National Certificate of Educational Achievement · Mathematics and Statistics

Algebraic Procedures, Quadratic and Exponential Equations and Graphs

40 practice questions, each with the answer and an explanation.

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All 40 questions

  1. What are the solutions to the quadratic equation x^2 - 7x + 12 = 0?
  2. Simplify the algebraic fraction (x^2 - 9) / (x^2 + 5x + 6).
  3. What is the discriminant of the quadratic equation 2x^2 + 5x - 3 = 0, and what does it indicate about the roots?
  4. Express 2^(3x - 1) = 32 as a linear equation and solve for x.
  5. What are the coordinates of the vertex of the parabola y = (x - 4)^2 - 9?
  6. Solve the simultaneous equations: y = 2x + 1 and x + y = 7.
  7. Which expression represents 5x^(-2) with positive indices?
  8. What are the x-intercepts of the function f(x) = 2x^2 - 8x?
  9. Solve the inequality 3 - 2x <= 11 for x.
  10. A population of kiwi on an offshore island sanctuary grows exponentially according to N(t) = 50 * 2^(0.1t), where t is in years. What is the kiwi population when t = 20 years?
  11. Expand and simplify the algebraic expression (3x - 2)(2x + 5).
  12. Complete the square for the quadratic expression x^2 + 6x - 7.
  13. If log_2(x) + log_2(x - 2) = 3, what is the valid real value of x?
  14. What is the equation of the line passing through points (2, 5) and (6, 13)?
  15. What is the solution set for the quadratic inequality x^2 - 5x - 6 < 0?
  16. Simplify (3a^4 b^2)^3 / (9a^5 b^4).
  17. What is the vertex form of the parabola y = 2x^2 - 12x + 11?
  18. Solve for x: 9^x - 10 * 3^x + 9 = 0.
  19. What are the asymptotes of the rational function f(x) = (3x + 2) / (x - 4)?
  20. If a line is perpendicular to y = -3x + 4 and passes through (6, 2), what is its equation?
  21. For what values of k does the quadratic equation x^2 + (k - 2)x + (k + 1) = 0 have exactly one repeated real root?
  22. Solve the non-linear simultaneous system: y = x^2 - 3x - 4 and y = 2x + 2. What are the points of intersection?
  23. If 2^(2x) - 5 * 2^(x + 1) + 24 = 0, what are the exact real values of x?
  24. The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is 5, and the product of the roots is -14. What is the quadratic equation?
  25. Find the domain and range of the function f(x) = sqrt(16 - x^2).
  26. For what values of m does the line y = mx - 5 not intersect the parabola y = x^2 + 3x + 4?
  27. Solve the fractional rational equation: 2 / (x - 3) + 1 / (x + 1) = 5 / (x^2 - 2x - 3).
  28. Solve for x: (2x - 1) / 3 - (x + 2) / 4 = 1.
  29. Simplify the radical expression sqrt(75) - sqrt(27) + sqrt(12).
  30. What is the value of x if 5^(x + 2) = 125^(2x - 1)?
  31. If f(x) = 3x - 5, what is the inverse function f^(-1)(x)?
  32. Factorise fully: 16x^4 - 81.
  33. What is the range of the quadratic function g(x) = -2(x + 1)^2 + 8?
  34. Solve for x: |2x - 5| = 9.
  35. Which of the following functions has a graph that is symmetrical about the y-axis?
  36. Solve the quadratic equation x^2 - 5x + 6 = 0.
  37. Solve the exponential equation 2^(x + 1) = 32.
  38. For which values of k does the equation x^2 + kx + 9 = 0 have exactly one real solution?
  39. Simplify (x^2 - 9) / (x + 3), for x not equal to -3.
  40. A town of 2000 people grows by 5% each year. To the nearest whole number, what is its population after 3 years?

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