umber":"14","questionType":"LA","specification":"The curve with equation $y = px^2 - 4px - 5p$, where $p$ is a constant does not intersect the line with equation $y = 2x - 12$.","difficulty":1,"difficultyBand":"medium","totalMarks":8,"sourceTitle":"Question Set 3: Equations and Inequalities","calculatorUse":null,"parts":[{"id":"22228924-40ff-4ba5-b863-60788262b597","label":"a","content":"Show that $9p^2 - 8p + 1 < 0$","markscheme":"$$px^2 - 4px - 5p = 2x - 12$\n$px^2 + (-4p - 2)x + (12 - 5p) = 0$\nFor no intersection, $b^2 - 4ac < 0$\n$(-4p - 2)^2 - 4(p)(12 - 5p) < 0$\n$16p^2 + 16p + 4 - 48p + 20p^2 < 0$\n$36p^2 - 32p + 4 < 0$\n$9p^2 - 8p + 1 < 0$ 4","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Equates curve and line and rearranges to $ax^2 + bx + c = 0$.","code":"M"},{"id":"p2","marks":1,"description":"States or uses $b^2 - 4ac < 0$ for no intersection.","code":"M"},{"id":"p3","marks":1,"description":"Correct expansion of $(-4p - 2)^2$.","code":"A"},{"id":"p4","marks":1,"description":"Simplifies to the required inequality $9p^2 - 8p + 1 < 0$.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.552Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"discriminant","label":"Use the discriminant","steps":[{"type":"text","order":0,"title":"Set the expressions equal","content":"At an intersection, both equations have the same value of $y$, so set the two expressions for $y$ equal.","highlights":[{"text":"does not intersect the line with equation $y = 2x - 12$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"p x^2-4p x-5p=2x-12"},{"latex":"p x^2+\\htmlData{anchor=coefficient}{\\color{@sky}{(-4p-2)}}x+(12-5p)=0"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"coefficient"}]},"order":1,"title":"Form a quadratic","description":"Equating the two expressions for y earns the first mark. Rearranging into this quadratic form earns the second mark."},{"type":"text","order":2,"title":"Check p is non-zero","content":"First, $p$ cannot be $0$: then the curve is $y=0$, which meets $y=2x-12$ at $x=6$. Hence $p\\ne0$, so the equation is genuinely quadratic in $x$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=condition}{b^2-4ac<0}"},{"latex":"\\htmlData{anchor=substituted}{\\left(-4p-2\\right)^2-4\\left(p\\right)\\left(12-5p\\right)<0}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"condition"}]},"order":3,"title":"Apply the discriminant","description":"No intersections mean no real roots, so use the negative-discriminant condition. This earns the third mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\left(-4p-2\\right)^2-4(p)(12-5p)<0"},{"latex":"16p^2+16p+4-48p+20p^2<0"},{"latex":"\\htmlData{anchor=collected}{36p^2-32p+4<0}"},{"latex":"\\color{@green}{\\htmlData{anchor=final}{9p^2-8p+1<0}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"final"}],"connections":[{"to":"final","from":"collected","color":"green","label":"divide by 4","style":"solid"}]},"order":4,"title":"Expand and simplify","description":"Expand the square and brackets, then collect like terms. Dividing by 4 gives the required inequality. This earns the fourth mark."}]}],"generatedAt":"2026-07-21T04:06:10.893Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"9bf9634c-1a4d-4c72-b1ab-20ada6a542ce","label":"b","content":"Find the set of possible values for $p$.","markscheme":"Solve $9p^2 - 8p + 1 = 0$\n$p = \\frac{8 \\pm \\sqrt{64 - 36}}{18} = \\frac{8 \\pm \\sqrt{28}}{18} = \\frac{8 \\pm 2\\sqrt{7}}{18} = \\frac{4 \\pm \\sqrt{7}}{9}$\n$\\frac{4 - \\sqrt{7}}{9} < p < \\frac{4 + \\sqrt{7}}{9}$ 4","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Uses quadratic formula to find critical values of $p$.","code":"M"},{"id":"p2","marks":1,"description":"Finds critical values $\\frac{4 \\pm \\sqrt{7}}{9}$.","code":"A"},{"id":"p3","marks":1,"descri

Equations and Inequalities

Question 5

Equations and Inequalities Questions

Equations and Inequalities

Question 5

Equations and Inequalities Questions