1.6 Simple deductive proof Quiz

1. Which of these is an identity?

• A. $x^2 - 1 = 0$
• B. $(x+2)^2 \equiv x^2+4x+4$
• C. $2x + 1 = 7$
• D. $x^2 = 2x$

2. Which expression represents the sum of two consecutive even integers?

• A. $4n+2$
• B. $4n$
• C. $2n+1$
• D. $4n+1$

3. Simplify $(x+5)^2 - (x-5)^2$.

• A. $0$
• B. $50$
• C. $20x$
• D. $10x$

4. A student checks that $n^2 - n + 11$ is prime for $n = 1, 2, 3$. What can they conclude?

• A. It is prime for all positive integers
• B. Nothing is proved for all $n$; in fact $n = 11$ gives $121 = 11^2$
• C. It is never prime
• D. It is prime for all odd $n$

5. Which first line is correct for proving $(x-3)^2+5 \equiv x^2-6x+14$ from LHS to RHS?

• A. LHS $= x^2 - 9 + 5$
• B. $(x-3)^2+5 = x^2-6x+14$, so $x^2-6x+14 = x^2-6x+14$
• C. LHS $= x^2-6x+9+5$
• D. RHS $- 14 = x^2-6x$