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M1\n- Simplify to obtain $3x^2-20x+12=0$. A1\n- Factorise and solve to obtain $x=\\frac{2}{3}$ or $x=6$. M1\n- Find the corresponding values of $y$, giving $(x,y)=\\left(\\frac{2}{3},\\frac{7}{3}\\right)$ or $(6,-3)$. A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Substitutes $y=3-x$ into the quadratic equation.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to the quadratic equation $3x^2-20x+12=0$.","code":"A"},{"id":"p3","marks":1,"description":"Solves the quadratic to obtain $x=\\frac{2}{3}$ and $x=6$.","code":"M"},{"id":"p4","marks":1,"description":"Gives both solutions as $\\left(\\frac{2}{3},\\frac{7}{3}\\right)$ and $(6,-3)$.","code":"A"}],"finalAnswers":["$$\\left(\\frac{2}{3},\\frac{7}{3}\\right)$","$$(6,-3)$"],"generatedAt":"2026-08-31T09:25:35.304Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"a85b0d3d-1b96-4231-8457-8b7746fe4e7c","revisionId":"92ac70d1-7433-4cef-a5a0-a9e8052bb2d7","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":null,"totalMarks":3,"sourceTitle":"3.5 Quadratic Inequalities","calculatorUse":null,"isCalculation":true,"parts":[{"id":"948531b6-e375-4319-ae8a-77f75ee7c444","label":"","content":"Solve the inequality\n\n$$x(x + 1) \\leq 12$$","markscheme":"- Expand and rearrange to obtain $x^2+x-12\\leq0$. M1\n- Factorise as $(x+4)(x-3)\\leq0$, identifying the critical values $-4$ and $3$. M1\n- State $-4\\leq x\\leq3$. A1\n\n![Graph of the inequality (x+4)(x-3) <= 0.](https://assets.mathsgenie.co.uk/booklet/2a80a074a7dbc4e4210c4b1275842a7dc2d802e93f1b2a08d99e10569fd9a429.jpg)","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Expands and rearranges the inequality to $x^2+x-12\\leq0$.","code":"M"},{"id":"p2","marks":1,"description":"Factorises the quadratic as $(x+4)(x-3)$ and identifies the critical values $-4$ and $3$.","code":"M"},{"id":"p3","marks":1,"description":"States the solution $-4\\leq x\\leq3$.","code":"A","acceptable":["$$x\\in[-4,3]$"]}],"finalAnswers":["$$-4 \\leq x \\leq 3$"],"generatedAt":"2026-08-31T09:25:35.304Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"88b23b6f-8a71-4cdf-94bc-261c4d958f1c","revisionId":"0a4e39a8-3387-4b96-8409-300859c8f95f","questionNumber":"5","questionType":"LA","specification":"","difficulty":2,"difficultyBand":null,"totalMarks":4,"sourceTitle":"3.5 Quadratic Inequalities","calculatorUse":null,"isCalculation":true,"parts":[{"id":"dca4eea5-9d78-4049-a095-6bd9f262366d","label":"","content":"Find the coordinates of the points where the circle C with equation $x^2+y^2-2x=19$ meets the line L with equation $y=3x-1$.","markscheme":"- Substitute $y=3x-1$ into the equation of the circle. M1\n- Simplify to obtain $5x^2-4x-9=0$. A1\n- Factorise as $(5x-9)(x+1)=0$ to obtain $x=\\frac{9}{5}$ or $x=-1$. M1\n- Find the corresponding $y$-coordinates and state $\\left(\\frac{9}{5},\\frac{22}{5}\\right)$ and $(-1,-4)$. A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Substitutes $y=3x-1$ into the circle equation.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to the quadratic equation $5x^2-4x-9=0$.","code":"A","acceptable":["$$10x^2-8x-18=0$"]},{"id":"p3","marks":1,"description":"Solves the quadratic to obtain $x=\\frac{9}{5}$ and $x=-1$.","code":"M"},{"id":"p4","marks":1,"description":"States both intersection points as $\\left(\\frac{9}{5},\\frac{22}{5}\\right)$ and $(-1,-4)$.","code":"A"}],"finalAnswers":["$$\\left(\\frac{9}{5},\\frac{22}{5}\\right)$","$$(-1,-4)$"],"generatedAt":"2026-08-31T09:25:35.304Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"as_level_maths","summaries":[{"id":"3ae08f67-616b-46a1-b1d0-5ea73b8544f3","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c150f986-3cc3-4a24-8ac0-8c34c7c47c05","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f23dcedf-2a4c-4de0-847d-e018d9bab020","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a85b0d3d-1b96-4231-8457-8b7746fe4e7c","difficultyBand":null,"calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"88b23b6f-8a71-4cdf-94bc-261c4d958f1c","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0255da01-e68c-4e51-b514-545fa8f8f9ce","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"26cac68e-94a4-4d2d-8895-1baed6671fea","difficultyBand":null,"calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"be9975e4-05a7-44bf-97cf-d78a6b2efa9d","difficultyBand":null,"calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true}],"title":"3.5 Quadratic Inequalities","topicId":"02913333-61fa-49da-a356-ee3dfb3af099","topicSidebar":"$L3c","topicSlug":"as-pure-mathematics-3-5-quadratic-inequalities","topicTitle":"3.5 Quadratic Inequalities"}]}],"$L3d","$L3e"]}]