term, then the $x^2$ term. For a fifth power, the first three binomial coefficients are $1$, $5$, and $10$. Identifying these earns the **B1** mark.","highlights":[{"text":"Find the first three terms, in ascending powers of $x$, of the binomial expansion of $(3 + 2x)^5, giving each term in its simplest form.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\text{First three terms: }(3+2x)^5=\\htmlData{anchor=first}{1}(3)^5+\\htmlData{anchor=second}{5}(3)^4(2x)+\\htmlData{anchor=third}{10}(3)^3(2x)^2+\\cdots"},{"latex":"\\text{The powers progress as }3^5,\\ 3^4,\\ 3^3\\text{ and }(2x)^0,\\ (2x)^1,\\ (2x)^2."}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"first"},{"color":"sky","style":"text-color-with-background","anchor":"second"},{"color":"sky","style":"text-color-with-background","anchor":"third"}]},"order":1,"title":"Set up the expansion","description":"Use the first three coefficients, decreasing the power of 3 and increasing the power of 2x by one each time. This correct structure earns the M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"1(3)^5=243"},{"latex":"5(3)^4(2x)=810x"},{"latex":"10(3)^3(2x)^2=1080x^2"},{"latex":"(3+2x)^5=243+810x+1080x^2+\\cdots"}]},"order":3,"title":"Simplify each term","description":"Evaluate each numerical factor and retain its power of x. These simplified terms earn the A2 marks."}]}],"generatedAt":"2026-07-21T05:05:40.568Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"faa8d023-b6ed-4408-a3cf-b666708b7ab7","revisionId":"fc084885-a31e-4a52-bf6e-a5b08c389f53","questionNumber":"2","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":4,"sourceTitle":"The Binomial Expansion","calculatorUse":null,"isCalculation":true,"parts":[{"id":"bcc661dc-10d0-4b52-9046-af7ae6025a9d","label":"0","content":"Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $(3 - 2x)^5$, giving each term in its simplest form.","markscheme":"- Identifies binomial coefficients 1, 5, 10 B1\n- Correct structure for expansion: $1(3)^5 + 5(3)^4(-2x) + 10(3)^3(-2x)^2$ M1\n- Simplifies terms to $243 - 810x + 1080x^2$ A2","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Identifies binomial coefficients 1, 5, 10.","code":"B"},{"id":"p2","marks":1,"description":"Attempts expansion with correct powers of 3 and $(-2x)$.","code":"M"},{"id":"p3","marks":2,"description":"Correct simplified terms $243 - 810x + 1080x^2$.","code":"A"}],"finalAnswers":["243 - 810x + 1080x^2"],"generatedAt":"2026-05-06T15:27:02.764Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"8fcc4ea4-2fec-4733-bc62-3f9e78cc85b4","revisionId":"90f1be51-c783-4189-b6df-1b3d0ecc39d3","questionNumber":"3","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"The Binomial Expansion","calculatorUse":null,"isCalculation":true,"parts":[{"id":"e2cce8d4-1a75-4ed6-82a0-4375ab8b7133","label":"a","content":"Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $$(2 + kx)^7$$ where $k$ is a constant. Give each term in its simplest form.","markscheme":"- Identifies binomial coefficients 1, 7, 21 B1\n- Correct structure: $1(2)^7 + 7(2)^6(kx) + 21(2)^5(kx)^2$ M1\n- Simplifies to $128 + 448kx + 672k^2x^2$ A2","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Identifies binomial coefficients 1, 7, 21.","code":"B"},{"id":"p2","marks":1,"description":"Attempts expansion with correct powers of 2 and $(kx)$.","code":"M"},{"id":"p3","marks":2,"description":"Correct simplified terms $128 + 448kx + 672k^2x^2$.","code":"A"}],"finalAnswe

The Binomial Expansion

Question 2

The Binomial Expansion Questions

The Binomial Expansion

Question 2

The Binomial Expansion Questions