Edexcel FP1 (Further Pure Mathematics 1) 2014 January

1. \(\mathrm { f } ( x ) = 2 x - 5 \cos x , \quad\) where \(x\) is in radians.
   1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1,1.4 ]\).
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   2. Starting with the interval [1,1.4], use interval bisection twice to find an interval of width 0.1 which contains \(\alpha\).
      

2.

1. $$\mathbf { A } = \left( \begin{array} { c c } - 4 & 10
   - 3 & k \end{array} \right) , \quad \text { where } k \text { is a constant. }$$ The triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { A }\). Given that the area of triangle \(T ^ { \prime }\) is twice the area of triangle \(T\), find the possible values of \(k\).
2. Given that $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 2 & 3
   - 2 & 5 & 1 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { r r } 2 & 8
   0 & 2
   1 & - 2 \end{array} \right)$$ find \(\mathbf { B C }\).
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3. A rectangular hyperbola has parametric equations $$x = 2 t , \quad y = \frac { 2 } { t } , \quad t \neq 0$$ Points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 2 }\) and \(t = 4\) respectively.
The line \(L\), which passes through the origin \(O\), is perpendicular to the chord \(P Q\).
Find an equation for \(L\).

4. $$f ( x ) = 2 x ^ { \frac { 1 } { 2 } } - \frac { 6 } { x ^ { 2 } } - 3 , \quad x > 0$$ A root \(\beta\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,4 ]\).
Taking 3.5 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 3 decimal places.

5. $$z = 5 + \mathrm { i } \sqrt { 3 } , \quad w = \sqrt { 3 } - \mathrm { i }$$

1. Find the value of \(| w |\). Find in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real constants,
2. \(z w\), showing clearly how you obtained your answer,
3. \(\frac { z } { w }\), showing clearly how you obtained your answer. Given that $$\arg ( z + \lambda ) = \frac { \pi } { 3 } , \quad \text { where } \lambda \text { is a real constant, }$$
4. find the value of \(\lambda\).

6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n - 1 ) ( n + a )$$ where \(a\) is an integer to be determined.
(b) Hence find the value of \(n\), where \(n > 1\), that satisfies $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = 10 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

7. $$\mathbf { P } = \left( \begin{array} { c c } 3 a & - 2 a
- b & 2 b \end{array} \right) , \quad \mathbf { M } = \left( \begin{array} { c c } - 6 a & 7 a
2 b & - b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.

1. Find \(\mathbf { P } ^ { - 1 }\), leaving your answer in terms of \(a\) and \(b\). Given that $$\mathbf { M } = \mathbf { P Q }$$
2. find the matrix \(\mathbf { Q }\), giving your answer in its simplest form.
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8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a p ^ { 2 } , 2 a p \right)\) lies on the parabola \(C\).

1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = a p ^ { 3 } + 2 a p$$ The normal to \(C\) at the point \(P\) meets the \(x\)-axis at the point \(( 6 a , 0 )\) and meets the directrix of \(C\) at the point \(D\). Given that \(p > 0\),
2. find, in terms of \(a\), the coordinates of \(D\). Given also that the directrix of \(C\) cuts the \(x\)-axis at the point \(X\),
3. find, in terms of \(a\), the area of the triangle XPD, giving your answer in its simplest form.

9. Given that \(z = x + \mathrm { i } y\), where \(x \in \mathbb { R } , y \in \mathbb { R }\), find the value of \(x\) and the value of \(y\) such that $$( 3 - i ) z ^ { * } + 2 i z = 9 - i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

10. (i) A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by $$u _ { n + 1 } = 5 u _ { n } + 3 , \quad u _ { 1 } = 3$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = \frac { 3 } { 4 } \left( 5 ^ { n } - 1 \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 5 \left( 5 ^ { n } \right) - 4 n - 5 \text { is divisible by } 16 .$$ \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-32_109_127_2473_1818}
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