d32698a-f993-41c2-a82b-7ca0ba7c9222","label":"a","content":"Expand and simplify $(3 + \\sqrt{5})^2 - (3 - \\sqrt{5})^2$","markscheme":"- Expands both squared terms: $(14 + 6\\sqrt{5}) - (14 - 6\\sqrt{5})$ M1\n- Simplifies to $12\\sqrt{5}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Attempt to expand both binomial squared terms correctly, showing at least one expansion like $14 + 6\\sqrt{5}$ or $14 - 6\\sqrt{5}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct simplification to the final answer of $12\\sqrt{5}$.","code":"A"}],"finalAnswers":["12\\sqrt{5}"],"generatedAt":"2026-05-06T14:02:09.907Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"expand_each_bracket","label":"Expand each bracket","steps":[{"type":"text","order":0,"title":"Recall the square rule","content":"Use\n$$\n(a+b)^2=a^2+2ab+b^2\n$$\nand\n$$\n(a-b)^2=a^2-2ab+b^2.\n$$\nRemember that squaring a bracket creates the middle term $2ab$; it is not just $a^2+b^2$.","highlights":[{"text":"Expand and simplify $(3 + \\sqrt{5})^2 - (3 - \\sqrt{5})^2$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(3+\\sqrt5)^2-(3-\\sqrt5)^2$"},{"latex":"$$\\htmlData{anchor=expanded}{\\left(3^2+2(3)(\\sqrt5)+(\\sqrt5)^2\\right)-\\left(3^2-2(3)(\\sqrt5)+(\\sqrt5)^2\\right)}$"},{"latex":"$$\\htmlData{anchor=simplified}{(14+6\\sqrt5)-(14-6\\sqrt5)}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"expanded"},{"color":"green","style":"text-color-with-background","anchor":"simplified"}],"connections":[{"to":"simplified","from":"expanded","color":"sky","label":"evaluate squares"}]},"order":1,"title":"Expand both squares","description":"Apply the square rule to each bracket. Reaching the final line in this step earns the method mark M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(14+6\\sqrt{5})-\\color{@red}{\\htmlData{anchor=second_bracket}{(14-6\\sqrt{5})}}$"},{"latex":"$$14+6\\sqrt{5}\\color{@red}{\\htmlData{anchor=changed_signs}{-14+6\\sqrt{5}}}$"}],"highlights":[{"color":"red","style":"text-color-with-background","anchor":"second_bracket"},{"color":"red","style":"text-color-with-background","anchor":"changed_signs"}],"connections":[{"to":"changed_signs","from":"second_bracket","color":"red","style":"solid"}]},"order":2,"title":"Remove the brackets","description":"The subtraction sign changes both signs in the second bracket. This is the key sign to handle carefully."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$14-14+\\htmlData{anchor=surds}{\\color{@green}{6\\sqrt5+6\\sqrt5}}$"},{"latex":"$$\\boxed{\\htmlData{anchor=result}{\\color{@green}{12\\sqrt5}}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"surds"},{"color":"green","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"result","from":"surds","color":"green","label":"add coefficients"}]},"order":3,"title":"Collect like terms","description":"The ordinary numbers cancel, then the two like surd terms combine. This correct simplification earns A1."}]}],"generatedAt":"2026-07-19T07:49:04.734Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":37,"sampleSize":156}},{"id":"6e4c04aa-9ba8-4c8c-b824-e3368cb54219","revisionId":"c4dddded-e2e8-4642-9a73-2cdcaa52a357","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"d26aa31f-7f1c-4cc3-b707-7a371ecce927","label":"a","content":"Show that $\\frac{4}{\\frac{1}{\\sqrt{5}} + 1}$ can be written as $5 - \\sqrt{5}$","markscheme":"- Simplifies denominator: $\\frac{1 + \\sqrt{5}}{\\sqrt{5}}$ M1\n- Rewrites expression: $4 \\times \\frac{\\sqrt{5}}{1 + \\sqrt{5}} = \\frac{4\\sqrt{5}}{1 + \\sqrt{5}}$ M1\n- Rationalises: $\\frac{4\\sqrt{5}(1 - \\sqrt{5})}{1 - 5} = \\frac{4\\sqrt{5} - 20}{-4} = 5 - \\sqrt{5}$ A1","ma

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