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Algebraic Operations · CAPE Pure Mathematics Unit 1

105 past-paper questions on Algebraic Operations, part of Basic Algebra and Functions, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

Practise Algebraic Operations
  1. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The polynomial P(x) = 2x^3 + x^2 - 13x + 6, when divided by (x - 1), gives a remainder of
  2. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1(4x)^3 - (4y)^3 can be expressed in the form
  3. Q111 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The expression 2 - 4x + 3x^2 can be written as
  4. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If a remainder of 3 is obtained when 8x^3 + 4x + k is divided by x - 1, then k equals
  5. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Given that the roots of x^2 - 5x + a = 0 are equal, then a =
  6. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1If \alpha and \beta are the roots of the quadratic equation -x^2 + 10x + 2 = 0, then \alpha^2 + \beta^2 =
  7. 1(b)9 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Determine the values of a, b, and c, such that 2x^2 - 7x + 12 = a(x - 2)(x - 1) - b(x - 3) + c.
  8. 2(c)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2The function f(x) = 2x^3 - px^2 + qx - 10 is divisible by x - 1 and has a remainder of -6 when divided by x + 1. Calculate the values of p and q.
  9. 1(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Hence, or otherwise, find the value of EACH of the constants h and k.
  10. 1(c)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Factorise f(x) completely.
  11. 3(b)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Given that x + 1/x = 1, by considering (x + 1/x)^2, show that x^2 + 1/x^2 = -1.
  12. 3(b)(ii)5 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Hence, or otherwise, find the value of x^3 + 1/x^3.
  13. 8(a)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the range of values of k for which the quadratic equation x^2 + 2kx + 9 = 0 has complex roots.
  14. 8(b)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Express the complex number (2 + 3i) / (3 + 4i) in the form x + yi, where x and y are real numbers.
  15. 1(a)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Show that (x - 1) is a factor of f(x) for all values of p.
  16. 1(a)(ii)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1If (x - 2) is a factor of f(x), find the value of p.
  17. 2(b)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Let x, y, k in R such that (x + 1/2 y)^2 + ky^2 = x^2 + xy + y^2. Find the value of k.
  18. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Given (x - p)^3 - q^2(x - p) = 0, the values of x are
  19. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1x - 2 is a factor of
  20. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1If px^2 + 5x + 2 = 0 has 2 real and distinct roots, the range of possible values of p is
  21. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The rational expression \frac{x}{x^2 + 3x - 4} + \frac{2}{x - 1} simplified as a single fraction is
  22. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1a^5 - b^5 =
  23. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If a remainder of 7 is obtained when x^3 - 3x + k is divided by x - 3, then k equals
  24. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1If a remainder of 7 is obtained when x^3 - 3x + k is divided by x - 3, then k equals
  25. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1x - 2 is a factor of
  26. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1a^5 - b^5 =
  27. Q111 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1Given that the roots of x^2 - 5x + a = 0 are equal, then a =
  28. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The expression 2 - 4x + 3x^2 can be written as
  29. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If a remainder of 7 is obtained when x^3 - 3x + k is divided by x - 3, then k equals
  30. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Which of the following are factors of 4x^4 + 8x^3 - 2x^2 - 6x - 4? I. x + 1 II. x - 1 III. x + 2 IV. x - 2
  31. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1a^5 - b^5 =
  32. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1If a remainder of 7 is obtained when x^3 - 3x + k is divided by x - 3, then k equals
  33. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1If \alpha and \beta represent the roots of the equation x^2 - px + q = 0, then the value of \alpha^2 + \beta^2 is
  34. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1a^4 - b^4 =
  35. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Given (x - p)^3 - q^2(x - p) = 0, the values of x are
  36. Q421 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Based on the diagram above, which of the following statements is NOT correct?
  37. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1a^5 - b^5 =
  38. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1If a remainder of 3 is obtained when 8x^3 + 4x + k is divided by x - 1, then k equals
  39. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1Which of the following are factors of 4x^4 + 8x^3 - 2x^2 - 6x - 4? I. x + 1 II. x - 1 III. x + 2 IV. x - 2
  40. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The general quadratic equation with roots \alpha and \beta may be written as
  41. 1(c)9 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2The expression f(x) = 6x^3 + px^2 + qx + 2 is divisible by 2x - 1 and has a remainder of 2 when divided by x - 1. Calculate the values of p and q.
