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\\sqrt{6})}{\\sqrt{3}}$","markscheme":"- Award method mark for multiplying by $\\frac{\\sqrt{3}}{\\sqrt{3}}$: $\\frac{3\\sqrt{3} + \\sqrt{18}}{3}$ M1\n- Award method mark for simplifying $\\sqrt{18}$ to $3\\sqrt{2}$ M1\n- Final answer $\\sqrt{3} + \\sqrt{2}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies numerator and denominator by $\\sqrt{3}$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies $\\sqrt{18}$ to $3\\sqrt{2}$.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of $\\sqrt{3} + \\sqrt{2}$.","code":"A"}],"finalAnswers":["\\sqrt{3} + \\sqrt{2}"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":59},"walkthroughVideo":null},{"id":"8589d66b-7925-4d40-8356-acde4f97efda","revisionId":"0d32a15e-50b5-40a8-be57-5f67afb3d624","questionNumber":"14","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Surds","calculatorUse":null,"isCalculation":true,"parts":[{"id":"477c71cb-96e8-456c-97f9-f8bc943008b8","label":"0","content":"Simplify fully $\\frac{(4 + 2\\sqrt{3})(4 - 2\\sqrt{3})}{\\sqrt{11}}$\n\nYou must show all your working.","markscheme":"- Award method mark for expanding numerator: $16 - 4(3) = 4$ M1\n- Award method mark for rationalising: $\\frac{4}{\\sqrt{11}} \\times \\frac{\\sqrt{11}}{\\sqrt{11}}$ M1\n- Final answer $\\frac{4\\sqrt{11}}{11}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly expands the numerator to get 4.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies numerator and denominator by $\\sqrt{11}$.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of $\\frac{4\\sqrt{11}}{11}$.","code":"A"}],"finalAnswers":["\\frac{4\\sqrt{11}}{11}"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"difference_of_two_squares","label":"Difference of two squares","steps":[{"type":"text","order":0,"title":"Spot the conjugates","content":"The numerator contains a pair of **conjugates**: the terms are identical except for the signs between them. Use the difference-of-two-squares identity\n\n$$ (a+b)(a-b)=a^2-b^2. $$\n\nHere, $a=4$ and $b=2\\sqrt{3}$.","highlights":[{"text":"You must show all your working.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{\\htmlData{anchor=conjugates}{\\color{@sky}{(4+2\\sqrt3)(4-2\\sqrt3)}}}{\\sqrt{11}}$"},{"latex":"$$\\dfrac{\\htmlData{anchor=squares}{\\color{@sky}{4^2-(2\\sqrt3)^2}}}{\\sqrt{11}}$"},{"latex":"$$\\dfrac{16-4(3)}{\\sqrt{11}}$"},{"latex":"$$\\dfrac{4}{\\sqrt{11}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"conjugates"},{"color":"sky","style":"text-color","anchor":"squares"}],"connections":[{"to":"squares","from":"conjugates","color":"sky","label":"difference of two squares"}]},"order":1,"title":"Expand the numerator","description":"Use the difference of two squares, then square each term. This expansion earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{4}{\\sqrt{11}}\\times \\htmlData{anchor=multiplier}{\\color{@orange}{\\dfrac{\\sqrt{11}}{\\sqrt{11}}}}$"},{"latex":"$$\\htmlData{anchor=final_result}{\\boxed{\\dfrac{4\\sqrt{11}}{11}}}$"}],"highlights":[{"color":"orange","style":"text-color-with-background","anchor":"multiplier"}],"connections":[{"to":"final_result","from":"multiplier","color":"orange","label":"multiply and simplify"}]},"order":3,"title":"Rationalise the denominator","description":"Multiply by √11/√11 = 1 to remove the surd from the denominator. Setting up this multiplication earns M1; the boxed final answer earns A1."}]}],"generatedAt":"2026-07-20T16:49:51.723Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":52},"walkthroughVideo":null}],"subjectCode":"igcse_maths__cie","summaries":[{"id":"c484ee1f-1bf6-47f3-8890-f0e0c9be7909","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"0dacce83-74f3-4f99-8117-8330978282ef","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"72342b57-0e03-4584-aeb4-c241090951c7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"cb761f3d-d598-47d0-8d14-da3783309823","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"8e84081a-aed1-439b-8908-c1900fffcb4a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"8f495886-5503-40d6-8287-a440cec45cef","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"fd3b66f0-2b80-4928-9527-ffa5c326123b","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"96b5c553-923a-4ae0-a9cb-0e38ba01685d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"459ea282-538c-47a6-851e-37e43fa737e7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"58da9761-01de-4848-8c89-1b88aeabaae6","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"c00c601a-faa1-4fe7-90a3-343c6368b2cb","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"a79070ee-d294-4a3d-a2d3-fbcad7907073","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"45b051e4-21f9-4ca0-8299-deb8d6b8daf7","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8589d66b-7925-4d40-8356-acde4f97efda","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"33397bbc-7bcc-4813-8a4b-b51313185314","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a2be29b3-42b1-4a7b-b349-cded36ca8985","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"18eb1e0e-a1ea-4b6b-8395-5fe2b0f1da54","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1f743cac-719e-4a41-b668-629d4a63f8e0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3ef7bcea-af90-4560-9d02-414c974f9ba5","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e56f66f9-08d3-4347-bc34-22336ea478f4","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true}],"title":"Surds","topicId":"b4d9f117-99dd-4545-8c12-15910ffad122","topicSidebar":"$L3b","topicSlug":"grade-7-7-1-surds","topicTitle":"Surds"}]}],"$L3c","$L3d"]}]