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DSE M1 Binomial Expansion Practice

DSE M1 · 60 practice questions with instant marking and worked solutions.

Binomial Expansion is an important part of the DSE M1 mathematics curriculum. This topic develops the general term, particular coefficients, sums of coefficients and approximations. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Binomial Expansion has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Binomial Expansion; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 12 of them are DSE Extended Part style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Binomial Expansion set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Binomial Expansion is closely linked with the rest of the Extended Part: algebraic and trigonometric skills underpin calculus, and calculus in turn supports probability distributions and statistical inference. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

Start practising

  1. 1 Basic
    Consider the expansion of $(1+x)^{6}$. Find the coefficient of $x$. Choose the correct answer.
  2. 2 Basic
    Consider the expansion of $(x-\frac{1}{x})^{8}$. Find the constant term. Give the answer only.

  3. 3 Basic
    Let $(2-2x)^{6}=a_0+a_1x+\dots+a_{6}x^{6}$. Find the value of $a_0+a_1+\dots+a_{6}$. Show the essential working and give the answer.

  4. 4 Basic
    Use the first three terms of the expansion of $(1+x)^{6}$ to estimate $(1.01)^{6}$. The estimate is 1.06. Is this true?
  5. 5 Basic
    Consider the expansion of $(2+3x)^{4}$. Find the coefficient of $x^{3}$. Choose the correct answer.
  6. 6 Basic
    Consider the expansion of $(x+\frac{1}{x})^{6}$. Find the constant term. Give the answer only.

  7. 7 Basic
    Let $(3+x)^{3}=a_0+a_1x+\dots+a_{3}x^{3}$. Find the value of $a_0+a_1+\dots+a_{3}$. Show the essential working and give the answer.

  8. 8 Basic
    Use the first three terms of the expansion of $(1+x)^{7}$ to estimate $(1.02)^{7}$. The estimate is 1.14. Is this true?
  9. 9 Basic
    Consider the expansion of $(3+x)^{5}$. Find the coefficient of $x^{2}$. Choose the correct answer.
  10. 10 Basic
    Consider the expansion of $(x+\frac{2}{x})^{6}$. Find the constant term. Give the answer only.

  11. 11 Basic
    Let $(2-x)^{3}=a_0+a_1x+\dots+a_{3}x^{3}$. Find the value of $a_0+a_1+\dots+a_{3}$. Show the essential working and give the answer.

  12. 12 Basic
    Use the first three terms of the expansion of $(1+x)^{5}$ to estimate $(1.01)^{5}$. The estimate is 1.05. Is this true?
  13. 13 Standard
    Consider the expansion of $(2+2x)^{5}$. Find the coefficient of $x^{3}$. Choose the correct answer.
  14. 14 Standard
    Let $(3+x)^{4}=a_0+a_1x+\dots+a_{4}x^{4}$. Find the value of $a_0+a_1+\dots+a_{4}$. Give the answer only.

  15. 15 Standard
    Consider the expansion of $(3+3x)^{5}$. Find the coefficient of $x$. Show the essential working and give the answer.

  16. 16 Standard DSE long question
    Consider the expansion of $(1+3x)^{5}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  17. 17 Standard
    Consider the expansion of $(2-2x)^{6}$. Find the coefficient of $x^{3}$. Choose the correct answer.
  18. 18 Standard
    Consider the expansion of $(x+\frac{3}{x})^{6}$. Find the constant term. Give the answer only.

  19. 19 Standard
    Let $(1+2x)^{4}=a_0+a_1x+\dots+a_{4}x^{4}$. Find the value of $a_0+a_1+\dots+a_{4}$. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    Consider the expansion of $(1+2x)^{5}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  21. 21 Standard
    Consider the expansion of $(2+x)^{6}$. Find the coefficient of $x^{3}$. Choose the correct answer.
  22. 22 Standard
    Consider the expansion of $(x+\frac{3}{x})^{8}$. Find the constant term. Give the answer only.

  23. 23 Standard
    Let $(1-x)^{6}=a_0+a_1x+\dots+a_{6}x^{6}$. Find the value of $a_0+a_1+\dots+a_{6}$. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    Consider the expansion of $(-x+3)^{7}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all coefficients.
    4. (d) Find the coefficient of $x^3$.
  25. 25 Advanced
    Use the first three terms of the expansion of $(1+x)^{8}$ to estimate $(1.02)^{8}$. Find the estimate. Choose the correct answer.
  26. 26 Advanced
    Consider the expansion of $(2-2x)^{6}$. Find the coefficient of $x$. Give the answer only.

  27. 27 Advanced
    Consider the expansion of $(x+\frac{1}{x})^{8}$. The constant term is 70. Is this true?
  28. 28 Advanced DSE long question
    Consider the expansion of $(1+2x)^{6}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  29. 29 Advanced
    Consider the expansion of $(1-x)^{4}$. Find the coefficient of $x^{3}$. Choose the correct answer.
  30. 30 Advanced
    Consider the expansion of $(1-x)^{4}$. Find the coefficient of $x^{2}$. Give the answer only.

