Complete or critique given proof

10. (a) A student's attempt to answer the question
"Prove by contradiction that if \(n ^ { 3 }\) is even, then \(n\) is even" is shown below. Line 5 of the proof is missing. Assume that there exists a number \(n\) such that \(n ^ { 3 }\) is even, but \(n\) is odd. If \(n\) is odd then \(n = 2 p + 1\) where \(p \in \mathbb { Z }\)
So \(n ^ { 3 } = ( 2 p + 1 ) ^ { 3 }\) $$\begin{aligned} & = 8 p ^ { 3 } + 12 p ^ { 2 } + 6 p + 1 \\ & = \end{aligned}$$ This contradicts our initial assumption, so if \(n ^ { 3 }\) is even, then \(n\) is even. Complete this proof by filling in line 5.
(b) Hence, prove by contradiction that \(\sqrt [ 3 ] { 2 }\) is irrational.

7．The variable \(y\) is defined by $$y = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \text { for } 0 < x < \frac { \pi } { 2 } .$$ A student was asked to prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - 4 \cot 2 x .$$ The attempted proof was as follows： $$\begin{aligned} y & = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \\ & = \ln \left( \sec ^ { 2 } x \right) + \ln \left( \operatorname { cosec } ^ { 2 } x \right) \\ & = 2 \ln \sec x + 2 \ln \operatorname { cosec } x \\ \frac { \mathrm {~d} y } { \mathrm {~d} x } & = 2 \tan x - 2 \cot x \\ & = \frac { 2 \left( \sin ^ { 2 } x - \cos ^ { 2 } x \right) } { \sin x \cos x } \\ & = \frac { - 2 \cos 2 x } { \frac { 1 } { 2 } \sin 2 x } \\ & = - 4 \cot 2 x \end{aligned}$$ （a）Identify the error in this attempt at a proof．
（b）Give a correct version of the proof．
（c）Find and simplify a general relationship between \(p\) and \(q\) ，where \(p\) and \(q\) are variables that depend on \(x\) ，such that the student would obtain the correct result when differentiating \(\ln ( p + q )\) with respect to \(x\) by the above incorrect method．
（d）Given that \(p ( x ) = k \sec r x\) and \(q ( x ) = \operatorname { cosec } ^ { 2 } x\) ，where \(k\) and \(r\) are positive integers，find the values of \(k\) and \(r\) such that \(p\) and \(q\) satisfy the relationship found in part（c）． \section*{END} Marks for presentation： 7
TOTAL MARKS： 100

2．A student is attempting to prove that there are infinitely many prime numbers．
The student＇s attempt to prove this is in the box below． Assume there are only finitely many prime numbers，then there is a biggest prime number，\(p\) ． Let \(n = 2 p + 1\) ．Then \(n\) is bigger than \(p\) and since \(2 p + 1\) is not divisible by \(p\) ， \(n\) is a prime number． Hence \(n\) is a prime number bigger than \(p\) ，contradicting the initial assumption． So we conclude there are infinitely many prime numbers．
（a）Use \(p = 7\) to show that the following claim made in the student＇s proof is not true： since \(2 p + 1\) is not divisible by \(p , n\) is a prime number． The student changes their proof to use \(n = 6 p + 1\) instead of \(n = 2 p + 1\)
（b）Show，by counter example，that this does not correct the student＇s proof．
（c）Write out a correct proof by contradiction to show that there are infinitely many prime numbers．

1. A student attempts to answer the following question:

Given that \(x\) is an obtuse angle, use algebra to prove by contradiction that $$\sin x - \cos x \geqslant 1$$ The student starts the proof with: Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\begin{aligned} & \Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1 \\ & \Rightarrow \ldots \end{aligned}$$ The start of the student's proof is reprinted below.
Complete the proof. Assume that \(\sin x - \cos x < 1\) when \(x\) is an obtuse angle $$\Rightarrow ( \sin x - \cos x ) ^ { 2 } < 1$$