Edexcel F1 (Further Pure Mathematics 1) 2018 January

1. $$f ( x ) = 3 x ^ { 2 } - \frac { 5 } { 3 \sqrt { x } } - 6 , \quad x > 0$$ The single root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [1.5, 1.6].

1. Taking 1.5 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.
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2. Use linear interpolation once on the interval [1.5, 1.6] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.

2. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + 38 z ^ { 2 } - 94 z + 221$$

1. Given that \(z = 2 + 3 i\) is a root of the equation \(f ( z ) = 0\), use algebra to find the three other roots of \(f ( z ) = 0\)
2. Show the four roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.
(b) Given that $$\sum _ { r = 5 } ^ { 25 } r ^ { 2 } ( r + 1 ) + \sum _ { r = 1 } ^ { k } 3 ^ { r } = 140543$$ find the value of the integer \(k\).

1. The quadratic equation

$$3 x ^ { 2 } + 2 x + 5 = 0$$ has roots \(\alpha\) and \(\beta\). Without solving the equation,

1. find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
2. show that \(\alpha ^ { 3 } + \beta ^ { 3 } = \frac { 82 } { 27 }\)
3. find a quadratic equation which has roots $$\left( \alpha + \frac { \alpha } { \beta ^ { 2 } } \right) \text { and } \left( \beta + \frac { \beta } { \alpha ^ { 2 } } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.

5. (i) Given that $$\frac { 2 z + 3 } { z + 5 - 2 i } = 1 + i$$ find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants.
(ii) Given that $$w = ( 3 + \lambda \mathrm { i } ) ( 2 + \mathrm { i } )$$ where \(\lambda\) is a real constant, and that $$| w | = 15$$ find the possible values of \(\lambda\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 32 x\) and the point \(S\) is the focus of this parabola. The point \(P ( 2,8 )\) lies on \(C\) and the point \(T\) lies on the directrix of \(C\). The line segment \(P T\) is parallel to the \(x\)-axis.
   1. Write down the coordinates of \(S\).
   2. Find the length of \(P T\).
   3. Using calculus, show that the tangent to \(C\) at the point \(P\) has equation
   $$y = 2 x + 4$$ The hyperbola \(H\) has equation \(x y = 4\). The tangent to \(C\) at \(P\) meets \(H\) at the points \(L\) and \(M\).
2. Find the exact coordinates of the points \(L\) and \(M\), giving your answers in their simplest form.
   

7. (i) $$\mathbf { A } = \left( \begin{array} { r r } 6 & k
- 3 & - 4 \end{array} \right) , \text { where } k \text { is a real constant, } k \neq 8$$ Find, in terms of \(k\),

1. \(\mathbf { A } ^ { - 1 }\)
2. \(\mathbf { A } ^ { 2 }\) Given that \(\mathbf { A } ^ { 2 } + 3 \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 5 & 9
   - 3 & - 5 \end{array} \right)\)
3. find the value of \(k\).
   (ii) $$\mathbf { M } = \left( \begin{array} { c c } - \frac { 1 } { 2 } & - \sqrt { 3 }
   \frac { \sqrt { 3 } } { 2 } & - 1 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents a one way stretch, parallel to the \(y\)-axis, scale factor \(p\), where \(p > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about \(( 0,0 )\).
4. Find the value of \(p\).
5. Find the value of \(\theta\).

8. (i) A sequence of numbers is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = u _ { n } + 3 n - 2 \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for all positive integers \(n\), $$u _ { n } = \frac { 3 } { 2 } n ^ { 2 } - \frac { 7 } { 2 } n + 5$$ (ii) Prove by induction that, for all positive integers \(n\), $$f ( n ) = 3 ^ { 2 n + 3 } + 40 n - 27 \text { is divisible by } 64$$