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Form 5 Quadratic Inequalities and Linear Programming Practice

Form 5 · 60 practice questions with instant marking and worked solutions.

Quadratic Inequalities and Linear Programming is an important part of the Form 5 mathematics curriculum. This topic develops inequality direction, solution sets, graphs and constraints. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Quadratic Inequalities and Linear Programming has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Quadratic Inequalities and Linear Programming; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 8 of them are DSE Paper 1 style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Quadratic Inequalities and Linear Programming set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Quadratic Inequalities and Linear Programming should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    Solve $3x+5<32$ and give the boundary value of the solution set. Choose the correct answer.
  2. 2 Basic
    Solve $4x+4<16$ and give the boundary value of the solution set. Give the numerical value only.

  3. 3 Basic
    Solve $5x+5<25$ and give the boundary value of the solution set. Show the essential working and give the numerical value.

  4. 4 Basic
    The boundary value for $6x+4<34$ is 6. Is this true?
  5. 5 Basic
    Consider the inequality $x^{2}+11x+28<0$. Find the number of integers satisfying it. Choose the correct answer.
  6. 6 Basic
    Consider the inequality $x^{2}-5x-14\le 0$. Find the largest integer satisfying it. Give the answer only.

  7. 7 Basic
    The equation $x^2+kx+6=0$ has no real roots and $k$ is an integer. Find the greatest possible value of $k$. Show the essential working and give the answer.

  8. 8 Basic
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(9,0)$, $(6,5)$, $(0,6)$ (boundary included). The maximum value of $P=6x+3y$ is 18. Is this true?
  9. 9 Basic
    Consider the inequality $x^{2}+3x-10<0$. Find the number of integers satisfying it. Choose the correct answer.
  10. 10 Basic
    Consider the inequality $x^{2}+x-2\le 0$. Find the largest integer satisfying it. Give the answer only.

  11. 11 Basic
    The equation $x^2+kx+15=0$ has no real roots and $k$ is an integer. Find the greatest possible value of $k$. Show the essential working and give the answer.

  12. 12 Basic
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(6,0)$, $(5,4)$, $(0,5)$ (boundary included). The maximum value of $P=7x+6y$ is 30. Is this true?
  13. 13 Standard
    Solve $2x+8<22$ and give the boundary value of the solution set. Choose the correct answer.
  14. 14 Standard
    Solve $3x+7<31$ and give the boundary value of the solution set. Give the numerical value only.

  15. 15 Standard
    Solve $4x+8<44$ and give the boundary value of the solution set. Show the essential working and give the numerical value.

  16. 16 Standard
    The boundary value for $5x+7<57$ is 10. Is this true?
  17. 17 Standard
    Consider the quadratic equation $x^{2}-9x+1=0$. Find the value of the discriminant $\Delta$. Choose the correct answer.
  18. 18 Standard
    Consider the inequality $x^{2}+3x<0$. Find the number of integers satisfying it. Give the answer only.

  19. 19 Standard
    Consider the inequality $x^{2}+4x+3\le 0$. Find the largest integer satisfying it. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(8,0)$, $(2,6)$ and $(0,6)$. $P=5x+8y$. Answer each part; give the answer only.
    1. (a) Find $P$ at $(8,0)$.
    2. (b) Find $P$ at $(2,6)$.
    3. (c) Find the maximum value of $P$.
  21. 21 Standard
    Consider the quadratic equation $x^{2}-9x-8=0$. Find the value of the discriminant $\Delta$. Choose the correct answer.
  22. 22 Standard
    Consider the inequality $x^{2}-9x-10<0$. Find the number of integers satisfying it. Give the answer only.

  23. 23 Standard
    Consider the inequality $x^{2}+10x+24\le 0$. Find the largest integer satisfying it. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(7,0)$, $(3,8)$ and $(0,8)$. $P=6x+8y$. Answer each part; give the answer only.
    1. (a) Find $P$ at $(7,0)$.
    2. (b) Find $P$ at $(3,8)$.
    3. (c) Find the maximum value of $P$.
  25. 25 Advanced
    Solve $6x+11<41$ and give the boundary value of the solution set. Choose the correct answer.
  26. 26 Advanced
    Solve $2x+10<22$ and give the boundary value of the solution set. Give the numerical value only.

  27. 27 Advanced
    Solve $3x+11<32$ and give the boundary value of the solution set. Show the essential working and give the numerical value.

  28. 28 Advanced
    The boundary value for $4x+10<42$ is 9. Is this true?
  29. 29 Advanced
    The equation $x^2+kx+5=0$ has no real roots and $k$ is an integer. Find the greatest possible value of $k$. Choose the correct answer.
  30. 30 Advanced
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(5,0)$, $(2,5)$, $(0,6)$ (boundary included). Find the maximum value of $P=6x+5y$. Give the answer only.

  31. 31 Advanced
    Consider the quadratic equation $x^{2}+5x+12=0$. The value of the discriminant $\Delta$ is -23. Is this true?
  32. 32 Advanced DSE long question
    It is given that $f(x)=x^2+bx+c$, where the roots of $f(x)=0$ are $-2$ and $1$. Answer each part; give the answer only.
    1. (a) Find $b$.
    2. (b) Find $c$.
    3. (c) Find $f(5)$.
    4. (d) Find the minimum value of $f(x)$ (as a fraction in lowest terms).
  33. 33 Advanced
    The equation $x^2+kx+14=0$ has no real roots and $k$ is an integer. Find the greatest possible value of $k$. Choose the correct answer.
  34. 34 Advanced
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(7,0)$, $(5,4)$, $(0,4)$ (boundary included). Find the maximum value of $P=6x+4y$. Give the answer only.

