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The Real Number System – ℝ · CAPE Pure Mathematics Unit 1

117 past-paper questions on The Real Number System – ℝ, part of Basic Algebra and Functions, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

Practise The Real Number System – ℝ
  1. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1In the real number system, the inverse of addition is represented by
  2. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The (k + 1)\text{th} term in \sum_{r=1}^{n} r(r - 1) is
  3. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1Rationalising \frac{\sqrt{2} - 1}{\sqrt{2} + 1} gives
  4. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1\sqrt{8} + \sqrt{32} - \sqrt{162} can be simplified as
  5. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The range of values for x such that 5x + 7 > 10x - 13 is
  6. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1\sum_{r=1}^{50} (r + 2) is equal to
  7. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Rationalising \frac{\sqrt{2} - 1}{\sqrt{2} + 1} gives
  8. 2(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Show that if x and y are real numbers such that x < y, then for any real number k < 0, kx > ky.
  9. 3(a)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Without using calculators or tables, show that (sqrt(11) + sqrt(7)) = 4 / (sqrt(11) - sqrt(7)).
  10. 12(b)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Show that the equation x^3 = 8 + 4x has a root in the closed interval [2, 3].
  11. 1(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Given that sum_(r=1)^n r = n/2(n + 1), show that sum_(r=1)^n (3r + 1) = 1/2 n(3n + 5).
  12. 3(a)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find a, b in R such that (3x)/(x + 1) - 2 = (ax + b)/(x + 1), where x != -1.
  13. 5(a)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the values of m, n in R for which the system of equations possesses a unique solution.
  14. 5(b)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the values of m, n in R for which the system of equations is inconsistent.
  15. 5(c)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the values of m, n in R for which the system of equations possesses infinitely many solutions.
  16. 9(a)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1The roots of the quadratic equation x^2 + 6x + k = 0 are -3 + 2i and -3 - 2i. Find the value of the constant k.
  17. 9(b)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find the real numbers u and v such that (u + 2i)/(3 - 4i) = 1 + vi.
  18. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Given that x and y are negative integers, and that x > y, which of the following is true?
  19. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 14\sqrt{x} + \frac{4\sqrt{3x}}{\sqrt{48}} =
  20. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The range of values of x for which \frac{2x + 3}{x} \ge 8 is
  21. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1If p and q are positive integers such that p < q, then which of the following statement(s) is/are correct? I. -p > -q II. p^2 > pq III. p - 1 < q - 1
  22. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} + \sqrt{y}} may be expressed as
  23. Q111 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Given \frac{8}{x^2} = \frac{\sqrt{x}}{4}, the value of x is
  24. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1\sqrt{8} + \sqrt{32} - \sqrt{162} can be simplified as
  25. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If p and q are positive integers such that p < q, then which of the following statements is/are correct? \begin{align*} \text{I.} & \quad -p > -q \\ \text{II.} & \quad p^2 > pq \\ \text{III.} & \quad p - 1 < q - 1…
  26. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The (k + 1) term in \sum_{r=1}^n r(r - 1) is
  27. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The range of values of x, (x > 0), for which \frac{2x + 3}{x} \ge 8 is
  28. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The range of real values of x for which x^2 - 3x - 28 < 0 is
  29. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1In the real number system the inverse of addition is represented by
  30. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1If p and q are positive integers such that p < q, then which of the following statement(s) is/are correct? I. -p > -q II. p^2 > pq III. p - 1 < q - 1
  31. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1\sqrt{8} + \sqrt{32} - \sqrt{162} can be simplified as
  32. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If p and q are positive integers such that p < q, then which of the following statements is/are correct? I. -p > -q II. p^2 > pq III. p - 1 < q - 1
  33. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Rationalising \frac{\sqrt{2}-1}{\sqrt{2}+1} gives
  34. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1A vector equation is given as s\begin{pmatrix} -2 \\ 1 \end{pmatrix} + t\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -5 \\ 1 \end{pmatrix}. The values of s and t are, respectively
  35. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1In the real number system the inverse of addition is represented by
  36. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Rationalising \frac{\sqrt{2}-1}{\sqrt{2}+1} gives
  37. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 14\sqrt{x} + \frac{4\sqrt{3x}}{\sqrt{48}} =
  38. 1(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Let x and y be negative real numbers and let z be any real number. Use a counter example to show that the statement "if x > y then xz > yz" is false.
