des":[{"latex":"$$f(2)=\\htmlData{anchor=substitution}{\\color{@sky}{2^3-4(2)^2+9(2)-10}}$"},{"latex":"$$f(2)=\\htmlData{anchor=evaluated}{\\color{@orange}{8-16+18-10}}$"},{"latex":"$$f(2)=\\htmlData{anchor=zero}{\\color{@green}{0}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substitution"},{"color":"orange","style":"text-color","anchor":"evaluated"},{"color":"green","style":"text-color","anchor":"zero"}],"connections":[{"to":"evaluated","from":"substitution","color":"orange","label":"evaluate each term"}]},"order":1,"title":"Evaluate the function","description":"Substitute 2 into every occurrence of x. This substitution earns the method mark."},{"type":"text","order":2,"title":"Apply the factor theorem","content":"The factor theorem says that if $f(a)=0$, then $(x-a)$ is a factor of $f(x)$. Here, $f(2)=0$, so **$(x-2)$ is a factor of $f(x)$**. Showing $f(2)=0$ and stating this conclusion earns the accuracy mark (A1).","highlights":[{"text":"(x - 2)","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:56:53.158Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"5627ec69-4b05-4a9f-9ae1-25d24a4fc27d","label":"b","content":"Show that $f(x) = 0$ only has one real root.","markscheme":"- Divide $f(x)$ by $(x-2)$ to get $x^2 - 2x + 5$. M1 A1\n- Check discriminant of $x^2 - 2x + 5$: $b^2 - 4ac = (-2)^2 - 4(1)(5) = -16$. M1\n- Since $-16 < 0$, there are no real roots for the quadratic, so $x=2$ is the only real root. A1","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":2,"description":"Finds the quadratic factor $x^2 - 2x + 5$.","code":"M"},{"id":"p2","marks":1,"description":"Calculates the discriminant of the quadratic factor.","code":"M"},{"id":"p3","marks":1,"description":"Concludes there is only one real root because the discriminant is negative.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.612Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"factor_then_discriminant","label":"Factor then use discriminant","steps":[{"type":"text","order":0,"title":"Check the known root","content":"Check whether $x=2$ is a root:\n\n$$f(2)=2^3-4(2)^2+9(2)-10=0.$$\n\nSo $(x-2)$ is a factor of $f(x)$.","highlights":[{"text":"$$f(x) = x^3 - 4x^2 + 9x - 10$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{f(x)}{x-2}=\\htmlData{anchor=quotient}{\\color{@sky}{x^2-2x+5}}"},{"latex":"f(x)=(x-2)\\bigl(\\color{@sky}{x^2-2x+5}\\bigr)"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"quotient"}]},"order":1,"title":"Divide by the factor","description":"Dividing by (x - 2) earns M1, and obtaining the quotient x² - 2x + 5 earns A1. Now test whether this quadratic factor can have real roots."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x^2-2x+5=0"},{"latex":"b^2-4ac=(-2)^2-4(1)(5)=-16"}]},"order":2,"title":"Calculate the discriminant","description":"For ax² + bx + c, the discriminant is b² - 4ac. Here a = 1, b = -2 and c = 5. Substituting these values gives -16, earning M1."},{"type":"text","order":3,"title":"State the conclusion","content":"Since $-16<0$, the quadratic $x^2-2x+5=0$ has no real roots. Therefore the only real root of $f(x)=0$ is $x=2$. This conclusion earns A1."}]}],"generatedAt":"2026-07-21T04:05:08.742Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0f7d9277-8301-4478-b715-a8d241530560","revisionId":"a08a4aee-6da7-4001-8cb6-a9f1655b1684","questionNumber":"23","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"Question Set 7: Algebraic Methods","calculatorUse":null,"isCalculation":true,"parts":[{"id":"6a35f301-0384-415c-b230-8ea798b6d770","label":"a","content":"Express $n^3 - n$ as a product of three factors.","markscheme":"- Factor out $n$: $n(n^2 - 1)$. M1\n- Fac

Algebraic Methods

Question 96

Algebraic Methods Questions

Algebraic Methods

Question 96

Algebraic Methods Questions