is an identity, the coefficients of the terms on the left-hand side must equal the coefficients of the terms on the right-hand side. This gives us three simultaneous equations:\n\n1. **For $x^2$ coefficients:**\n $$A + B = 5$$\n\n2. **For $x$ coefficients:**\n $$-2A + 2B + C = -12$$\n\n3. **For the constant terms:**\n $$A - 3B + 3C = -1$$"},{"type":"text","order":4,"title":"Solve the system","content":"Let's solve the system of linear equations:\n\nFrom the first equation, express $B$ in terms of $A$:\n$$B = 5 - A$$\n\nSubstitute $B = 5 - A$ into the other two equations:\n* **Equation 2:**\n $$\\begin{aligned} -2A + 2(5 - A) + C &= -12 \\\\ -2A + 10 - 2A + C &= -12 \\\\ -4A + C &= -22 \\quad \\Rightarrow \\quad C = 4A - 22 \\end{aligned}$$\n\n* **Equation 3:**\n $$\\begin{aligned} A - 3(5 - A) + 3C &= -1 \\\\ A - 15 + 3A + 3C &= -1 \\\\ 4A + 3C &= 14 \\end{aligned}$$\n\nNow substitute $C = 4A - 22$ into this simplified Equation 3:\n$$\\begin{aligned} 4A + 3(4A - 22) &= 14 \\\\ 4A + 12A - 66 &= 14 \\\\ 16A &= 80 \\\\ A &= 5 \\end{aligned}$$\n\nThis gives **1 mark** (A1) for finding $A = 5$."},{"type":"text","order":5,"title":"Find B and C","content":"Now we can easily find the remaining constants $B$ and $C$:\n\n* Substitute $A = 5$ into $B = 5 - A$:\n $$B = 5 - 5 = 0$$\n This earns **1 mark** (A1) for $B = 0$.\n\n* Substitute $A = 5$ into $C = 4A - 22$:\n $$C = 4(5) - 22 = -2$$\n This earns **1 mark** (A1) for $C = -2$."},{"type":"text","order":6,"title":"Write the final expression","content":"Substitute the values of $A = 5$, $B = 0$, and $C = -2$ back into our initial partial fraction form:\n\n$$\\frac{5}{x+3} + \\frac{0}{x-1} + \\frac{-2}{(x-1)^2}$$\n\nSimplifying this, we get our final answer which earns the final **1 mark** (A1):\n\n$$\\frac{5}{x+3} - \\frac{2}{(x-1)^2}$$"}]}],"generatedAt":"2026-06-11T03:23:01.462Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"93ccb000-f0d0-441f-84b0-3e490ef0fca8","revisionId":"8c40de61-627a-4a28-8452-3b5d69d11229","questionNumber":"12","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"Question Set 1: Algebraic Methods","calculatorUse":null,"parts":[{"id":"90d8577b-aa01-4e54-afda-f5fb0c50d2c4","label":"","content":"Express $\\frac{4x^2 + 12x + 11}{(x-1)(x+2)^2}$ in partial fractions","markscheme":"- Sets up the identity $4x^2 + 12x + 11 = A(x+2)^2 + B(x-1)(x+2) + C(x-1)$ M1\n- Finds $A = 3$ and $C = -1$ A1\n- Finds $B = 1$ A1\n- Final answer: $\\frac{3}{x-1} + \\frac{1}{x+2} - \\frac{1}{(x+2)^2}$ A1\n- Correct use of partial fraction form A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Sets up the correct identity for repeated linear factors.","code":"M"},{"id":"p2","marks":1,"description":"Correctly calculates $A$.","code":"A"},{"id":"p3","marks":1,"description":"Correctly calculates $C$.","code":"A"},{"id":"p4","marks":1,"description":"Correctly calculates $B$.","code":"A"},{"id":"p5","marks":1,"description":"States the final expression correctly.","code":"A"}],"finalAnswers":["\\frac{3}{x-1} + \\frac{1}{x+2} - \\frac{1}{(x+2)^2}"],"generatedAt":"2026-05-06T02:36:06.047Z","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 check the denominator. It has a linear factor $(x-1)$ and a repeated linear factor $(x+2)^2$. \n\nWe set up the partial fraction form as:\n$$\\frac{4x^2 + 12x + 11}{(x-1)(x+2)^2} \\equiv \\frac{A}{x-1} + \\frac{B}{x+2} + \\frac{C}{(x+2)^2}$$\n\nTo find the constants $A$, $B$, and $C$, we multiply both sides of the identity by the common denominator $(x-1)(x+2)^2$ to obtain the basic identity:\n$$4x^2 + 12x + 11 \\equiv A(x+2)^2 + B(x-1)(x+2) + C(x-1)$$\n\nSetting up this identity correctly earns the first meth

3.1.8 Algebra and functions (A-level only)

Question 18

3.1.8 Algebra and functions (A-level only) Questions

3.1.8 Algebra and functions (A-level only)

Question 18

3.1.8 Algebra and functions (A-level only) Questions