Cubic Functions and Equations · CAPE Pure Mathematics Unit 1

31 past-paper questions on Cubic Functions and Equations, part of Basic Algebra and Functions, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

1. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1If \alpha and \beta represent roots of the equation x^2 - px + q = 0, then the value of \alpha^2 + \beta^2 is
2. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The general quadratic equation with roots \alpha and \beta may be written as
3. 2(d)6 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2The roots of the equation 2x^3 - x^2 + 3x - 1 = 0 are \alpha, \beta, and \gamma. Given that \alpha\beta + \alpha\gamma + \beta\gamma = \frac{3}{2} and…
4. Q111 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Which of the following is true if \alpha, \beta and \gamma are roots of the cubic equation 3x^3 - 4x^2 - 7x - 10 = 0?
5. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1Given that \alpha, 2\alpha and 3\alpha are the roots of the equation x^3 + kx^2 + 48 = 0, the value of the constant k is
6. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The cubic equation whose roots \alpha, \beta and \gamma satisfy the following \alpha + \beta + \gamma = \frac{-2}{7}, \alpha\beta + \beta\gamma + \gamma\alpha = -1 and \alpha\beta\gamma = -\frac{3}{7} is
7. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1If \alpha, \beta and \gamma are the three roots of the cubic equation x^3 - 2x^2 - 4x + 5 = 0, then the value of \alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2 is
8. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If \alpha and \beta represent roots of the equation x^2 - px + q = 0, then the value of \alpha^2 + \beta^2 is
9. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Which of the following is true if \alpha, \beta and \gamma are roots of the cubic equation 3x^3 - 4x^2 - 7x - 10 = 0?
10. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The cubic equation whose roots \alpha, \beta and \gamma satisfy the following \alpha + \beta + \gamma = \frac{-2}{7} \alpha\beta + \beta\gamma + \gamma\alpha = -1 \alpha\beta\gamma = -\frac{3}{7} is
11. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1Given that \alpha, 2\alpha and 3\alpha are the roots of the equation x^3 + kx^2 + 48 = 0, the value of the constant k is
12. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The general quadratic equation with roots \alpha and \beta may be written as
13. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Two roots of the cubic equation 2x^3 + 3x^2 - 5x - 6 are -1 and -2. The THIRD root is
14. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1Which of the following is true if \alpha, \beta and \gamma are roots of the cubic equation 3x^3 - 4x^2 - 7x - 10 = 0?
15. Q101 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The cubic equation whose roots \alpha, \beta and \gamma satisfy the following conditions \alpha + \beta + \gamma = \frac{-2}{7},\ \alpha\beta + \beta\gamma + \gamma\alpha = -1\ \text{and}\ \alpha\beta\gamma =…
16. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1Two roots of the cubic equation 2x^3 + 3x^2 - 5x - 6 are -1 and -2. The THIRD root is
17. 2(b)8 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2The roots of the cubic equation 3x^3 - x^2 - 2x + 1 = 0 are \alpha, \beta and \gamma. Determine the equation whose roots are 1/\alpha, 1/\beta and 1/\gamma.
18. 2(a)8 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Determine the cubic equation with roots (alpha - 1), (beta - 1) and (gamma - 1), given that (alpha - 1)(beta - 1) + (alpha - 1)(gamma - 1) + (beta - 1)(gamma - 1) = 12.
19. 1(a)7 marks· Pure Mathematics · Unit 1 Q1 1(a)Find the values of the real constants p, q and r.
20. 1(a)(ii)7 marks· Pure Mathematics · Unit 1 Q1 1(a)(ii)The roots of the cubic equation x³ - 15x² + px – 105 = 0 are 5 - k, 5 and 5 + k. Find the values of the constants p and k.
21. 1(b)(i)2 marks· Pure Mathematics · Unit 1 Q1 1(b)(i)state the values of α + β and αβ
22. 1(b)(ii)3 marks· Pure Mathematics · Unit 1 Q1 1(b)(ii)find the value of α² + β²
23. 1(b)(iii)7 marks· Pure Mathematics · Unit 1 Q1 1(b)(iii)find the equation whose roots are 1 + 1/α and 1 + 1/β
24. 1(b)(iii)2 marks· Pure Mathematics · Unit 1 Q1 1(b)(iii)Find the x-coordinate of the point Q.
25. 1(c)10 marks· Pure Mathematics · Unit 1 Q1 1(c)Solve the equation x³ – 6x² – 69x + 154 = 0.
26. 2(c)8 marks· Pure Mathematics · Unit 1 Q2 2(c)Determine the equation whose roots are 1/(αβ), 1/(αγ) and 1/(βγ).
27. 2(c)8 marks· Pure Mathematics · Unit 1 Q2 2(c)Given that α, β and γ are the roots of the equation x³ + 3x + 2 = 0, form an equation whose roots are βγ, αγ and αβ.
28. 2(c)8 marks· Pure Mathematics · Unit 1 Q2 2(c)Determine the equation whose roots are 1/α, 1/β, 1/γ.
29. 2(c)(i)3 marks· Pure Mathematics · Unit 1 Q2 2(c)(i)State the values of α + β + γ, αβ + αγ + βγ and αβγ.
30. 2(c)(ii)8 marks· Pure Mathematics · Unit 1 Q2 2(c)(ii)Hence, or otherwise, determine an equation with integer coefficients which has roots 1/α², 1/β² and 1/γ². Note: (αβ)² + (αγ)² + (βγ)² = (αβ + αγ + βγ)² – 2αβγ (α + β + γ) and α² + β² + γ² = (α + β + γ)² – 2 (αβ + αγ +…
31. 4(b)(ii)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Find the quadratic equation whose roots are z₁ and z₂.