$x^2 + 4x - 6 = 0$","markscheme":"- Substitute $k = 4$ into the expression from part (a). M1\n- $x = -\\frac{4}{2} \\pm \\sqrt{\\frac{16}{4} + 6} = -2 \\pm \\sqrt{10}$. A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Substitutes $k=4$ into the formula.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $-2 \\pm \\sqrt{10}$.","code":"A"}],"finalAnswers":["-2 \\pm \\sqrt{10}"],"generatedAt":"2026-05-12T02:14:06.561Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"substitution","label":"Substitute into the formula","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"x=\\frac{-4}{2}\\pm\\sqrt{\\frac{16}{4}+6}"},{"latex":"x=-2\\pm\\sqrt{\\frac{16}{4}+6}"},{"latex":"x=-2\\pm\\sqrt{10}"},{"text":"The final result x = -2 plus or minus square root of 10 gives the two exact roots and earns the accuracy mark (A1)."}]},"order":1,"title":"Substitute and simplify","description":"Substitute k = 4 into the expression from part (a). This substitution earns the method mark (M1)."}]}],"generatedAt":"2026-07-21T04:09:42.038Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":83}},{"id":"04027f4f-6fe0-423e-8a28-e6ae748907bf","revisionId":"fa56a8e2-bb26-4e4f-bfa7-3b0278bcc109","questionNumber":"16","questionType":"LA","specification":"The curve $C$ has the equation $y = \\frac{2}{x} + k \\quad x \\in \\mathbb{R}, x \\neq 0$\n\nThe line $L$ has the equation $y = -3x + 2$","difficulty":2,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Question Set 2: Quadratics","calculatorUse":null,"parts":[{"id":"909bcf64-ad6a-4d26-bcb4-fe190b388120","label":"a","content":"Show that the $x$-coordinate of any point of intersection of $L$ with $C$ is given by a solution of the equation\n\n$$3x^2 + (k - 2)x + 2 = 0$$","markscheme":"- Set equations equal: $\\frac{2}{x} + k = -3x + 2$. M1\n- Multiply by $x$: $2 + kx = -3x^2 + 2x$. M1\n- Rearrange: $3x^2 + kx - 2x + 2 = 0 \\Rightarrow 3x^2 + (k-2)x + 2 = 0$. A1","marks":3,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Equates the two expressions for $y$.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies by $x$ and rearranges to the given form.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.561Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"equate_and_rearrange","label":"Equate and rearrange","steps":[{"type":"text","order":0,"title":"Match the y-values","content":"At an intersection, the point lies on both the curve and the line, so their $y$-values are equal. Write\n\n$$\\frac{2}{x}+k=-3x+2.$$\n\nThis earns the first method mark (M1).","highlights":[{"text":"point of intersection of $L$ with $C$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{2}{\\htmlData{anchor=denominator}{x}}+k=-3x+2"},{"latex":"2+kx=-3x^2+2x"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"denominator"}]},"order":1,"title":"Clear the denominator","description":"Since x ≠ 0, multiplying every term by x is valid. This removes the fraction. M1: this multiplication gives 2 + kx = -3x² + 2x."},{"type":"hyper_latex","block":{"nodes":[{"latex":"2+kx=-3x^2+2x"},{"latex":"3x^2+\\htmlData{anchor=collect_source}{\\color{@green}{kx-2x}}+2=0"},{"latex":"3x^2+\\htmlData{anchor=collect_result}{\\color{@green}{(k-2)x}}+2=0"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"collect_source"},{"color":"green","style":"text-color-with-background","anchor":"collect_result"}]},"order":3,"title":"Rearrange into a quadratic","description":"Add 3x² and subtract 2x on both sides to obtain the expanded quadratic. Then collect the terms containing x. A1: Correctly rearranges and collects to obtain the required quadratic."}]}],"generatedAt":"2026-07-21T04:07:25.726Z","modelVersion":"gpt-5.6-terra","expl

Quadratics

Question 7

Quadratics Questions

Quadratics

Question 7

Quadratics Questions