{1}{x-2} - \\frac{2}{(x-2)^2}$$"}]}],"generatedAt":"2026-06-09T04:56:15.731Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"acf362fa-77c0-4622-bfbb-e81ffd30f439","revisionId":"10b0d85f-46d7-4fc5-ae11-b32b004f1755","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"ea8c203a-4740-4713-aa39-c3913a1e72f5","label":"a","content":"Express $\\frac{4x^2 + 13x + 8}{(x-2)(x+3)^2}$ in partial fractions","markscheme":"- Sets up the identity $4x^2 + 13x + 8 = A(x+3)^2 + B(x-2)(x+3) + C(x-2)$ M1\n- Finds $A = 2$ and $C = -1$ A1\n- Finds $B = 2$ A1\n- Final answer: $\\frac{2}{x-2} + \\frac{2}{x+3} - \\frac{1}{(x+3)^2}$ A1\n- Correct use of partial fraction form A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Sets up a correct identity with three constants, accommodating the repeated factor.","code":"M"},{"id":"p2","marks":1,"description":"Correctly determines two constants (e.g., A and C) using substitution.","code":"A"},{"id":"p3","marks":1,"description":"Correctly determines the third constant (B).","code":"A"},{"id":"p4","marks":1,"description":"Provides the final expression using the calculated constants correctly.","code":"A"},{"id":"p5","marks":1,"description":"Consistent use of the partial fraction form throughout the working.","code":"A"}],"finalAnswers":["\\frac{2}{x-2} + \\frac{2}{x+3} - \\frac{1}{(x+3)^2}"],"generatedAt":"2026-05-06T14:03:45.563Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"substitution","label":"Substitution Method","steps":[{"type":"text","order":0,"title":"Set up the identity","content":"To express the algebraic fraction in partial fractions, we first write down the correct form. Since we have a repeated linear factor $(x+3)^2$, the decomposition takes the form:\n\n$$\\frac{4x^2 + 13x + 8}{(x-2)(x+3)^2} \\equiv \\frac{A}{x-2} + \\frac{B}{x+3} + \\frac{C}{(x+3)^2}$$\n\nMultiplying both sides by the denominator $(x-2)(x+3)^2$ gives us the identity:\n\n$$4x^2 + 13x + 8 \\equiv A(x+3)^2 + B(x-2)(x+3) + C(x-2)$$\n\nThis is worth our first method mark **(M1)**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4\\htmlData{anchor=x-val1}{\\color{@sky}{x}}^2 + 13\\htmlData{anchor=x-val1-2}{\\color{@sky}{x}} + 8 \\equiv A(\\htmlData{anchor=x-val1-3}{\\color{@sky}{x}}+3)^2 + B(\\htmlData{anchor=x-val1-4}{\\color{@sky}{x}}-2)(\\htmlData{anchor=x-val1-5}{\\color{@sky}{x}}+3) + C(\\htmlData{anchor=x-val1-6}{\\color{@sky}{x}}-2)$"},{"latex":"$$4(\\color{@sky}{2})^2 + 13(\\color{@sky}{2}) + 8 = A(\\color{@sky}{2}+3)^2 + B(0) + C(0)$"},{"latex":"$$16 + 26 + 8 = 25\\htmlData{anchor=a-const}{\\color{@amber}{A}}$"},{"latex":"$$50 = 25\\color{@amber}{A} \\implies \\htmlData{anchor=a-final}{\\color{@amber}{A}} = 2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x-val1"},{"color":"amber","style":"text-color","anchor":"a-final"}],"connections":[{"to":"x-val1-4","from":"x-val1","color":"sky","label":"Eliminate B and C","style":"dashed"}]},"order":1,"title":"Find the constant A","description":"We can strategically substitute $x = 2$ into the identity to eliminate the terms involving $B$ and $C$ because the factor $(x-2)$ becomes zero."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4\\htmlData{anchor=x-val2}{\\color{@pink}{x}}^2 + 13\\htmlData{anchor=x-val2-2}{\\color{@pink}{x}} + 8 \\equiv A(\\htmlData{anchor=x-val2-3}{\\color{@pink}{x}}+3)^2 + B(\\htmlData{anchor=x-val2-4}{\\color{@pink}{x}}-2)(\\htmlData{anchor=x-val2-5}{\\color{@pink}{x}}+3) + C(\\htmlData{anchor=x-val2-6}{\\color{@pink}{x}}-2)$"},{"latex":"$$4(\\color{@pink}{-3})^2 + 13(\\color{@pink}{-3}) + 8 = A(0) + B(0) + C(\\color{@pink}{-3}-2)$"},{"latex":"$$36 - 39 + 8 = -5\\htmlData{anchor=c-const}{\\color{@purple}{C}}$"},{"latex":"$$5 = -5\\color{@purple}{C} \\implies \\htmlData{anchor=c-final}{\\color{@purple}{C}}

3.1.8 Algebra and functions (A-level only)

Question 14

3.1.8 Algebra and functions (A-level only) Questions

3.1.8 Algebra and functions (A-level only)

Question 14

3.1.8 Algebra and functions (A-level only) Questions