1.01d Proof by contradiction

6. As a substance cools its temperature, \(T ^ { \circ } \mathrm { C }\), is related to the time ( \(t\) minutes) for which it has been cooling. The relationship is given by the equation $$T = 20 + 60 \mathrm { e } ^ { - 0.1 t } , t \geq 0$$

1. Find the value of \(T\) when the substance started to cool.
2. Explain why the temperature of the substance is always above \(20 ^ { \circ } \mathrm { C }\).
3. Sketch the graph of \(T\) against \(t\).
4. Find the value, to 2 significant figures, of \(t\) at the instant \(T = 60\).
5. Find \(\frac { \mathrm { d } T } { \mathrm {~d} t }\).
6. Hence find the value of \(T\) at which the temperature is decreasing at a rate of \(1.8 ^ { \circ } \mathrm { C }\) per minute.

3. Prove by contradiction that there is no greatest odd integer.

8. Use proof by contradiction to prove that, for all positive real numbers \(x\) and \(y\), $$\frac { 9 x } { y } + \frac { y } { x } \geqslant 6$$

1. Three consecutive terms in a sequence of real numbers are given by

$$k , 1 + 2 k \text { and } 3 + 3 k$$ where \(k\) is a constant. Use proof by contradiction to show that this sequence is not a geometric sequence.

1. A student was asked to prove, for \(p \in \mathbb { N }\), that
   "if \(p ^ { 3 }\) is a multiple of 3 , then \(p\) must be a multiple of 3 "

The start of the student's proof by contradiction is shown in the box below. Assumption:
There exists a number \(p , p \in \mathbb { N }\), such that \(p ^ { 3 }\) is a multiple of 3 , and \(p\) is NOT a multiple of 3 Let \(p = 3 k + 1 , k \in \mathbb { N }\). $$\text { Consider } \begin{aligned} p ^ { 3 } = ( 3 k + 1 ) ^ { 3 } & = 27 k ^ { 3 } + 27 k ^ { 2 } + 9 k + 1 \\ & = 3 \left( 9 k ^ { 3 } + 9 k ^ { 2 } + 3 k \right) + 1 \quad \text { which is not a multiple of } 3 \end{aligned}$$

1. Show the calculations and statements that are required to complete the proof.
2. Hence prove, by contradiction, that \(\sqrt [ 3 ] { 3 }\) is an irrational number.

1. Use proof by contradiction to prove that the curve with equation

$$y = 2 x + x ^ { 3 } + \cos x$$ has no stationary points.

9. (i) Relative to a fixed origin \(O\), the points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively. Points \(A , B\) and \(C\) lie in a straight line, with \(B\) lying between \(A\) and \(C\).
Given \(A B : A C = 1 : 3\) show that $$\mathbf { c } = 3 \mathbf { b } - 2 \mathbf { a }$$ (ii) Given that \(n \in \mathbb { N }\), prove by contradiction that if \(n ^ { 2 }\) is a multiple of 3 then \(n\) is a multiple of 3

1. Use proof by contradiction to show that, when \(n\) is an integer,

$$n ^ { 2 } - 2$$ is never divisible by 4

1. Given that \(n\) is an integer, use algebra, to prove by contradiction, that if \(n ^ { 3 }\) is even then \(n\) is even.
   

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.

1. (a) Prove by contradiction that for all positive numbers \(k\)

$$k + \frac { 9 } { k } \geqslant 6$$ (b) Show that the result in part (a) is not true for all real numbers.

6. Prove by contradiction that, if \(a , b\) are positive real numbers, then \(a + b \geqslant 2 \sqrt { a b }\) \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-20_2655_1943_114_118}

1. (i) Find, as natural logarithms, the solutions of the equation

$$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$ (ii) Use proof by contradiction to prove that \(\log _ { 2 } 3\) is irrational.

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.

1. 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 \(n = 2 p + 1\)
2. Show,by counter example,that this does not correct the student's proof.
3. Write out a correct proof by contradiction to show that there are infinitely many prime numbers.

4 Prove by contradiction that there is no greatest multiple of 5 .

5 Charlie claims to have proved the following statement.
"The sum of a square number and a prime number cannot be a square number."

1. Give an example to show that Charlie's statement is not true. Charlie's attempt at a proof is below.
   Assume that the statement is not true.
   ⇒ There exist integers \(n\) and \(m\) and a prime \(p\) such that \(n ^ { 2 } + p = m ^ { 2 }\). \(\Rightarrow p = m ^ { 2 } - n ^ { 2 }\) \(\Rightarrow p = ( m - n ) ( m + n )\) \(\Rightarrow p\) is the product of two integers. \(\Rightarrow p\) is not prime, which is a contradiction.
   ⇒ Charlie's statement is true.
2. Explain the error that Charlie has made.
3. Given that 853 is a prime number, find the square number \(S\) such that \(S + 853\) is also a square number.

6 Shona makes the following claim.
" \(n\) is an even positive integer greater than \(2 \Rightarrow 2 ^ { n } - 1\) is not prime"
Prove that Shona's claim is true.

7. (i) Given that \(a\) and \(b\) are integers such that $$a + b \text { is odd }$$ Use algebra to prove by contradiction that at least one of \(a\) and \(b\) is odd.
(ii) A student is trying to prove that $$( p + q ) ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 } \quad \text { where } p < 0$$ The student writes: $$\begin{gathered} \qquad \begin{array} { c } p ^ { 2 } + 2 p q + q ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 } \\ 2 p q < 12 p ^ { 2 } \\ \text { so as } p < 0 \quad 2 q < 12 p \\ q < 6 p \end{array} \end{gathered}$$ a. Identify the error made in the proof.
b. Write out the correct solution.

16. Prove by contradiction that if \(n ^ { 2 }\) is a multiple of \(3 , n\) is a multiple of 3 .

11. Prove, using algebra that $$n ^ { 2 } + 1$$ is not divisible by 4 .

1. (i) Given that \(p\) and \(q\) are integers such that

use algebra to prove by contradiction that at least one of \(p\) or \(q\) is even.
(ii) Given that \(x\) and \(y\) are integers such that

• \(x < 0\)
• \(( x + y ) ^ { 2 } < 9 x ^ { 2 } + y ^ { 2 }\) show that \(y > 4 x\)

1. (i) Show that \(k ^ { 2 } - 4 k + 5\) is positive for all real values of \(k\).
   (ii) A student was asked to prove by contradiction that "There are no positive integers \(x\) and \(y\) such that \(( 3 x + 2 y ) ( 2 x - 5 y ) = 28\) " The start of the student's proof is shown below.

Assume that positive integers \(x\) and \(y\) exist such that $$\left. \begin{array} { c } ( 3 x + 2 y ) ( 2 x - 5 y ) = 28 \\ \text { If } 3 x + 2 y = 14 \text { and } 2 x - 5 y = 2 \\ 3 x + 2 y = 14 \\ 2 x - 5 y = 2 \end{array} \right\} \Rightarrow x = \frac { 74 } { 19 } , y = \frac { 22 } { 19 } \text { Not integers }$$ Show the calculations and statements needed to complete the proof.