Edexcel F1 (Further Pure Mathematics 1) 2023 June

1. 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 \(n\)

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 ) = \frac { 1 } { 12 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Given that \(x = 2 + 3 \mathrm { i }\) is a root of the equation $$2 x ^ { 4 } - 8 x ^ { 3 } + 29 x ^ { 2 } - 12 x + 39 = 0$$

1. write down another complex root of this equation.
2. Use algebra to determine the other 2 roots of the equation.
3. Show all 4 roots on a single Argand diagram.

1. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 9\)

The point \(P\) with coordinates \(\left( 3 t , \frac { 3 } { t } \right)\), where \(t \neq 0\), lies on \(H\)

1. Use calculus to determine an equation for the normal to \(H\) at the point \(P\) Give your answer in the form \(t y - t ^ { 3 } x = \mathrm { f } ( t )\) Given that \(t = 2\)
2. determine the coordinates of the point where the normal meets \(H\) again. Give your answer in simplest form.

1. (i) \(\mathbf { A } = \left( \begin{array} { c c } - 3 & 8
   - 3 & k \end{array} \right) \quad\) where \(k\) is a constant The transformation represented by \(\mathbf { A }\) transforms triangle \(T\) to triangle \(T ^ { \prime }\) The area of triangle \(T ^ { \prime }\) is three times the area of triangle \(T\)

Determine the possible values of \(k\)
(ii) \(\mathbf { B } = \left( \begin{array} { r r } a & - 4
2 & 3 \end{array} \right)\) and \(\mathbf { B C } = \left( \begin{array} { l l l } 2 & 5 & 1
1 & 4 & 2 \end{array} \right)\) where \(a\) is a constant Determine, in terms of \(a\), the matrix \(\mathbf { C }\)

5. $$f ( x ) = x ^ { 2 } - 6 x + 3$$ The equation \(\mathrm { f } ( x ) = 0\) has roots \(\alpha\) and \(\beta\)
Without solving the equation,

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

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$z _ { 1 } = 3 + 2 i \quad z _ { 2 } = 2 + 3 i \quad z _ { 3 } = a + b i \quad a , b \in \mathbb { R }$$

1. Determine the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\) Given that \(w = \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\)
2. determine \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y\), where \(x , y \in \mathbb { R }\) Given also that \(w = \frac { 4 } { 13 } + \frac { 58 } { 13 } \mathrm { i }\)
3. determine the value of \(a\) and the value of \(b\)
4. determine arg \(w\), giving your answer in radians to 4 significant figures.

7. $$f ( x ) = x ^ { \frac { 3 } { 2 } } + x - 3$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval \([ 1,2 ]\)
   [0pt]
2. Starting with the interval [1, 2], use interval bisection twice to show that \(\alpha\) lies in the interval [1.25, 1.5]
3. 1. Determine \(\mathrm { f } ^ { \prime } ( x )\)
   2. Using 1.375 as a first approximation for \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to determine a second approximation for \(\alpha\), giving your answer to 3 decimal places.
      [0pt]
4. Use linear interpolation once on the interval [1.25,1.5] to obtain a different approximation for \(\alpha\), giving your answer to 3 decimal places.

1. The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) lies on the parabola with equation \(y ^ { 2 } = 8 x\)
   1. Show that the point \(Q \left( \frac { 2 } { p ^ { 2 } } , \frac { - 4 } { p } \right)\), where \(p \neq 0\), lies on the parabola.
   2. Show that the chord \(P Q\) passes through the focus of the parabola.
   The tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) meet at the point \(R\)
2. Determine, in simplest form, the coordinates of \(R\)

1. Prove, by induction, that for \(n \in \mathbb { Z } , n \geqslant 2\)

$$4 ^ { n } + 6 n - 10$$ is divisible by 18