  39. 1(d)(ii)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Show that \sqrt{320x^3} + \sqrt{125x^3} simplifies to 13x\sqrt{5x}.
  40. 2(c)8 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Provide a proof by contradiction for this proposition, by assuming n is odd and showing the assumption is incorrect.
  41. 1(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Rationalize the denominator of (√3 + √7) / (√5 + √2), expressing your answer in surd form.
  42. 1(c)8 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Prove, by mathematical induction, that for all natural numbers n, 1 + 5 + 9 + 13 + ... + (4n - 3) = n(2n - 1).
  43. 1(a)5 marks· Pure Mathematics · Unit 1 Q1 1(a)Without the use of tables or a calculator, simplify √28 + √343 in the form k√7, where k is an integer.
  44. 1(a)(i)3 marks· Pure Mathematics · Unit 1 Q1 1(a)(i)Find the exact value of (√75 + √12)² - (√75 - √12)²
  45. 1(a)(i)8 marks· Pure Mathematics · Unit 1 Q1 1(a)(i)Determine the values of the real number h for which the roots of the quadratic equation 4x² - 2hx + (8 – h) = 0 are real.
  46. 1(a)(ii)3 marks· Pure Mathematics · Unit 1 Q1 1(a)(ii)Find the exact value of 27^(3/4) x 9^(3/8) x 81^(1/8)
  47. 1(b)8 marks· Pure Mathematics · Unit 1 Q1 1(b)Use mathematical induction to prove that 10ⁿ⁺¹ + 3(10ⁿ) + 5 is divisible by 9 for all natural numbers.
  48. 1(b)6 marks· Pure Mathematics · Unit 1 Q1 1(b)Use mathematical induction to prove that 6ⁿ - 1 is divisible by 5 for all natural numbers n.
  49. 1(b)5 marks· Pure Mathematics · Unit 1 Q1 1(b)Solve the equation 2 ⊕ x = 0.
  50. 1(b)8 marks· Pure Mathematics · Unit 1 Q1 1(b)Find positive integers x and y such that (√x + √y)² = 16 + √240.
  51. 1(b)(i)5 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)Without using calculators or tables, show that (√6 + √2) / (√6 - √2) = 2 + √3.
  52. 1(b)(i)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)State, giving a reason for your answer, if ⊕ is commutative in R.
  53. 1(b)(i)1 mark· Pure Mathematics · Unit 1 Q1 1(b)(i)Calculate 5 ⊗ 2.
  54. 1(b)(i)1 mark· Pure Mathematics · Unit 1 Q1 1(b)(i)Prove that * is commutative.
  55. 1(b)(ii)5 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Without using calculators or tables, show that (√6 + √2) / (√6 - √2) + (√6 - √2) / (√6 + √2) = 4.
  56. 1(b)(ii)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Prove that * is closed in ℝ.
  57. 1(b)(ii)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Prove that ⊗ is closed in R.
  58. 1(b)(ii)2 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Show that the identity element of * is 3.
  59. 1(b)(iii)2 marks· Pure Mathematics · Unit 1 Q1 1(b)(iii)Deduce that (y+1)⁴ - y⁴ < 4(y + 1)³.
  60. 1(b)(iii)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(iii)Determine whether ⊗ is commutative.
  61. 1(b)(iii)2 marks· Pure Mathematics · Unit 1 Q1 1(b)(iii)Determine whether the operation * is commutative.
  62. 1(c)10 marks· Pure Mathematics · Unit 1 Q1 1(c)Use mathematical induction to prove that 1² + 3² + 5² + ..... + (2n - 1)² = n/3 (4n² - 1) for n ∈ N.
  63. 1(c)8 marks· Pure Mathematics · Unit 1 Q1 1(c)Use mathematical induction to prove that 5ⁿ + 3 is divisible by 2 for all values of n ∈ N.
