1.01c Disproof by counter example

5. (i) Use algebra to prove that for all \(x \geqslant 0\) $$3 x + 1 \geqslant 2 \sqrt { 3 x }$$ (ii) Show that the following statement is not true.
"The sum of three consecutive prime numbers is always a multiple of 5 "

10. (i) Prove by counter example that the statement
"if \(p\) is a prime number then \(2 p + 1\) is also a prime number" is not true.
(ii) Use proof by exhaustion to prove that if \(n\) is an integer then $$5 n ^ { 2 } + n + 12$$ is always even.

1. (i) Use a counter example to show that the following statement is false

$$\text { " } n ^ { 2 } + 3 n + 1 \text { is prime for all } n \in \mathbb { N } \text { " }$$ (ii) Use algebra to prove by exhaustion that for all \(n \in \mathbb { N }\) $$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 \text { " }$$

3. (i) Use algebra to prove that for all real values of \(x\) $$( x - 4 ) ^ { 2 } \geqslant 2 x - 9$$ (ii) Show that the following statement is untrue. $$2 ^ { n } + 1 \text { is a prime number for all values of } n , n \in \mathbb { N }$$

1. (i) A student writes the following statement:
   "When \(a\) and \(b\) are consecutive prime numbers, \(a ^ { 2 } + b ^ { 2 }\) is never a multiple of 10 "
   Prove by counter example that this statement is not true.
   (ii) Given that \(x\) and \(y\) are even integers greater than 0 and less than 6 , prove by exhaustion, that

$$1 < x ^ { 2 } - \frac { x y } { 4 } < 15$$

9. (a) Prove that for all positive values of \(x\) and \(y\), $$\frac { x + y } { 2 } \geqslant \sqrt { x y }$$ (b) Prove by counter-example that this inequality does not hold when \(x\) and \(y\) are both negative.
(1)
\includegraphics[max width=\textwidth, alt={}, center]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-29_61_54_2608_1852}

3 The converse of the statement ' \(P \Rightarrow Q\) ' is ' \(Q \Rightarrow P\) '.
Write down the converse of the following statement.
' \(n\) is an odd integer \(\Rightarrow 2 n\) is an even integer.'
Show that this converse is false.

5

1. Verify the following statement: $$\text { ' } 2 ^ { p } - 1 \text { is a prime number for all prime numbers } p \text { less than } 11 \text { '. }$$
2. Calculate \(23 \times 89\), and hence disprove this statement: $$\text { ' } 2 ^ { p } - 1 \text { is a prime number for all prime numbers } p ^ { \prime } \text {. }$$

2. (i) Solve the equation $$\ln ( 3 x + 1 ) = 2$$ giving your answer in terms of e.
(ii) 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.

5 Prove that the following statement is false.
For all integers \(n\) greater than or equal to \(1 , n ^ { 2 } + 3 n + 1\) is a prime number.

1 John asserts that the expression \(n ^ { 2 } + n + 11\) is prime for all positive integer values of \(n\). Show that John is wrong in his assertion.

6

1. Disprove the following statement. $$\text { 'If } p > q \text {, then } \frac { 1 } { p } < \frac { 1 } { q } \text {. }$$
2. State a condition on \(p\) and \(q\) so that the statement is true.

2

1. Given that \(a\) and \(b\) are real numbers, find a counterexample to disprove the statement that, if \(a > b\), then \(a ^ { 2 } > b ^ { 2 }\).
2. A student writes the statement that \(\sin x ^ { \circ } = 0.5 \Longleftrightarrow x ^ { \circ } = 30 ^ { \circ }\).
   1. Explain why this statement is incorrect.
   2. Write a corrected version of this statement.
3. Prove that the sum of four consecutive multiples of 4 is always a multiple of 8 .

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.

1. (a) Prove that for all positive values of \(a\) and \(b\)

$$\frac { 4 a } { b } + \frac { b } { a } \geqslant 4$$ (b) Prove, by counter example, that this is not true for all values of \(a\) and \(b\).

1. (i) A student states
   "if \(x ^ { 2 }\) is greater than 9 then \(x\) must be greater than 3 "

Determine whether or not this statement is true, giving a reason for your answer.
(ii) Prove that for all positive integers \(n\), $$n ^ { 3 } + 3 n ^ { 2 } + 2 n$$ is divisible by 6

1. In this question \(p\) and \(q\) are positive integers with \(q > p\)

Statement 1: \(q ^ { 3 } - p ^ { 3 }\) is never a multiple of 5

1. Show, by means of a counter example, that Statement 1 is not true. Statement 2: When \(p\) and \(q\) are consecutive even integers \(q ^ { 3 } - p ^ { 3 }\) is a multiple of 8
2. Prove, using algebra, that Statement 2 is true.

1. A student is investigating the following statement about natural numbers.

\begin{displayquote} " \(n ^ { 3 } - n\) is a multiple of 4 "

1. Prove, using algebra, that the statement is true for all odd numbers.
2. Use a counterexample to show that the statement is not always true. \end{displayquote}

1. (i) Use a counter example to show that the following statement is false.

$$" n ^ { 2 } - n - 1 \text { is a prime number, for } 3 \leqslant n \leqslant 10 \text {." }$$ (ii) Prove that the following statement is always true.
"The difference between the cube and the square of an odd number is even."
For example \(5 ^ { 3 } - 5 ^ { 2 } = 100\) is even. \includegraphics[max width=\textwidth, alt={}, center]{fa7abe9f-f5c0-4578-afd1-73176c717536-12_2255_51_314_1978}

3. a. "If \(p\) and \(q\) are irrational numbers, where \(p \neq q , q \neq 0\), then \(\frac { p } { q }\) is also irrational." Disprove this statement by means of a counter example.
b. (i) Sketch the graph of \(y = | x | - 2\).
(ii) Explain why \(| x - 2 | \geq | x | - 2\) for all real values of \(x\).

1. (a) "If \(m\) and \(n\) are irrational numbers, where \(m \neq n\), then \(m n\) is also irrational."

Disprove this statement by means of a counter example.
(b) (i) Sketch the graph of \(y = | x | + 3\) (ii) Explain why \(| x | + 3 \geqslant | x + 3 |\) for all real values of \(x\).

5

1. Prove that the following statement is not true. \(m\) is an odd number greater than \(1 \Rightarrow m ^ { 2 } + 4\) is prime.
2. By considering separately the case when \(n\) is odd and the case when \(n\) is even, prove that the following statement is true. \(n\) is a positive integer \(\Rightarrow n ^ { 2 } + 1\) is not a multiple of 4 .

8

1. Prove that the following statement is not true. $$p \text { is a positive integer } \Rightarrow 2 ^ { p } \geqslant p ^ { 2 }$$
2. Prove that the following statement is true. \(m\) and \(n\) are consecutive positive odd numbers \(\Rightarrow m n + 1\) is the square of an even number

3 Give a counter example to disprove the following statement.
If \(x\) and \(y\) are both irrational then \(x + y\) is irrational.