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From $y=\\frac12x+5$, the gradient is $m=\\frac12$. This identification earns **M1**.","highlights":[{"text":"parallel to $y = \\frac{1}{2}x + 5$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=general}{y=\\frac{1}{2}x+c}"},{"latex":"\\htmlData{anchor=substituted}{2=\\frac{1}{2}(0)+c}"},{"latex":"\\htmlData{anchor=intercept}{c=2}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substituted"}],"connections":[{"to":"substituted","from":"general","color":"sky","label":"substitute"},{"to":"intercept","from":"substituted","color":"orange","label":"simplify","style":"dashed"}]},"order":1,"title":"Use the given point","description":"For the point (0, 2), use x = 0 and y = 2. Substitute these into the line equation to find c."},{"type":"text","order":2,"title":"State the equation","content":"Using the intercept found, the equation is $$y=\\frac{1}{2}x+2.$$ This correct equation earns **A1**."}]}],"generatedAt":"2026-07-21T02:57:40.031Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":86},"walkthroughVideo":null},{"id":"88754053-ce58-485c-a539-99de2a7000a2","revisionId":"1cde5455-a532-483f-bea2-b2d518dd8465","questionNumber":"3","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"b7e386fc-bf37-49c6-b506-df31c69380a2","label":"0","content":"Write down the equation of the line parallel to $y = -x + 1$ which passes through $(0,-4)$","markscheme":"- Identifies gradient $m = -1$ M1\n- Correct equation $y = -x - 4$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies that the gradient must be $-1$.","code":"M"},{"id":"p2","marks":1,"description":"Correct final equation $y = -x - 4$.","code":"A"}],"finalAnswers":["y=-x-4"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"gradient_intercept","label":"Use gradient and intercept","steps":[{"type":"text","order":0,"title":"Keep the gradient","content":"Parallel lines have the same gradient (the coefficient of $x$). In $y=-x+1$, the coefficient of $x$ is $-1$, so the required line has gradient $m=-1$. This earns the **M1** mark.","highlights":[{"text":"parallel to $y = -x + 1$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = \\htmlData{anchor=mx_general}{\\color{@sky}{mx}} + \\htmlData{anchor=c_general}{\\color{@amber}{c}}$"},{"latex":"$$-4 = \\htmlData{anchor=mx_sub}{\\color{@sky}{(-1)(0)}} + \\htmlData{anchor=c_sub}{\\color{@amber}{c}}$"},{"latex":"$$\\htmlData{anchor=c_value}{\\color{@amber}{c = -4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"mx_general"},{"color":"amber","style":"text-color","anchor":"c_general"},{"color":"sky","style":"text-color","anchor":"mx_sub"},{"color":"amber","style":"text-color","anchor":"c_sub"},{"color":"amber","style":"text-color","anchor":"c_value"}]},"order":1,"title":"Use the point","description":"Use y = mx + c and substitute m = -1 and (x, y) = (0, -4) to find c."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = \\htmlData{anchor=substituted}{(-1)x + (-4)}$"},{"latex":"$$\\htmlData{anchor=final_equation}{\\color{@green}{y = -x - 4}}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"final_equation"}]},"order":2,"title":"Write the equation","description":"Substitute m = -1 and c = -4 into y = mx + c. Its y-intercept is -4, so it passes through (0, -4). This correct equation earns the A1 mark."}]}],"generatedAt":"2026-07-20T16:31:45.790Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":89},"walkthroughVideo":null},{"id":"99e9a58e-58ce-4c5b-bcf8-e1fc35241405","revisionId":"7f6f6fed-115c-4152-ba4d-4c8bf2bc47d5","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":1,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d255ddef-a5a4-4b8c-950a-a80e2e981e3b","label":"0","content":"Write down the equation of a line perpendicular to $y = 3x + 3$","markscheme":"- Identifies perpendicular gradient $m = -\\frac{1}{3}$ B1\n- Example: $y = -\\frac{1}{3}x$","marks":1,"order":0,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States any equation with a gradient of $-\\frac{1}{3}$.","code":"B"}],"finalAnswers":["y=-1/3x"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"negative_reciprocal","label":"Use the perpendicular gradient","steps":[{"type":"text","order":0,"title":"Read the gradient","content":"A straight line in the form $y=mx+c$ has gradient $m$. Therefore, the gradient of the given line is $3$.","highlights":[{"text":"$$y = 3x + 3$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$m=\\htmlData{anchor=original_gradient}{\\color{@orange}{\\frac{3}{1}}}$"},{"latex":"$$\\htmlData{anchor=operation}{\\color{@green}{\\boxed{\\text{take the negative reciprocal}}}}$"},{"latex":"$$\\htmlData{anchor=result_gradient}{m_{\\perp}=\\mathord{-}\\!\\frac{1}{3}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"original_gradient"},{"color":"green","style":"text-color-with-background","anchor":"operation"},{"color":"green","style":"text-color","anchor":"result_gradient"}],"connections":[{"to":"operation","from":"original_gradient","color":"orange","style":"solid"},{"to":"result_gradient","from":"operation","color":"green","style":"solid"}]},"order":1,"title":"Find the new gradient","description":"Perpendicular gradients are negative reciprocals: flip the fraction and change its sign. Identifying this gradient earns the B1 mark."},{"type":"text","order":2,"title":"Write a valid equation","content":"No point or intercept is specified, so any line with gradient $-\\dfrac{1}{3}$ is valid. The simplest choice is\n\n$$y=-\\dfrac{1}{3}x.$$","highlights":[{"text":"Write down the equation of a line perpendicular","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:32:30.471Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":86},"walkthroughVideo":null}],"subjectCode":"igcse_maths","summaries":[{"id":"acbe8ebb-46d9-478e-8f4d-932b2885f04c","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"00df6036-9aab-4beb-a1f9-478482a7e43b","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"88754053-ce58-485c-a539-99de2a7000a2","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"99e9a58e-58ce-4c5b-bcf8-e1fc35241405","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"5aa9fb76-4e85-46d4-9eb8-828c02ccc7a6","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9b5feb40-0aa0-44e4-8ce6-453faebd020a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e0cb84a5-5545-4566-9b6a-7e816ee630e0","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"d258283c-865e-47c0-89da-0e5a1bcf9a81","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"91efa1cb-1437-4764-928f-184d64d102eb","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"6277a69d-6c9b-4e4d-a024-c4c43440a622","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"e3704fe4-d3b3-4d05-a5be-25f12b806bf9","difficultyBand":"medium","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"3e371dc2-7407-45d1-80db-f83b8384f7ae","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"279118c0-d7d5-42b6-8a6a-0acb5f408d8f","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c5bf0b85-9272-4d46-9a99-b4887b1793ec","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4bb0fdaa-3145-4c69-b31a-7058b11ddbd8","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a8a661f3-569f-4c77-a493-be9716343848","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c22eabf2-eada-4b44-bdb5-ceda01fdd864","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f5aac4ae-4ceb-4ac8-9cdc-ca59354e843c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2db8e411-8c21-448e-ae58-f85a6e6f3962","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Parallel and Perpendicular Lines","topicId":"f80ee1fd-8ae1-4f86-a4e8-a8296113fb74","topicSidebar":"$L3b","topicSlug":"grade-6-6-12-parallel-and-perpendicular-lines","topicTitle":"Parallel and Perpendicular Lines"}]}],"$L3c","$L3d"]}]