1.01c Disproof by counter example

1 Celia states that \(n ^ { 2 } + 2 n + 10\) is always odd when \(n\) is a prime number. Prove that Celia's statement is false.

4

1. Show that \(4 ! < 4 ^ { 4 }\).
2. Nina believes that the statement \(n ! < n ^ { n }\) is true for all positive integers \(n\). Prove that Nina is not correct.

3 A student makes the following conjecture.
For all positive integers \(n , 6 n - 1\) is always prime. Use a counter example to disprove this conjecture.

1 A student states that \(1 + x ^ { 2 } < ( 1 + x ) ^ { 2 }\) for all values of \(x\).
Using a counter example, show that the student is wrong.

1 Beth states that for all real numbers \(p\) and \(q\), if \(p ^ { 2 } > q ^ { 2 }\) then \(p > q\). Prove that Beth is not correct.

9

1. Show that if \(a = 1\) and \(b > 1\) then \(\mathrm { a } ^ { \mathrm { b } } < \mathrm { b } ^ { \mathrm { a } }\).
2. Find integer values of \(a\) and \(b\) with \(b > a > 1\) and \(\mathrm { a } ^ { \mathrm { b } }\) not greater than \(\mathrm { b } ^ { \mathrm { a } }\) (a counter example to the conjecture given in lines 7-8).

7 By finding a counter example, disprove the following statement. If \(p\) and \(q\) are non-zero real numbers with \(p < q\), then \(\frac { 1 } { p } > \frac { 1 } { q }\).

1. (a) Prove, by counter-example, that the statement

$$\text { "cosec } \theta - \sin \theta > 0 \text { for all values of } \theta \text { in the interval } 0 < \theta < \pi \text { " }$$ is false.
(b) Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that $$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.

3. (a) Solve the equation $$\ln ( 3 x + 1 ) = 2$$ giving your answer in terms of e.
(b) Prove, by counter-example, that the statement $$\text { "ln } \left( 3 x ^ { 2 } + 5 x + 3 \right) \geq 0 \text { for all real values of } x \text { " }$$ is false.

2 Angela makes the following claim. \begin{displayquote} " \(n\) is an odd positive integer greater than \(1 \Rightarrow 2 ^ { n } - 1\) is prime" \end{displayquote} Prove that Angela's claim is false.

8 Kai is proving that \(n ^ { 3 } - n\) is a multiple of 3 for all positive integer values of \(n\). Kai begins a proof by exhaustion.
Step 1 $$n ^ { 3 } - n = n \left( n ^ { 2 } - 1 \right)$$ Step 2 When \(n = 3 m\), where \(m\) is a \(n ^ { 3 } - n = 3 m \left( 9 m ^ { 2 } - 1 \right)\) non-negative integer which is a multiple of 3 Step 3 When \(n = 3 m + 1\), $$\begin{aligned} & n ^ { 3 } - n = ( 3 m + 1 ) \left( ( 3 m + 1 ) ^ { 2 } - 1 \right) \\ & = ( 3 m + 1 ) \left( 9 m ^ { 2 } \right) \\ & = 3 ( 3 m + 1 ) \left( 3 m ^ { 2 } \right) \end{aligned}$$ Step 5 Therefore \(n ^ { 3 } - n\) is a multiple of 3 for all positive integer values of \(n\) 8

1. Explain the two mistakes that Kai has made after Step 3. Step 4 \section*{which is a multiple of 3
   which is a multiple of 3}
   \includegraphics[max width=\textwidth, alt={}]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-10_67_134_964_230}
   别
   all positive integer values of \(n\) \section*{a} \includegraphics[max width=\textwidth, alt={}]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-10_58_49_1037_370} 墐 pount \(\_\_\_\_\) \(\_\_\_\_\) " \(\_\_\_\_\) 8
2. Correct Kai's argument from Step 4 onwards.

6

1. Asif notices that \(24 ^ { 2 } = 576\) and \(2 + 4 = 6\) gives the last digit of 576 He checks two more examples: $$\begin{array} { l c } 27 ^ { 2 } = 729 & 29 ^ { 2 } = 841 \\ 2 + 7 = 9 & 2 + 9 = 11 \\ \text { Last digit } 9 & \text { Last digit } 1 \end{array}$$ Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared. Give a counter example to show that Asif's conclusion is not correct. 6
2. Claire tells Asif that he should look only at the last digit of the number being squared. $$\begin{array} { c c } 27 ^ { 2 } = 729 & 24 ^ { 2 } = 576 \\ 7 ^ { 2 } = 49 & 4 ^ { 2 } = 16 \\ \text { Last digit } 9 & \text { Last digit } 6 \end{array}$$ Using Claire's method determine the last digit of \(23456789 { } ^ { 2 }\) [0pt] [1 mark] 6
3. Given Claire's method is correct, use proof by exhaustion to show that no square number has a last digit of 8

3 The function min \(( a , b )\) is defined by: $$\begin{aligned} \min ( a , b ) & = a , a < b \\ & = b , \text { otherwise } \end{aligned}$$ For example, \(\min ( 7,2 ) = 2\) and \(\min ( - 4,6 ) = - 4\). Gary claims that the binary operation \(\Delta\), which is defined as $$x \Delta y = \min ( x , y - 3 )$$ where \(x\) and \(y\) are real numbers, is associative as finding the smallest number is not affected by the order of operation. Disprove Gary's claim.
[0pt] [2 marks]

1. (i) Prove by counter example that the statement
   "If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number." is false.
   (ii) Use proof by exhaustion to prove that if \(m\) is an integer that is not divisible by 3 , then

$$m ^ { 2 } - 1$$ is divisible by 3

Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x^2 = 25$$ [2]

Prove or disprove the following statement: 'No cube of an integer has 2 as its units digit.' [2]

Either prove or disprove each of the following statements.

1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.' [2]
2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]

Either prove or disprove each of the following statements.

1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.' [2]
2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]