Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2020 Specimen

6 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - 2 x y + y ^ { 2 } = 9 .$$

7

1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).

9 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find

1. \(2 z + z ^ { * }\),
2. \(\frac { 5 } { z }\).
3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.

10

1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).

12

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
2. Find
   1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
   2. \(\int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).