ltiple of 4.","difficulty":1,"difficultyBand":"easy","totalMarks":1,"sourceTitle":"Question Set 7: Algebraic Methods","calculatorUse":null,"parts":[{"id":"253046be-4aea-4d49-8509-2f7dc6f06100","label":"0","content":"Which value of $n$ shown below is a counter example to Dan’s belief?\n\n$$n = 2 \\quad n = 3 \\quad n = 4 \\quad n = 5$$","markscheme":"- Test $n=4$: $4^3 + 4 + 2 = 64 + 4 + 2 = 70$. B1\n- 70 is not a multiple of 4.","marks":1,"order":0,"markingRubric":null,"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"ef0ef36e-8ebb-4d3a-93df-910252d01ca2","revisionId":"6b6ddec7-7ab8-4f72-a22a-7eeb4d258dee","questionNumber":"35","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":4,"sourceTitle":"Question Set 7: Algebraic Methods","calculatorUse":null,"parts":[{"id":"9a6ae26f-c2ea-4187-8971-33dbd1633f8f","label":"a","content":"Find a counter example to disprove the statement: When $n$ is a prime number $n^2$ is always odd","markscheme":"- Let $n = 2$ (the only even prime). M1\n- $2^2 = 4$, which is even. A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies $n=2$ as the counterexample.","code":"M"},{"id":"p2","marks":1,"description":"Shows $2^2 = 4$ is even.","code":"A"}],"finalAnswers":["n = 2"],"generatedAt":"2026-05-12T02:14:06.612Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"counterexample","label":"Use a counterexample","steps":[{"type":"text","order":0,"title":"Test the exception","content":"A **counterexample** is one value that satisfies the starting condition but makes the conclusion false. The statement claims this works for *every* prime number, so test $n=2$, the only even prime number. This choice earns **M1**.","highlights":[{"text":"When $n$ is a prime number $n^2$ is always odd","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=chosen}{\\color{@sky}{n=2}}"},{"latex":"\\htmlData{anchor=squared}{\\color{@amber}{n^2=4}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"chosen"},{"color":"amber","style":"text-color-with-background","anchor":"squared"}],"connections":[{"to":"squared","from":"chosen","color":"sky","label":"square"}]},"order":1,"title":"Square the prime","description":"Now calculate the square of the chosen prime."},{"type":"text","order":2,"title":"State the disproof","content":"Since $4$ is even, this prime value of $n$ does **not** give an odd square. Therefore the word “always” is false, so the statement is disproved. Stating that $2^2=4$ is even earns **A1**.","highlights":[{"text":"$$n^2$ is always odd","location":"specification"}]}]}],"generatedAt":"2026-07-21T04:00:31.452Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"ef829299-ba74-4764-8df2-f13f0305d2a1","label":"b","content":"Find a counter example to disprove the statement: $(x + 2)^2$ is always greater than $x^2 + 2$","markscheme":"- Let $x = -2$. M1\n- LHS: $(-2+2)^2 = 0$. A1\n- RHS: $(-2)^2 + 2 = 6$. A1\n- $0$ is not $> 6$.","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Selects a value for $x$ (e.g., $x=-2$).","code":"M"},{"id":"p2","marks":1,"description":"Evaluates both sides and shows the statement is false.","code":"A"}],"finalAnswers":["x = -2"],"generatedAt":"2026-05-12T02:14:06.612Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"counterexample","label":"Choose a counterexample","steps":[{"type":"text","order":0,"title":"Choose a value that makes the bracket zero","content":"Choose $x=-2$, because it makes the bracket $x+2$ equal to zero. This is a valid counterexample value, and writing it down earns the M1 mark.","highlights":[{"text":"Find a counter example to disprove the statement","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$

Algebraic Methods

Question 2

Algebraic Methods Questions

Algebraic Methods

Question 2

Algebraic Methods Questions