  64. 1(c)8 marks· Pure Mathematics · Unit 1 Q1 1(c)Use mathematical induction to prove that 4S(n) = 5ⁿ⁺¹ – 5 for n ∈ N.
  65. 1(c)(i)5 marks· Pure Mathematics · Unit 1 Q1 1(c)(i)Show that Σ (from r=1 to n) r(r + 1) = (1/3)n(n + 1)(n + 2), n ∈ N.
  66. 1(c)(ii)3 marks· Pure Mathematics · Unit 1 Q1 1(c)(ii)Hence, or otherwise, evaluate Σ (from r=31 to 50) r(r + 1).
  67. 1(c)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(c)(ii)Without using calculators or tables, evaluate (√2 + 1)³ - (√2 - 1)³.
  68. 1(d)7 marks· Pure Mathematics · Unit 1 Q1 1(d)Use mathematical induction to prove that 8+16+24+ 32 + ... + 8n = 4n(n + 1) for all n ∈ N.
  69. 2(a)6 marks· Pure Mathematics · Unit 1 Q2 2(a)Without solving the equation, find a quadratic equation with roots 2/α and 2/β.
  70. 2(a)10 marks· Pure Mathematics · Unit 1 Q2 2(a)Prove, by Mathematical Induction, that Σ (from r=1 to n) r = (1/2)n(n + 1).
  71. 2(a)9 marks· Pure Mathematics · Unit 1 Q2 2(a)Prove, by Mathematical Induction, that 10ⁿ - 1 is divisible by 9 for all positive integers n.
  72. 2(a)8 marks· Pure Mathematics · Unit 1 Q2 2(a)Without using calculators or tables, evaluate (27¹⁰ + 9¹⁰) / (27⁴ + 9¹¹).
  73. 2(a)4 marks· Pure Mathematics · Unit 1 Q2 2(a)Find the value of n for which 3S_{2n} = 11 S_n. Note: Σr = n(n+1)/2.
  74. 2(a)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Write down the values of α + β and αβ.
  75. 2(a)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Use the fact that Sₙ = ∑r = 1 to n of r = (1/2) n (n + 1) to express S₂ₙ = ∑r = 1 to 2n of r in terms of n.
  76. 2(a)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Express f(x) in the form p(x-q)² + r, where p, q, r ∈ R.
  77. 2(a)(i)7 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)Express f(x) = 12x-2x² in the form A + B(x+p)² where A, B and p are real numbers, and find the maximum value of 12x - 2x².
  78. 2(a)(i)a)1 mark· Pure Mathematics · Unit 1 Q2 2(a)(i)a)Express α + β in terms of p
  79. 2(a)(i)b)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(i)b)Express α² + β² in terms of p
  80. 2(a)(ii)5 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Find constants p and q such that S₂ₙ - Sₙ = pn² + qn.
  81. 2(a)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)Given that α² + β² = 33, find the possible values of p.
  82. 2(a)(ii)a)2 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)a)Calculate α² + β².
  83. 2(a)(ii)b)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(ii)b)Calculate α³ + β³.
  84. 2(a)(iii)5 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Hence, or otherwise, find n such that S₂ₙ - Sₙ = 260.
  85. 2(a)(iii)4 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Find a quadratic equation whose roots are α³ and β³.
  86. 2(b)1 mark· Pure Mathematics · Unit 1 Q2 2(b)Express, in terms of n and in the SIMPLEST form,
  87. 2(b)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Σ (from r=1 to 2n) r
  88. 2(b)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Σ (from r=n+1 to 2n) r.
  89. 2(c)7 marks· Pure Mathematics · Unit 1 Q2 2(c)Prove, by the principle of mathematical induction, that f(n) = 7ⁿ - 1 is divisible by 6, for all n ∈ N.
  90. 2(c)4 marks· Pure Mathematics · Unit 1 Q2 2(c)Find n if Σ (from r=n+1 to 2n) r = 100.
  91. 2(c)8 marks· Pure Mathematics · Unit 1 Q2 2(c)Prove, by Mathematical Induction, that n^2 > 2n for all integers n ≥ 3.
