Disproof by counter example Quiz

1. Which value is a counterexample to 'for all real $x$, $x^2\ge x$'?

• A. $x=2$
• B. $x=-1$
• C. $x=\frac12$
• D. $x=10$

2. A statement is true for $n=1,2,3,4,5$. What can be said about it?

• A. It is true for all positive integers n
• B. It has been proved
• C. It has not been disproved by these values, but it is not proved
• D. It is false

3. 'If $n$ is a positive integer then $n^2+n+1$ is prime.' Which $n$ is the smallest counterexample?

• A. $n=2$
• B. $n=3$
• C. $n=5$
• D. $n=4$

4. Which is a counterexample to 'the sum of two prime numbers is even'?

• A. $3+5$
• B. $2+3$
• C. $7+11$
• D. $13+17$

5. What is the problem with using $x=0$ to disprove '$\sqrt{x^2+9}=x+3$ for all real $x$'?

• A. It does not satisfy the conditions
• B. Both sides equal $3$, so the statement is not disproved
• C. It makes the square root negative
• D. It makes the right side zero