2
3 & - 1
4 & 5 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r r } 2 & - 1 & 4
1 & 3 & 1 \end{array} \right)$$

1. find \(\mathbf { A B }\).
2. Explain why \(\mathbf { A B } \neq \mathbf { B A }\).
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { c r } 2 k & - 2
   3 & k \end{array} \right) \text {, where } k \text { is a real number }$$ find \(\mathbf { C } ^ { - 1 }\), giving your answer in terms of \(k\).

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ (b) Hence show that $$\sum _ { r = 2 n + 1 } ^ { 4 n } ( 2 r - 1 ) ^ { 2 } = a n \left( b n ^ { 2 } - 1 \right)$$ where \(a\) and \(b\) are constants to be found.

6. The rectangular hyperbola \(H\) has cartesian equation \(x y = c ^ { 2 }\). The point \(P \left( c t , \frac { c } { t } \right) , t > 0\), is a general point on \(H\).

1. Show that an equation of the tangent to \(H\) at the point \(P\) is $$t ^ { 2 } y + x = 2 c t$$ An equation of the normal to \(H\) at the point \(P\) is \(t ^ { 3 } x - t y = c t ^ { 4 } - c\) Given that the normal to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and the tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(B\),
2. find, in terms of \(c\) and \(t\), the coordinates of \(A\) and the coordinates of \(B\). Given that \(c = 4\),
3. find, in terms of \(t\), the area of the triangle \(A P B\). Give your answer in its simplest form.

7. (i) In each of the following cases, find a \(2 \times 2\) matrix that represents

1. a reflection in the line \(y = - x\),
2. a rotation of \(135 ^ { \circ }\) anticlockwise about \(( 0,0 )\),
3. a reflection in the line \(y = - x\) followed by a rotation of \(135 ^ { \circ }\) anticlockwise about \(( 0,0 )\).
   (ii) The triangle \(T\) has vertices at the points \(( 1 , k ) , ( 3,0 )\) and \(( 11,0 )\), where \(k\) is a constant. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 6 & - 2
   1 & 2 \end{array} \right)$$ Given that the area of triangle \(T ^ { \prime }\) is 364 square units, find the value of \(k\).

8. The points \(P \left( 4 k ^ { 2 } , 8 k \right)\) and \(Q \left( k ^ { 2 } , 4 k \right)\), where \(k\) is a constant, lie on the parabola \(C\) with equation \(y ^ { 2 } = 16 x\). The straight line \(l _ { 1 }\) passes through the points \(P\) and \(Q\).

1. Show that an equation of the line \(l _ { 1 }\) is given by $$3 k y - 4 x = 8 k ^ { 2 }$$ The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the focus of the parabola \(C\). The line \(l _ { 2 }\) meets the directrix of \(C\) at the point \(R\).
2. Find, in terms of \(k\), the \(y\) coordinate of the point \(R\).

9. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 8 ^ { n } - 2 ^ { n }$$ is divisible by 6