Stage 9 - Q1 - 3 marks
Do these rules agree for every real input? Justify.
Hint
Prove an identity for “yes”; one counterexample suffices for “no”.
Stage 9 - Q2 - 3 marks
Do these rules agree for every real input? Justify.
Hint
Prove an identity for “yes”; one counterexample suffices for “no”.
Stage 9 - Q3 - 3 marks
Is \(y=4x^{4}\) strictly positive for every real \(x\)?
Hint
Strictly positive excludes zero.
Stage 9 - Q4 - 4 marks
For the rule \(y=ax^2\), find a and give two machines with the same rule.
Hint
The factor applied before squaring must itself be squared.
Stage 9 - Q5 - 4 marks
Find a rule of the form \(n^2+c\) for \(5,\ 8,\ 13,\ 20\) and its twentieth term.
Hint
Do not extend the term list one term at a time.
Stage 9 - Q6 - 3 marks
For \(y=2x^2+1\), find every real input giving \(y=51\).
Hint
No domain restriction has removed either sign.
Stage 9 - Q7 - 3 marks
Explain why “add 4, then square” cannot be replaced by “square, then add 16”.
Hint
Matching only one input does not prove identical functions.
Stage 9 - Q8 - 4 marks
A linear rule and \(n^2+3\) agree at \(n=1\) and \(n=2\). Find the linear rule and the first position where they differ.
Hint
Two points fix a linear rule, but not a general non-linear rule.
Stage 9 - Q9 - 4 marks
Two machines agree at \(x=0\) and \(x=4\). Does that prove they agree for every input? Give a counterexample.
Hint
Construct a second curve through the same two input-output pairs.
Stage 9 - Q10 - 5 marks
Challenge: A machine is tested only at inputs −3, 0 and 3, giving outputs 9, 0 and 9. Construct two different power-based functions fitting all tests and name one extra input that distinguishes them.
Hint
Even powers agree in sign; scale one power to match the chosen test magnitudes.