  42. 5(c)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Show that the volume of the container is 4x^3 - 58x^2 + 208x.
  43. 2(b)9 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Calculate the values of p and q.
  44. 1(d)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Given that x + 3 is a factor of x³ - 7x + q, determine the value of q.
  45. 1(d)(ii)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Hence, factorize the expression x³ - 7x + q.
  46. 1(e)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the remainder when x⁴ - 2x² + x + 1 is divided by 2x - 1.
  47. 6(a)(i)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the coordinates of the points where the curve y = f(x) intersects the x-axes and y-axes.
  48. 1(a)5 marks· Pure Mathematics · Unit 1 Q1 1(a)Find the values of the constant p such that x - p is a factor of f(x) = 4x³ - (3p + 2) x² - (p² – 1) x + 3.
  49. 1(a)1 mark· Pure Mathematics · Unit 1 Q1 1(a)Find
  50. 1(a)3 marks· Pure Mathematics · Unit 1 Q1 1(a)Find p, q ∈ R such that f(x) = p + q / (x - 2).
  51. 1(a)7 marks· Pure Mathematics · Unit 1 Q1 1(a)Find p and q. Hence, find the remainder when x² + px + q is divided by (x + 1).
  52. 1(a)10 marks· Pure Mathematics · Unit 1 Q1 1(a)Find the constants m and n, and the third factor of f(x).
  53. 1(a)3 marks· Pure Mathematics · Unit 1 Q1 1(a)Find all the real factors of g(x).
  54. 1(a)(i)7 marks· Pure Mathematics · Unit 1 Q1 1(a)(i)Show that p = -25 and q = -12.
  55. 1(a)(i)7 marks· Pure Mathematics · Unit 1 Q1 1(a)(i)the values of the constants p and q
  56. 1(a)(ii)6 marks· Pure Mathematics · Unit 1 Q1 1(a)(ii)Hence, solve the equation f(x) = 0.
  57. 1(a)(ii)1 mark· Pure Mathematics · Unit 1 Q1 1(a)(ii)Find all the real roots of g(x) = 0.
  58. 1(a)(ii)3 marks· Pure Mathematics · Unit 1 Q1 1(a)(ii)the factors of f(x).
  59. 1(b)8 marks· Pure Mathematics · Unit 1 Q1 1(b)Find the constants p, q and r such that 2y²-9y + 14 = p(y-1)(y-2) + q(y - 1) + r.
  60. 1(b)6 marks· Pure Mathematics · Unit 1 Q1 1(b)Simplify (x⁴ - y⁴) / (x - y).
  61. 1(b)4 marks· Pure Mathematics · Unit 1 Q1 1(b)Factorise completely the polynomial, x³ + 2x² - x - 2.
  62. 1(b)6 marks· Pure Mathematics · Unit 1 Q1 1(b)Find the value(s) of the real number, k, for which the equation k(x² + 5) = 6 + 12x - x² has equal roots.
  63. 1(b)(i)2 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)Find the value of p
  64. 1(b)(i)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)Express u^2 in terms of x.
  65. 1(b)(i)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)Find the values of p and q.
  66. 1(b)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Hence, or otherwise, show that (y+1)⁴ - y⁴ = (y+1)³ + (y+1)²y + (y+1)y² + y³.
  67. 1(b)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Find the values of m and n
  68. 1(b)(ii)6 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)By writing f(x) = x^2 [x^2 - 9x + 28 - 36/x + 16/x^2] and using the result from (b)(i) above, show that if f(x) = 0, then u^2 - 9u + 20 = 0.
  69. 1(b)(ii)5 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)Hence, factorize f(x) = x³ + px² – x + q completely.
  70. 1(b)(ii)a)4 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)a)find the value of a
  71. 1(b)(ii)b)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)b)factorize f(x) completely.
  72. 1(b)(iii)7 marks· Pure Mathematics · Unit 1 Q1 1(b)(iii)Hence, determine the values of x ∈ R for which f(x) = 0.
  73. 1(c)4 marks· Pure Mathematics · Unit 1 Q1 1(c)Find the value of r such that x⁴ - 7x + 4r has a remainder -2 when it is divided by x + 3.
