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The Binomial Theorem · CAPE Pure Mathematics Unit 2

70 past-paper questions on The Binomial Theorem, part of Sequences, Series and Approximations, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise The Binomial Theorem
  1. 3(c)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Show that sum_{r=0}^n binom(n, r) = 2^n.
  2. 3(c)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Show that sum_{r=0}^n binom(n, r) (-1)^r = 0.
  3. 5(b)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Calculate the probability that he will hit the target AT LEAST 8 times.
  4. 5(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Calculate the probability that he will hit the target NO MORE than seven times.
  5. 4(b)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Use the binomial theorem or Maclaurin's theorem to expand (1 + x)^{-1/2} in ascending powers of x as far as the term in x^3, stating the values of x for which the expansion is valid.
  6. 4(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Obtain a similar expansion for (1 - x)^{1/2}.
  7. 4(b)(iii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Prove that if x is so small that x^3 and higher powers of x can be neglected, then \sqrt{\frac{1 - x}{1 + x}} \approx 1 - x + \frac{1}{2}x^2.
  8. 4(b)(iv)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Hence, by taking x = \frac{1}{17}, show, without using calculators or tables, that \sqrt{2} is approximately equal to \frac{1635}{1156}.
  9. 4(b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Find the term independent of x in the binomial expansion of (x^2 - \frac{6}{x^3})^{15}. [You may leave your answer in the form of factorials and powers, for example, \frac{15!}{2!} \times 8^5.]
  10. 4(c)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Use the binomial theorem to find the difference between 2^{10} and (2.002)^{10} correct to 5 decimal places.
  11. 4(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Find n \in \mathbb{N} such that 5\binom{n}{2} = 2\binom{n+2}{2}.
  12. 4(a)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2The coefficient of x^2 in the expansion of (1 + 2x)^5 (1 + px)^4 is -26. Find the possible values of the real number p.
  13. 3(c)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Use the binomial theorem to expand (1 + 2x)^{\frac{1}{2}} as far as the term in x^3, stating the values of x for which the expansion is valid.
  14. 3(c)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Prove that \frac{x}{1 + x + \sqrt{1 + 2x}} = \frac{1}{x}(1 + x - \sqrt{1 + 2x}) for x > 0.
  15. 3(c)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Hence, or otherwise, show that, if x is small so that the term in x^3 and higher powers of x can be neglected, the expansion in (c)(ii) above is approximately equal to \frac{1}{2}x(1 - x).
  16. 4(a)(i)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2By expressing ^n\mathrm{C}_r and ^n\mathrm{C}_{r-1} in terms of factorials, prove that ^n\mathrm{C}_r + {}^n\mathrm{C}_{r-1} = {}^{n+1}\mathrm{C}_r.
  17. 3(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Obtain the first FOUR non-zero terms of the expansion of each of (1 - x)^{-1} and (1 - 2x)^{-1} as power series of x in ascending order.
  18. 3(b)(iii) a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the range of values of x for which the series expansion of \frac{2 - 3x}{(1 - x)(1 - 2x)} is valid.
  19. 3(b)(iii) b)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the coefficient of x^n in the series expansion of \frac{2 - 3x}{(1 - x)(1 - 2x)}.
  20. 5(a)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2From the definition, show that \binom{n}{r} = \binom{n}{n - r}.
  21. 5(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2From the definition, show that \binom{n+1}{r} = \binom{n}{r} + \binom{n}{r - 1}.
  22. 5(a)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Hence, prove that \left[ \binom{8}{6} + \binom{8}{5} \right] \times \left[ \binom{8}{3} + \binom{8}{2} \right] is a perfect square.
  23. 4(a)(i)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Express \binom{n}{r} in terms of factorials.
  24. 4(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Hence, show that \binom{n}{r} = \binom{n}{n - r}.
  25. 4(a)(iii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find the coefficient of x^4 in \left(x^2 - \frac{3}{x}\right)^8.
  26. 4(a)(iv)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Using the identity (1 + x)^{2n} = (1 + x)^n (1 + x)^n, show that \binom{2n}{n} = c_0^2 + c_1^2 + c_2^2 + \dots + c_{n-1}^2 + c_n^2, where c_r = \binom{n}{r}.
  27. 4(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Obtain the binomial expansion of \sqrt[4]{(1+x)} + \sqrt[4]{(1-x)} up to the term containing x^2.
  28. 4(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Hence, by letting x = \frac{1}{16}, compute an approximation of \sqrt[4]{17} + \sqrt[4]{15} to four decimal places.
  29. 4(b)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Show that the coefficient of the x^5 term of the product (x+2)^5(x-2)^4 is 96.
  30. 4(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2For the binomial expansion of (2x + 3)^{20}, show that the ratio of the term in x^6 to the term in x^7 is \frac{3}{4x}.
  31. 4(a)(ii)a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Determine the FIRST THREE terms of the binomial expansion of (1 + 2x)^{10}.
  32. 4(a)(ii)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Hence, obtain an estimate for (1.01)^{10}.
  33. 4(b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Show that \frac{n!}{(n - r)!r!} + \frac{n!}{(n - r + 1)!(r - 1)!} = \frac{(n + 1)!}{(n - r + 1)!r!}.
