ad-a1798c654dcb","revisionId":"37480cff-de70-4d53-9eff-6bb5cd5904bb","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"f5cb570d-ced4-4f52-913c-4bca2a7b571d","label":"a","content":"Write $3\\sqrt{48}$ in the form $k\\sqrt{3}$, where $k$ is an integer.","markscheme":"- Simplifies $\\sqrt{48}$ to $4\\sqrt{3}$: $3(4\\sqrt{3})$ M1\n- Simplifies to $12\\sqrt{3}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Simplifies the surd part by identifying a square factor, e.g., showing $3(4\\sqrt{3})$ or equivalent.","code":"M"},{"id":"p2","marks":1,"description":"Correctly evaluates the expression to $12\\sqrt{3}$.","code":"A"}],"finalAnswers":["12\\sqrt{3}"],"generatedAt":"2026-05-06T14:01:54.200Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"extract_square_factor","label":"Extract the square factor","steps":[{"type":"text","order":0,"title":"Find a square factor","content":"Look for the largest square number that is a factor of $48$. Since $48=16\\times3$, use $16$; its square root is an integer.","highlights":[{"text":"Write $3\\sqrt{48}$ in the form $k\\sqrt{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3\\sqrt{48}=3\\sqrt{\\htmlData{anchor=product}{\\color{@sky}{16\\times3}}}$"},{"latex":"$$3\\sqrt{48}=3\\left(\\htmlData{anchor=split}{\\color{@sky}{\\sqrt{16}\\sqrt{3}}}\\right)$"},{"latex":"$$3\\sqrt{48}=3\\left(\\htmlData{anchor=evaluated}{\\color{@orange}{4}}\\sqrt{3}\\right)$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"product"},{"color":"sky","style":"text-color-with-background","anchor":"split"},{"color":"orange","style":"text-color-with-background","anchor":"evaluated"}],"connections":[{"to":"split","from":"product","color":"sky","label":"split the root"}]},"order":1,"title":"Simplify the surd","description":"Split the square root across the product, then evaluate the square number. Reaching the final line earns the method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3\\left(\\htmlData{anchor=product}{4\\sqrt3}\\right)$"},{"latex":"$$\\htmlData{anchor=answer}{12\\sqrt3}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"product"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"product","color":"green","label":"multiply 3 by 4"}]},"order":2,"title":"Multiply the coefficient","description":"Multiply the numbers outside the square root. This exact simplification earns the accuracy mark (A1)."}]}],"generatedAt":"2026-07-19T18:53:44.979Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":67,"sampleSize":39}},{"id":"f25755cf-8c37-410f-8c1d-f574f1414673","revisionId":"bf98f6cd-667e-4aec-ba85-f1064ce0a0a7","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"c651ce46-1b3f-4afd-8829-293a2ac1d7cd","label":"a","content":"Show that $\\frac{5 + 4\\sqrt{2}}{\\sqrt{2} + 1}$ can be written as $3 + \\sqrt{2}$","markscheme":"- Multiplies by conjugate: $\\frac{(5 + 4\\sqrt{2})(\\sqrt{2} - 1)}{(\\sqrt{2} + 1)(\\sqrt{2} - 1)}$ M1\n- Expands numerator: $5\\sqrt{2} - 5 + 8 - 4\\sqrt{2} = 3 + \\sqrt{2}$ M1\n- Shows denominator is $2 - 1 = 1$ and concludes $3 + \\sqrt{2}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies numerator and denominator by the conjugate $(\\sqrt{2} - 1)$","code":"M"},{"id":"p2","marks":1,"description":"Correct expansion of the numerator into at least four terms before simplifying to $3 + \\sqrt{2}$","code":"M"},{"id":"p3","marks":1,"description":"Evaluates the denominator to 1 and completes the pr

Surds Questions

Surds Questions

Surds Questions

Surds Questions