  74. 1(c)7 marks· Pure Mathematics · Unit 1 Q1 1(c)Calculate the values of a, b and c.
  75. 1(c)7 marks· Pure Mathematics · Unit 1 Q1 1(c)Calculate the values of a and b.
  76. 1(c)(i)4 marks· Pure Mathematics · Unit 1 Q1 1(c)(i)Show that a = 2 and b = -30.
  77. 1(c)(ii)9 marks· Pure Mathematics · Unit 1 Q1 1(c)(ii)Hence, solve ax³ + 9x² – 11x + b = 0.
  78. 1(d)(i)2 marks· Pure Mathematics · Unit 1 Q1 1(d)(i)Given that (x + 1) is a factor of f(x), show that p = 6.
  79. 1(d)(ii)4 marks· Pure Mathematics · Unit 1 Q1 1(d)(ii)Factorise f(x) completely.
  80. 1(d)(iii)3 marks· Pure Mathematics · Unit 1 Q1 1(d)(iii)Hence, or otherwise, solve f(x) = 0.
  81. 2(a)(iii)3 marks· Pure Mathematics · Unit 1 Q2 2(a)(iii)Express f(x) in the form u(x - v)² + w, where u, v, w ∈ R.
  82. 2(b)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Find the value of p for which the system has an infinite number of solutions.
  83. 2(b)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Express p and q in terms of α and β.
  84. 2(b)(i)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)Write down the solution set of the inequality x² (3 - x) ≤ 0.
  85. 2(b)(i)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(i)write down the values of α + β and αβ
  86. 2(b)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Find the solutions for this value of p.
  87. 2(b)(ii)4 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Find the values of α and β.
  88. 2(b)(ii)3 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)Given that the equation x² (3 - x) = k has three real solutions for x, write down the set of possible values for k.
  89. 2(b)(ii)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(ii)find the value of α² + β²
  90. 2(b)(iii)2 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)Hence, determine the values of p and q.
  91. 2(b)(iii)5 marks· Pure Mathematics · Unit 1 Q2 2(b)(iii)obtain a quadratic equation whose roots are 2/α² and 2/β².
  92. 2(c)(i)4 marks· Pure Mathematics · Unit 1 Q2 2(c)(i)3x² + 4x +1 ≤ 5
  93. 3(b)(i)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)Calculate the range of values of k for which the equation f(x) = 0 has no real roots.
  94. 3(b)(ii)3 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)Solve the equation f(x) = 0 for k = 1, giving your answer in the form a ± bi, where a, b ∈ R.
  95. 3(b)(iii)6 marks· Pure Mathematics · Unit 1 Q3 3(b)(iii)Let α and β be roots of f(x) = 0 when k = 8. Without first solving f(x) = 0, determine the equation whose roots are respectively 1/α and 1/β.
  96. 4(a)8 marks· Pure Mathematics · Unit 1 Q4 4(a)Find the two square roots of the complex number 5 - 12i in the form x + yi, where x, y ∈ R.
  97. 4(a)8 marks· Pure Mathematics · Unit 1 Q4 4(a)Express the complex number, w = (z-1)/(z+2), in a similar form.
  98. 4(b)(i)4 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Find the equation connecting x and y in the form ax² + by² + cx + dy + f = 0 where a, b, c, d, f are integers.
  99. 4(b)(i)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)If z = x + yi, where x, y ∈ R, y ≠ 0, find the real and imaginary parts of z + 1/z.
  100. 4(b)(ii)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Find and identify the locus of the points for which the imaginary part of z + 1/z is zero.
  101. 4(c)4 marks· Pure Mathematics · Unit 1 Q4 4(c)Find the modulus of the complex number z = (25(2+3i))/(4+3i).
  102. 6(a)3 marks· Pure Mathematics · Unit 1 Q6 6(a)Find g(0), g(1) and g(-1).
  103. 6(a)5 marks· Pure Mathematics · Unit 1 Q6 6(a)Find the values of x for which h(x) = 0.
  104. 6(c)4 marks· Pure Mathematics · Unit 1 Q6 6(c)Show that the curve f(x) touches the x-axis at x = 1.
  105. 6(c)(i)5 marks· Pure Mathematics · Unit 1 Q6 6(c)(i)Show that the volume, V cm³, of the tray is given by V = 4(x³ - 13x² + 40x).

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