":"amber","label":"simplify"}]},"order":2,"title":"Simplify the constant","description":"Combine the constant terms. This gives the required completed-square form and earns A1."},{"type":"text","order":3,"title":"State the minimum","content":"For every real $x$, $(x-2)^2\\geq 0$. Therefore\n\n$$\n(x-2)^2+3\\geq 3>0.\n$$\n\nThe minimum value is $3$, attained when $x=2$. Hence $x^2-4x+7$ is positive for all real $x$. This complete conclusion earns the final A1 mark.","highlights":[{"text":"Prove that $x^2 - 4x + 7$ is positive for all values of $x$","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:22:56.657Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":52}},{"id":"4b3cd01e-c540-4bb2-9a33-323bb8589c17","revisionId":"5bfab2a1-a3de-440b-906a-4fb8a884c245","questionNumber":"8","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Question Set 7: Algebraic Methods","calculatorUse":null,"parts":[{"id":"1d6afdde-d3fc-4cbd-adff-1f6bba9752d8","label":"0","content":"Disprove the statement: $n^2 - n + 3$ is a prime number for all values of $n$","markscheme":"- Find a counterexample, e.g., let $n = 10$. M1\n- Evaluate: $10^2 - 10 + 3 = 93$. A1\n- Show $93 = 3 \\times 31$, so it is not prime. A1","marks":3,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies a value of $n$ that results in a non-prime number.","code":"M"},{"id":"p2","marks":1,"description":"Evaluates the expression and shows it is not prime (e.g., $n=10, 93 = 3 \\times 31$).","code":"A"}],"finalAnswers":[],"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":"Choose a counterexample","content":"To disprove a statement claiming something is true for **all** values of $n$, it is enough to find one value for which it fails. Choose $n=10$. This earns the method mark, M1.","highlights":[{"text":"Disprove the statement: $n^2 - n + 3$ is a prime number for all values of $n$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$10^2-10+3=\\htmlData{anchor=result}{93}$"},{"text":"Correct evaluation: A1."}],"highlights":[{"color":"green","style":"text-color","anchor":"result"}]},"order":1,"title":"Substitute n equals 10","description":"Substitute n = 10 and evaluate the expression."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$93=\\htmlData{anchor=factors}{3\\times31}$"},{"text":"93 = 3 × 31, so 93 is not prime: A1."},{"text":"Because n = 10 gives a non-prime value, the statement that the expression is prime for all values of n is disproved."}],"highlights":[{"color":"orange","style":"text-color-with-background","anchor":"factors"}]},"order":2,"title":"Show a factor","description":"A factorisation with factors other than 1 and the number itself shows that the result is not prime."}]}],"generatedAt":"2026-07-21T03:57:14.536Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":67}},{"id":"f28417c1-aa4d-4f6a-9015-8ec9fd291c5d","revisionId":"ed3c1ea2-5b62-4408-a1cf-892b4b29c382","questionNumber":"9","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Question Set 7: Algebraic Methods","calculatorUse":null,"parts":[{"id":"2cf60469-a66c-4b5a-ba2e-cf17800b0306","label":"0","content":"Prove that the sum of two consecutive odd numbers is a multiple of 4","markscheme":"- Define two consecutive odd numbers: $(2n + 1)$ and $(2n + 3)$. M1\n- Sum them: $(2n + 1) + (2n + 3) = 4n + 4$. M1\n- Factorise: $4(n + 1)$, which is a multiple of 4. A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly defines two consecutive

Algebraic Methods

Question 8

Algebraic Methods Questions

Algebraic Methods

Question 8

Algebraic Methods Questions