Edexcel F1 (Further Pure Mathematics 1) 2022 June

1. $$z _ { 1 } = 3 + 3 i \quad z _ { 2 } = p + q i \quad p , q \in \mathbb { R }$$ Given that \(\left| z _ { 1 } z _ { 2 } \right| = 15 \sqrt { 2 }\)

1. determine \(\left| z _ { 2 } \right|\) Given also that \(p = - 4\)
2. determine the possible values of \(q\)
3. Show \(z _ { 1 }\) and the possible positions for \(z _ { 2 }\) on the same Argand diagram.

2. $$f ( x ) = 10 - 2 x - \frac { 1 } { 2 \sqrt { x } } - \frac { 1 } { x ^ { 3 } } \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [0.4, 0.5]
2. Determine \(\mathrm { f } ^ { \prime } ( x )\).
3. Using \(x _ { 0 } = 0.5\) 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. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval [4.8, 4.9]
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4. Use linear interpolation once on the interval [4.8, 4.9] to find an approximation to \(\beta\), giving your answer to 3 decimal places.

1. \(\mathbf { M } = \left( \begin{array} { c c } k & k
   3 & 5 \end{array} \right) \quad\) where \(k\) is a non-zero constant
   1. Determine \(\mathbf { M } ^ { - 1 }\), giving your answer in simplest form in terms of \(k\).
   Hence, given that \(\mathbf { N } ^ { - 1 } = \left( \begin{array} { c c } k & k
   4 & - 1 \end{array} \right)\)
2. determine \(( \mathbf { M N } ) ^ { - 1 }\), giving your answer in simplest form in terms of \(k\).

4. $$f ( z ) = 2 z ^ { 4 } - 19 z ^ { 3 } + A z ^ { 2 } + B z - 156$$ where \(A\) and \(B\) are constants.
The complex number \(5 - \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
3. Determine the value of \(A\) and the value of \(B\).

1. The quadratic equation

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

1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
2. determine the value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
3. find a quadratic equation which has roots $$\left( \alpha ^ { 3 } - \beta \right) \text { and } \left( \beta ^ { 3 } - \alpha \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.

1. The parabola \(C\) has equation \(y ^ { 2 } = 36 x\)

The point \(P \left( 9 t ^ { 2 } , 18 t \right)\), where \(t \neq 0\), lies on \(C\)

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$y + t x = 9 t ^ { 3 } + 18 t$$
2. Hence find the equations of the two normals to \(C\) which pass through the point (54, 0), giving your answers in the form \(y = p x + q\) where \(p\) and \(q\) are constants to be determined. Given that
   • the normals found in part (b) intersect the directrix of \(C\) at the points \(A\) and \(B\)
   • the point \(F\) is the focus of \(C\)
   • determine the area of triangle \(A F B\)

7. $$A = \left( \begin{array} { c c } - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

1. Determine the matrix \(\mathbf { A } ^ { 2 }\)
2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\)
3. Hence determine the smallest positive integer value of \(n\) for which \(\mathbf { A } ^ { n } = \mathbf { I }\) The matrix \(\mathbf { B }\) represents a stretch scale factor 4 parallel to the \(x\)-axis.
4. Write down the matrix \(\mathbf { B }\) The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\)
5. Determine the matrix \(\mathbf { C }\) The parallelogram \(P\) is transformed onto the parallelogram \(P ^ { \prime }\) by the matrix \(\mathbf { C }\)
6. Given that the area of parallelogram \(P ^ { \prime }\) is 20 square units, determine the area of parallelogram \(P\)

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

$$\sum _ { r = 0 } ^ { n } ( r + 1 ) ( r + 2 ) = \frac { 1 } { 3 } ( n + 1 ) ( n + 2 ) ( n + 3 )$$ (b) Hence determine the value of $$10 \times 11 + 11 \times 12 + 12 \times 13 + \ldots + 100 \times 101$$

1. (i) A sequence of numbers is defined by

$$\begin{gathered} u _ { 1 } = 3
u _ { n + 1 } = 2 u _ { n } - 2 ^ { n + 1 } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { N }\) $$u _ { n } = 5 \times 2 ^ { n - 1 } - n \times 2 ^ { n }$$ (ii) Prove by induction that, for \(n \in \mathbb { N }\) $$f ( n ) = 5 ^ { n + 2 } - 4 n - 9$$ is divisible by 16