Counter example to disprove statement

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

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

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.

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}

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

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 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 \(\frac { 1 } { p } > \frac { 1 } { q }\).

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.

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]

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]