OCR FP1 AS (Further Pure 1 AS) 2021 June

1 In this question you must show detailed reasoning.
Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).

2 In this question you must show detailed reasoning.
The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).

1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
   1. \(3 z - 4 z ^ { * }\)
   2. \(( z + 1 - 3 \mathrm { i } ) ^ { 2 }\)
   3. \(\frac { z + 1 } { z - 1 }\)
2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.

3 In this question you must show detailed reasoning.
The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

4 Prove that \(n ! > 2 ^ { 2 n }\) for all integers \(n \geqslant 9\).