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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},{"id":"5aa9fb76-4e85-46d4-9eb8-828c02ccc7a6","revisionId":"330a0c13-16e4-4053-83f7-00804eda4489","questionNumber":"5","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"112dbbda-65ac-43f1-b1a4-ca72d4114f74","label":"0","content":"Write down the equation of the line perpendicular to $y = \\frac{1}{2}x - 4$ which passes through $(0,7)$","markscheme":"- Identifies perpendicular gradient $m = -2$ M1\n- Correct equation $y = -2x + 7$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies the perpendicular gradient as $-2$.","code":"M"},{"id":"p2","marks":1,"description":"Correct final equation $y = -2x + 7$.","code":"A"}],"finalAnswers":["y=-2x+7"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"gradient_intercept","label":"Perpendicular gradient and intercept","steps":[{"type":"text","order":0,"title":"Read the gradient","content":"A straight line written as $y=mx+c$ has gradient $m$. In $y=\\frac12x-4$, the gradient is therefore $\\frac12$.","highlights":[{"text":"$$y = \\frac{1}{2}x - 4$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$m_1=\\htmlData{anchor=original_gradient}{\\color{@sky}{\\frac{1}{2}}}$"},{"latex":"$$m_2=\\htmlData{anchor=perpendicular_gradient}{\\color{@green}{-2}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"original_gradient"},{"color":"green","style":"text-color-with-background","anchor":"perpendicular_gradient"}],"connections":[{"to":"perpendicular_gradient","from":"original_gradient","color":"green","label":"negative reciprocal"}]},"order":1,"title":"Find the perpendicular gradient","description":"Perpendicular gradients multiply to -1. Identifying the new gradient as -2 earns M1."},{"type":"text","order":2,"title":"Use the given point","content":"The point $(0,7)$ lies on the $y$-axis because its $x$-coordinate is $0$. Therefore, the line's $y$-intercept is $7$.","highlights":[{"text":"passes through $(0,7)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"Write the line in its simplest form."},{"latex":"$$\\htmlData{anchor=answer}{y=-2x+7}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":3,"title":"Write the equation","description":"Substituting m = -2 and c = 7 gives the correct equation and earns A1."}]}],"generatedAt":"2026-07-20T16:35:01.541Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":85},"walkthroughVideo":null},{"id":"9b5feb40-0aa0-44e4-8ce6-453faebd020a","revisionId":"66fe164d-9e78-4255-a782-dc48978ca2bd","questionNumber":"6","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Parallel and Perpendicular Lines","calculatorUse":null,"isCalculation":true,"parts":[{"id":"9d997246-9ca0-4bae-93c0-7486aee026db","label":"0","content":"Write down the equation of the line perpendicular to $y = -\\frac{3}{2}x - 1$ which passes through $(0,-8)$","markscheme":"- Identifies perpendicular gradient $m = \\frac{2}{3}$ M1\n- Correct equation $y = \\frac{2}{3}x - 8$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies the perpendicular gradient as $\\frac{2}{3}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct final equation $y = \\frac{2}{3}x - 8$.","code":"A"}],"finalAnswers":["y=2/3x-8"],"generatedAt":"2026-05-07T02:03:01.197Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"gradient_and_intercept","label":"Perpendicular gradient and intercept","steps":[{"type":"text","order":0,"title":"Find the new gradient","content":"The given line has gradient $-\\frac{3}{2}$. Perpendicular gradients have product $-1$, so take the **negative reciprocal**: turn the fraction upside down and change its sign.\n\nTherefore the perpendicular gradient is $m=\\frac{2}{3}$. Identifying this earns **M1**.","highlights":[{"text":"perpendicular to $y = -\\frac{3}{2}x - 1$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@orange}{\\htmlData{anchor=y_value}{y}}=\\frac{2}{3}\\color{@green}{\\htmlData{anchor=x_value}{x}}+c$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=minus_eight}{-8}}=\\frac{2}{3}\\color{@green}{\\htmlData{anchor=zero}{(0)}}+c$"},{"latex":"$$\\color{@purple}{\\htmlData{anchor=intercept}{c=-8}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"y_value"},{"color":"orange","style":"text-color","anchor":"minus_eight"},{"color":"green","style":"text-color","anchor":"x_value"},{"color":"green","style":"text-color","anchor":"zero"}]},"order":1,"title":"Use the given point","description":"Use the gradient-intercept form. At the point (0, -8), substitute x = 0 and y = -8 to find the intercept."},{"type":"text","order":2,"title":"State the equation","content":"Insert $m=\\frac{2}{3}$ and $c=-8$ into $y=mx+c$:\n\n$$\\boxed{y=\\frac{2}{3}x-8}$$\n\nThis correct equation earns **A1**. Notice that the $-1$ from the original line is not used: the required line has its own intercept because it passes through $(0,-8)$.","highlights":[{"text":"passes through $(0,-8)$","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:38:02.409Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":94},"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"]}]