  31. 31 Advanced
    Consider the expansion of $(x-\frac{2}{x})^{6}$. The constant term is -160. Is this true?
  32. 32 Advanced DSE long question
    Consider the expansion of $(1-x)^{6}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  33. 33 Advanced
    Use the first three terms of the expansion of $(1+x)^{5}$ to estimate $(1.02)^{5}$. Find the estimate. Choose the correct answer.
  34. 34 Advanced
    Consider the expansion of $(2+2x)^{8}$. Find the coefficient of $x$. Give the answer only.

  35. 35 Advanced
    Consider the expansion of $(x+\frac{1}{x})^{4}$. The constant term is 6. Is this true?
  36. 36 Advanced DSE long question
    Consider the expansion of $(-3x+3)^{8}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all coefficients.
    4. (d) Find the coefficient of $x^3$.
  37. 37 Challenging
    Let $(1+2x)^{4}=a_0+a_1x+\dots+a_{4}x^{4}$. Find the value of $a_0+a_1+\dots+a_{4}$. Choose the correct answer.
  38. 38 Challenging
    Use the first three terms of the expansion of $(1+x)^{4}$ to estimate $(1.01)^{4}$. Find the estimate. Show the essential working and give the answer.

  39. 39 Challenging
    Consider the expansion of $(2+2x)^{7}$. The coefficient of $x$ is 7. Is this true?
  40. 40 Challenging DSE long question
    Consider the expansion of $(1+2x)^{7}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  41. 41 Challenging
    Let $(3-2x)^{6}=a_0+a_1x+\dots+a_{6}x^{6}$. Find the value of $a_0+a_1+\dots+a_{6}$. Choose the correct answer.
  42. 42 Challenging
    Use the first three terms of the expansion of $(1+x)^{6}$ to estimate $(1.01)^{6}$. Find the estimate. Show the essential working and give the answer.

  43. 43 Challenging
    Consider the expansion of $(3-x)^{7}$. The coefficient of $x$ is 7. Is this true?
  44. 44 Challenging DSE long question
    Consider the expansion of $(1-x)^{8}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  45. 45 Challenging
    Let $(1+x)^{7}=a_0+a_1x+\dots+a_{7}x^{7}$. Find the value of $a_0+a_1+\dots+a_{7}$. Choose the correct answer.
  46. 46 Challenging
    Use the first three terms of the expansion of $(1+x)^{3}$ to estimate $(1.01)^{3}$. Find the estimate. Show the essential working and give the answer.

  47. 47 Challenging
    Consider the expansion of $(1-2x)^{6}$. The coefficient of $x$ is 6. Is this true?
  48. 48 Challenging DSE long question
    Consider the expansion of $(4x+3)^{5}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all coefficients.
    4. (d) Find the coefficient of $x^3$.
  49. 49 Elite
    Let $(1+2x)^{6}=a_0+a_1x+\dots+a_{6}x^{6}$. Find the value of $a_0+a_1+\dots+a_{6}$. Give the answer only.

  50. 50 Elite
    Let $(1+x)^{3}=a_0+a_1x+\dots+a_{3}x^{3}$. Find the value of $a_0+a_1+\dots+a_{3}$. Show the essential working and give the answer.

  51. 51 Elite
    Consider the expansion of $(2+3x)^{4}$. The coefficient of $x^{2}$ is 216. Is this true?
  52. 52 Elite DSE long question
    Consider the expansion of $(1-x)^{7}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  53. 53 Elite
    Consider the expansion of $(x-\frac{2}{x})^{8}$. Find the constant term. Give the answer only.

  54. 54 Elite
    Let $(3-2x)^{3}=a_0+a_1x+\dots+a_{3}x^{3}$. Find the value of $a_0+a_1+\dots+a_{3}$. Show the essential working and give the answer.

  55. 55 Elite
    Use the first three terms of the expansion of $(1+x)^{4}$ to estimate $(1.02)^{4}$. The estimate is 1.0824. Is this true?
  56. 56 Elite DSE long question
    Consider the expansion of $(1+2x)^{4}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all the coefficients.
  57. 57 Elite
    Consider the expansion of $(x-\frac{2}{x})^{6}$. Find the constant term. Give the answer only.

  58. 58 Elite
    Let $(3+2x)^{4}=a_0+a_1x+\dots+a_{4}x^{4}$. Find the value of $a_0+a_1+\dots+a_{4}$. Show the essential working and give the answer.

  59. 59 Elite
    Use the first three terms of the expansion of $(1+x)^{7}$ to estimate $(1.01)^{7}$. The estimate is 1.0721. Is this true?
  60. 60 Elite DSE long question
    Consider the expansion of $(-4x+1)^{10}$. Answer each part; give the answer only.
    1. (a) Find the coefficient of $x$.
    2. (b) Find the coefficient of $x^2$.
    3. (c) Find the sum of all coefficients.
    4. (d) Find the coefficient of $x^3$.

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Binomial Expansion

This topic needs particular care with the general term, particular coefficients, sums of coefficients and approximations. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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