  34. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The binomial coefficient \binom{n}{2} is equivalent to
  35. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The value of the term that is independent of x in the binomial expansion of \left(x^2 + \frac{1}{x}\right)^{12} is
  36. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Given that the coefficient of the term in b^3 in the binomial expansion of (a + b)^5 is 40, then a =
  37. 3(a)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Hence, use the binomial expansion with x = \frac{1}{16} to approximate the value of T_4 for terms up to and including x^3. Give your answer correct to two decimal places.
  38. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The expression \frac{n!}{(n - 2)!} can be simplified and written as
  39. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1^8C_3 equals
  40. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1If the coefficient of x^3 in the expression of (6 - ax)^9 is -84, then the value of a is
  41. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The values of x for which the expansion of \frac{1}{\sqrt{100 - 50x}} is valid are
  42. 3(c)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Determine the coefficient of the term in x^3 in the binomial expansion of (3x + 2)^5.
  43. 4(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Show that f(x) = (1 + 2x)^(1/3).
  44. 4(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Determine the series expansion of f up to and including the term in x^4.
  45. 4(a)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Hence, approximate f(0.4) correct to 2 decimal places.
  46. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The coefficient of a^2b^5 in the expansion of (a + b)^7 is
  47. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The value of the term that is independent of x in the binomial expansion of \left(x^2 + \frac{1}{x}\right)^{12} is
  48. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1If the coefficient of x^3 in the expansion of (6 - ax)^9 is -84, then the value of a is
  49. 4(b)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Show that the binomial expansion of \left(1 + \frac{1}{8}x\right)^8 up to and including the term in x^4 is 1 + x + \frac{7}{16}x^2 + \frac{7}{64}x^3 + \frac{35}{2048}x^4.
  50. 4(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Use the expansion to approximate the value of (1.0125)^8.
  51. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The binomial coefficient \begin{pmatrix} n \\ 4 \end{pmatrix} is equivalent to
  52. Q251 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The value of the term independent of x in the binomial expansion of \left(x^2 + \frac{1}{x}\right)^{12} is
  53. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1If the coefficient of x^3 in the expansion of (6 - ax)^9 is -84, then the value of a is
  54. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The values of x for which the expansion of \frac{1}{\sqrt{(100 - 50x)}} is valid are
  55. 4(a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Determine the coefficient of the term in x^7 in the expansion of (x^2 - 3/x)^8.
  56. 4(b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2By expressing ^n C_r and ^n C_(r-1) in terms of factorials, show that ^n C_r + ^n C_(r-1) = ^(n+1) C_r.
  57. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The value of the term that is independent of x in the binomial expansion of \left[x^2 + \frac{1}{x}\right]^{12} is
  58. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The binomial coefficient \binom{n}{4} is equivalent to
  59. 3(a)7 marks· Pure Mathematics · Unit 2 Q3 3(a)Determine the coefficient of x⁴ in the binomial expansion of (1 - 2x) / (1 + 3x)².
  60. 3(a)3 marks· Pure Mathematics · Unit 2 Q3 3(a)Determine the coefficient of the term in x³ in the binomial expansion of (3x + 2)⁵.
  61. 3(b)(i)4 marks· Pure Mathematics · Unit 2 Q3 3(b)(i)Show that the binomial expansion of (1 + x)^(1/2) + (1 - x)^(1/2) up to the term in x² is 2 - (3/16)x².
  62. 3(b)(ii)3 marks· Pure Mathematics · Unit 2 Q3 3(b)(ii)Hence, by letting x = 1/16, compute an approximation of √17 + √15, correct to 4 decimal places.
  63. 3(c)(i)6 marks· Pure Mathematics · Unit 2 Q3 3(c)(i)Obtain the binomial expansion of (16-5x)¹/⁴ up to and including the term in x².
  64. 3(c)(ii)2 marks· Pure Mathematics · Unit 2 Q3 3(c)(ii)Hence, state the values for which the expansion of (16 – 5x)¹/⁴ is valid.
  65. 3(c)(iii)2 marks· Pure Mathematics · Unit 2 Q3 3(c)(iii)Use the expansion obtained in (c)(i) to estimate the value of (16 – 5x)¹/⁴ when x = 3.
  66. 4(a)4 marks· Pure Mathematics · Unit 2 Q4 4(a)Determine the first three terms of the expansion of (1 – 8x)^(1/2).
  67. 4(a)(i)6 marks· Pure Mathematics · Unit 2 Q4 4(a)(i)Show that the binomial expansion of (1 + 5x)^(1/3) up to and including the term in x³ is 1 + x - 2x² + 6x³.
  68. 4(a)(ii)5 marks· Pure Mathematics · Unit 2 Q4 4(a)(ii)Hence, by letting x = -1/32, compute an estimate of ³√27.
  69. 4(b)6 marks· Pure Mathematics · Unit 2 Q4 4(b)Hence, by letting x = 1/100, determine √23.
  70. 4(b)6 marks· Pure Mathematics · Unit 2 Q4 4(b)Obtain the binomial expansion of (8 + x)^(1/3) in ascending powers of x, up to and including the term in x^2. State the values of x for which the expansion is valid.

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