  92. 2(c)1 mark· Pure Mathematics · Unit 1 Q2 2(c)Solve the following:
  93. 2(c)2 marks· Pure Mathematics · Unit 1 Q2 2(c)Prove that the product of any two consecutive integers k and k + 1 is an even integer.
  94. 2(c)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(c)(ii)∑_{r=1}^{99} log₁₀(r/(r+1)).
  95. 2(d)5 marks· Pure Mathematics · Unit 1 Q2 2(d)Without the use of a calculator, show that (√3-1)/(√3+1) + (√3+1)/(√3-1) + (√2-1)/(√2+1) + (√2+1)/(√2-1) = 10.
  96. 2(d)6 marks· Pure Mathematics · Unit 1 Q2 2(d)Prove, by mathematical induction, that n(n² + 5) is divisible by 6 for all positive integers n.
  97. 3(a)(i)3 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Find the value(s) of p for which the system of equations above has a unique solution.
  98. 3(a)(i)4 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Express the complex number z = (11 - 2i) / (3 + 4i) in the form a + ib, where a and b are real numbers.
  99. 3(a)(ii)3 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Hence, express z² and iz in a similar form.
  100. 3(a)(ii)3 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)Find the value(s) of p for which the system of equations above has an infinite number of solutions.
  101. 3(a)(iii)3 marks· Pure Mathematics · Unit 1 Q3 3(a)(iii)Find the modulus and principal value of the argument of z, where -π < arg z ≤ π.
  102. 3(a)(iii)3 marks· Pure Mathematics · Unit 1 Q3 3(a)(iii)Find the value(s) of p for which the system of equations above has no solution.
  103. 3(a)(iv)3 marks· Pure Mathematics · Unit 1 Q3 3(a)(iv)Find the exact distance between the points on the Argand diagram represented by z² and iz.
  104. 3(b)10 marks· Pure Mathematics · Unit 1 Q3 3(b)Solve for x ∈ R the inequality (2x + 3) / (3x + 4) < 1.
  105. 4(a)7 marks· Pure Mathematics · Unit 1 Q4 4(a)Express the complex number, (z-1)/(z+1), in a similar form.
  106. 4(a)(i)1 mark· Pure Mathematics · Unit 1 Q4 4(a)(i)Express the arc length ABC in terms of π.
  107. 4(a)(ii)a)3 marks· Pure Mathematics · Unit 1 Q4 4(a)(ii)a)Hence, show that r = 7/6.
  108. 4(a)(ii)b)2 marks· Pure Mathematics · Unit 1 Q4 4(a)(ii)b)If h cm is the height of the cone, then the exact value of h is (7√35)/6.
  109. 4(b)(i)1 mark· Pure Mathematics · Unit 1 Q4 4(b)(i)Find in the form a + bi, a, b ∈ R,
  110. 4(b)(i)a)1 mark· Pure Mathematics · Unit 1 Q4 4(b)(i)a)z₁ + z₂
  111. 4(b)(i)b)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)b)z₁z₂
  112. 4(b)(i)c)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)c)z₁/z₂
  113. 4(c)(iii)2 marks· Pure Mathematics · Unit 1 Q4 4(c)(iii)Evaluate Σ (from r=1 to n) (tan rθ sin 2rθ + cos 2rθ) where n is a positive integer.
  114. 5(a)7 marks· Pure Mathematics · Unit 1 Q5 5(a)Show that f(x) = 0 possesses a root in the interval [1/2, 1]. By considering suitable values of x greater than 1, show that there is another root of f(x) = 0 greater than 1.
  115. 6(a)7 marks· Pure Mathematics · Unit 1 Q6 6(a)Show that the sum, S, of the areas of the rectangular strips is 1/(n+1) + 1/(n+2) + ... + 1/(2n).
  116. 6(a)(i)6 marks· Pure Mathematics · Unit 1 Q6 6(a)(i)Show that the area S is approximately 1/n² + 2/n² + 3/n² + .... + (n-1)/n².
  117. 6(a)(ii)2 marks· Pure Mathematics · Unit 1 Q6 6(a)(ii)Given that Σ (from r=1 to n-1) r = (1/2)n(n - 1), show that S = (1/2)(1 - 1/n).

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