:"orange","style":"text-color","anchor":"second"}],"connections":[{"to":"scaled","from":"first","color":"sky","label":"multiply by two"}]},"order":1,"title":"Match the y-coefficients","description":"Multiply the first equation by two. Its y-coefficient is then the opposite of the second equation's y-coefficient, so addition removes y."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=eq}{5x+2y=4}"},{"latex":"\\htmlData{anchor=sub}{5(4y-1)+2y=4}"},{"latex":"\\htmlData{anchor=collect}{22y=9}"},{"latex":"\\htmlData{anchor=yval}{y=\\frac{9}{22}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"yval"}],"connections":[{"to":"sub","from":"eq","color":"sky","label":"substitute x=4y-1"},{"to":"collect","from":"sub","color":"orange","label":"expand and collect"},{"to":"yval","from":"collect","color":"green","label":"divide by 22"}]},"order":2,"title":"Substitute into the first line","description":"Replace x in the first equation with 4y minus 1, then collect the y terms. Solving for y earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{x-4y=-1}"},{"latex":"\\htmlData{anchor=substitute}{\\frac{7}{11}-4y=-1}"},{"latex":"\\htmlData{anchor=rearrange}{-4y=-\\frac{18}{11}}"},{"latex":"\\htmlData{anchor=yvalue}{y=\\frac{9}{22}}"}],"highlights":[{"color":"green","style":"text-color","anchor":"yvalue"}],"connections":[{"to":"substitute","from":"original","color":"sky","label":"substitute x value"},{"to":"rearrange","from":"substitute","color":"orange","label":"subtract seven elevenths"},{"to":"yvalue","from":"rearrange","color":"green","label":"divide by negative four"}]},"order":3,"title":"Find y","description":"Substitute the value of x into the simpler original equation to find the remaining coordinate."},{"type":"text","order":4,"title":"State the intersection","content":"The lines intersect at\n\n$$\\left(\\frac{7}{11},\\frac{9}{22}\\right).$$\n\nGiving the coordinates in this order earns the **A1** mark.","highlights":[{"text":"Find the coordinates of the point where $l_1$ and $l_2$ intersect.","location":"specification"}]}]}],"generatedAt":"2026-07-21T04:15:01.122Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":74},"walkthroughVideo":null},{"id":"4cbde072-4b41-4329-b727-53098bbae963","revisionId":"a0f87d53-d3c6-40e9-9366-7aceb4a51310","questionNumber":"7","questionType":"LA","specification":"A curve $C$ has parametric equations\n\n$$x = 2 \\cos t - 1, \\quad y = 4 \\sin^2 t + 3 \\quad 0 \\leq t \\leq 2\\pi$$","difficulty":1,"difficultyBand":"medium","totalMarks":8,"sourceTitle":"Question Set 8: Parametric Equations","calculatorUse":null,"isCalculation":false,"parts":[{"id":"1962a1de-4c5f-4994-a91d-84ca0c0b1d62","label":"a","content":"Find an equation for $C$ in the form $y = f(x)$ and state the domain for which the curve is defined.","markscheme":"- $\\cos t = \\frac{x+1}{2} \\Rightarrow \\cos^2 t = \\frac{(x+1)^2}{4}$ M1\n- $\\sin^2 t = \\frac{y-3}{4}$ M1\n- Use $\\cos^2 t + \\sin^2 t = 1$: $\\frac{(x+1)^2}{4} + \\frac{y-3}{4} = 1 \\Rightarrow (x+1)^2 + y - 3 = 4$ M1\n- $y = 7 - (x+1)^2$ A1\n- Domain: $-1 \\leq \\cos t \\leq 1 \\Rightarrow -2 \\leq 2\\cos t \\leq 2 \\Rightarrow -3 \\leq x \\leq 1$ B1","marks":5,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Uses the identity $\\cos^2 t + \\sin^2 t = 1$ to eliminate the parameter.","code":"M"},{"id":"p2","marks":1,"description":"Correct Cartesian equation $y = 7 - (x+1)^2$.","code":"A"},{"id":"p3","marks":1,"description":"Correct domain $-3 \\leq x \\leq 1$ based on the range of $\\cos t$.","code":"B"}],"finalAnswers":["y = 7 - (x+1)^2"],"generatedAt":"2026-05-06T03:46:57.180Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"pythagorean_identity","label":"Pythagorean Identity Elimination","steps":[{"type":"text","order":0,"title":"Express $\\cos^2

1.6 C: Coordinate geometry in the (x, y) plane

Question 51

1.6 C: Coordinate geometry in the (x, y) plane Questions

1.6 C: Coordinate geometry in the (x, y) plane

Question 51

1.6 C: Coordinate geometry in the (x, y) plane Questions