  35. 35 Advanced
    Consider the quadratic equation $3x^{2}-6x=0$. The value of the discriminant $\Delta$ is 36. Is this true?
  36. 36 Advanced DSE long question
    It is given that $f(x)=x^2+bx+c$, where the roots of $f(x)=0$ are $-2$ and $4$. Answer each part; give the answer only.
    1. (a) Find $b$.
    2. (b) Find $c$.
    3. (c) Find $f(3)$.
    4. (d) Find the minimum value of $f(x)$.
  37. 37 Challenging
    Solve $5x+14<64$ and give the boundary value of the solution set. Choose the correct answer.
  38. 38 Challenging
    Solve $6x+13<79$ and give the boundary value of the solution set. Give the numerical value only.

  39. 39 Challenging
    Solve $2x+14<38$ and give the boundary value of the solution set. Show the essential working and give the numerical value.

  40. 40 Challenging
    The boundary value for $3x+13<31$ is 6. Is this true?
  41. 41 Challenging
    Consider the inequality $x^{2}-8x-20<0$. Find the number of integers satisfying it. Choose the correct answer.
  42. 42 Challenging
    Consider the inequality $x^{2}+14x+48\le 0$. Find the largest integer satisfying it. Show the essential working and give the answer.

  43. 43 Challenging
    The equation $x^2+kx+3=0$ has no real roots and $k$ is an integer. The greatest possible value of $k$ is 4. Is this true?
  44. 44 Challenging DSE long question
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(8,0)$, $(6,9)$ and $(0,9)$. $P=6x+7y$. Answer each part; give the answer only.
    1. (a) Find $P$ at $(8,0)$.
    2. (b) Find $P$ at $(6,9)$.
    3. (c) Find the maximum value of $P$.
  45. 45 Challenging
    Consider the inequality $x^{2}+x-12<0$. Find the number of integers satisfying it. Choose the correct answer.
  46. 46 Challenging
    Consider the inequality $x^{2}-2x-3\le 0$. Find the largest integer satisfying it. Show the essential working and give the answer.

  47. 47 Challenging
    The equation $x^2+kx+11=0$ has no real roots and $k$ is an integer. The greatest possible value of $k$ is 7. Is this true?
  48. 48 Challenging DSE long question
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(7,0)$, $(3,7)$ and $(0,7)$. $P=8x+3y$. Answer each part; give the answer only.
    1. (a) Find $P$ at $(7,0)$.
    2. (b) Find $P$ at $(3,7)$.
    3. (c) Find the maximum value of $P$.
  49. 49 Elite
    Solve $4x+17<49$ and give the boundary value of the solution set. Choose the correct answer.
  50. 50 Elite
    Solve $5x+16<61$ and give the boundary value of the solution set. Give the numerical value only.

  51. 51 Elite
    Solve $6x+17<77$ and give the boundary value of the solution set. Show the essential working and give the numerical value.

  52. 52 Elite
    The boundary value for $2x+16<38$ is 12. Is this true?
  53. 53 Elite
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(9,0)$, $(7,5)$, $(0,5)$ (boundary included). Find the maximum value of $P=6x+2y$. Give the answer only.

  54. 54 Elite
    Consider the quadratic equation $x^{2}-9=0$. Find the value of the discriminant $\Delta$. Show the essential working and give the answer.

  55. 55 Elite
    Consider the inequality $x^{2}-3x-10<0$. The number of integers satisfying it is 6. Is this true?
  56. 56 Elite DSE long question
    It is given that $f(x)=x^2+bx+c$, where the roots of $f(x)=0$ are $-5$ and $2$. Answer each part; give the answer only.
    1. (a) Find $b$.
    2. (b) Find $c$.
    3. (c) Find $f(-2)$.
    4. (d) Find the minimum value of $f(x)$ (as a fraction in lowest terms).
  57. 57 Elite
    The feasible region is the convex quadrilateral with vertices $(0,0)$, $(4,0)$, $(2,4)$, $(0,4)$ (boundary included). Find the maximum value of $P=7x+3y$. Give the answer only.

  58. 58 Elite
    Consider the quadratic equation $3x^{2}+6x-9=0$. Find the value of the discriminant $\Delta$. Show the essential working and give the answer.

  59. 59 Elite
    Consider the inequality $x^{2}+2x-15<0$. The number of integers satisfying it is 7. Is this true?
  60. 60 Elite DSE long question
    It is given that $f(x)=x^2+bx+c$, where the roots of $f(x)=0$ are $-5$ and $2$. Answer each part; give the answer only.
    1. (a) Find $b$.
    2. (b) Find $c$.
    3. (c) Find $f(-1)$.
    4. (d) Find the minimum value of $f(x)$ (as a fraction in lowest terms).

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Quadratic Inequalities and Linear Programming

This topic needs particular care with inequality direction, solution sets, graphs and constraints. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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