ct to $2$ decimal places, so the rounding interval has half-width $0.005$ cm. The **upper bound** is therefore $4.28+0.005=4.285$ cm. This identifies the upper-bound radius, earning the method mark (M1).","highlights":[{"text":"radius of $4.28$ cm, correct to 2 decimal places","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=formula}{A=\\pi r^2}$"},{"latex":"$$\\htmlData{anchor=substitution}{A=\\pi\\times4.285^2}$"},{"latex":"$$\\htmlData{anchor=calculator}{A=57.68\\ldots\\ \\mathrm{cm^2}}$"},{"latex":"$$\\htmlData{anchor=answer}{A=57.7\\ \\mathrm{cm^2}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"formula"},{"color":"orange","style":"text-color","anchor":"substitution"},{"color":"green","style":"text-color","anchor":"calculator"},{"color":"purple","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"substitution","from":"formula","color":"sky","label":"substitute"},{"to":"calculator","from":"substitution","color":"green","label":"calculate"},{"to":"answer","from":"calculator","color":"purple","label":"3 s.f."}]},"order":1,"title":"Use the circle formula","description":"Substitute the upper-bound radius into the area formula. The calculator gives 57.68… cm². The fourth significant figure is 8, so this rounds to 57.7 cm². The correct calculation and rounded answer earn A1."}]}],"generatedAt":"2026-07-20T11:03:58.814Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"eb90b852-b185-4155-a150-4607aed2e09e","revisionId":"f5d53f60-3fd5-4a87-bc44-4cb3bd506a67","questionNumber":null,"questionType":"LA","specification":"$$$v^2 = u^2 + 2as$$\n\n$v = 28.4$ correct to 1 decimal place\n\n$a = 5.2$ correct to 1 decimal place\n\n$s = 70.45$ correct to 2 decimal places","difficulty":5,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":"calc","parts":[{"id":"7a424248-c2e8-42f0-bb08-be48b9f1a132","label":"a","content":"Work out the upper bound for $u$.\n\nGive your answer to 3 significant figures.","markscheme":"- Rearranges formula to $u = \\sqrt{v^2 - 2as}$ M1\n- Identifies upper bound for $v$ ($28.45$) B1\n- Identifies lower bounds for $a$ ($5.15$) and $s$ ($70.445$) B1\n- Substitutes into formula: $\\sqrt{28.45^2 - 2(5.15)(70.445)}$ M1\n- Calculates $9.1552...$ and rounds to $9.16$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Rearranges the formula to make u the subject.","code":"M"},{"id":"p2","marks":1,"description":"Correctly identifies the upper bound for v as 28.45.","code":"B"},{"id":"p3","marks":1,"description":"Correctly identifies the lower bounds for a (5.15) and s (70.445).","code":"B"},{"id":"p4","marks":1,"description":"Substitutes the correct bounds into the rearranged formula to find the upper bound of u.","code":"M"},{"id":"p5","marks":1,"description":"Calculates the value of u and rounds correctly to 3 significant figures.","code":"A"}],"finalAnswers":["9.16"],"generatedAt":"2026-05-06T14:01:48.412Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"bounds_and_rearrangement","label":"Bounds and rearrangement","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$v^2=u^2+\\htmlData{anchor=term}{\\color{@sky}{2as}}$"},{"latex":"$$\\htmlData{anchor=rearranged}{\\color{@sky}{u^2=v^2-2as}}$"},{"latex":"$$\\htmlData{anchor=root}{\\color{@orange}{u=\\sqrt{v^2-2as}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"term"},{"color":"orange","style":"text-color-with-background","anchor":"root"}],"connections":[{"to":"rearranged","from":"term","color":"sky","label":"subtract from both sides"},{"to":"root","from":"rearranged","color":"orange","label":"take the positive root"}]},"order":0,"title":"Make u the subject","description":"First subtract $2as$ from both sides. Taking the positive square root then gives the positive-root value of $u$, a

Bounds

Question 12

Bounds Questions

Bounds

Question 12

Bounds Questions