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Mathematics

UPCAT Mathematics practice questions

50 free UPCAT Mathematics practice questions with the correct answer and a worked explanation for every item. Read through them or run the section as a shuffled quiz.

  • 18 easy
  • 21 medium
  • 11 hard
Take the Mathematics quiz

Instructions: Solve each item and choose the best answer. Calculators are not required.

All 50 questions

Answers shown on each page

  1. 1Solve for x: 3(x − 4) = 2x + 5
    Answer: C. x = 17    Algebra
  2. 2Simplify: (2x²y)³
    Answer: A. 8x⁶y³    Algebra
  3. 3What is 15% of 240?
    Answer: B. 36    Arithmetic
  4. 4A shirt sells for ₱480 after a 20% discount. What was its original price?
    Answer: D. ₱600    Arithmetic
  5. 5What is the slope of the line passing through (2, −3) and (6, 5)?
    Answer: B. 2    Coordinate Geometry
  6. 6What are the roots of x² − 5x + 6 = 0?
    Answer: B. 2 and 3    Algebra
  7. 7If f(x) = 2x² − 3x + 1, what is f(−2)?
    Answer: D. 15    Algebra
  8. 8The average of five numbers is 18. If one of them, 22, is removed, what is the average of the remaining four?
    Answer: D. 17    Statistics
  9. 9A circle has a circumference of 12π. What is its area?
    Answer: A. 36π    Geometry
  10. 10The angles of a triangle are in the ratio 2 : 3 : 5. What is the measure of the largest angle?
    Answer: A. 90°    Geometry
  11. 11A right triangle has legs measuring 9 cm and 12 cm. What is the length of its hypotenuse?
    Answer: C. 15 cm    Geometry
  12. 12What is the value of sin 30°?
    Answer: A. 1/2    Trigonometry
  13. 13A quantity is increased by 20% and the result is then decreased by 20%. Compared with the original, the final value is
    Answer: A. 4% smaller    Arithmetic
  14. 14Two fair six-sided dice are rolled. What is the probability that the sum of the faces is 7?
    Answer: B. 1/6    Probability
  15. 15What is the 20th term of the arithmetic sequence 7, 11, 15, 19, ...?
    Answer: D. 83    Sequences
  16. 16What is the 8th term of the geometric sequence 3, 6, 12, 24, ...?
    Answer: C. 384    Sequences
  17. 17Solve the system: 2x + y = 11 and x − y = 1
    Answer: C. x = 4, y = 3    Algebra
  18. 18Simplify: √72
    Answer: B. 6√2    Algebra
  19. 19Expand: (x + 3)(x − 3)
    Answer: B. x² − 9    Algebra
  20. 20Factor completely: x² + 7x + 12
    Answer: A. (x + 3)(x + 4)    Algebra
  21. 21Pipe A can fill a tank in 6 hours and pipe B can fill the same tank in 12 hours. Working together, how long will they take to fill it?
    Answer: C. 4 hours    Word Problems
  22. 22A bus travels at a constant 60 km/h for 2.5 hours. How far does it go?
    Answer: C. 150 km    Word Problems
  23. 23A car leaves a terminal at 60 km/h. One hour later a second car leaves the same terminal along the same road at 80 km/h. How long after the second car departs will it overtake the first?
    Answer: B. 3 hours    Word Problems
  24. 24How much pure water must be added to 10 liters of a 30% acid solution to dilute it to 20% acid?
    Answer: A. 5 liters    Word Problems
  25. 2540 is what percent of 160?
    Answer: B. 25%    Arithmetic
  26. 26What is the volume of a cube with an edge of 4 cm?
    Answer: C. 64 cm³    Geometry
  27. 27What is the total surface area of a cube with an edge of 5 cm?
    Answer: D. 150 cm²    Geometry
  28. 28What is the volume of a cylinder with radius 3 cm and height 10 cm?
    Answer: A. 90π cm³    Geometry
  29. 29What is the median of the data set 4, 7, 9, 12, 15, 20?
    Answer: D. 10.5    Statistics
  30. 30What is the mode of the data set 3, 5, 5, 7, 8, 8, 8, 9?
    Answer: C. 8    Statistics
  31. 31Solve for x: |2x − 5| = 9
    Answer: B. x = 7 or x = −2    Algebra
  32. 32Solve the inequality: 3x − 4 < 11
    Answer: D. x < 5    Algebra
  33. 33If a : b = 3 : 4 and b : c = 6 : 7, what is a : c?
    Answer: A. 9 : 14    Arithmetic
  34. 34How much simple interest is earned on ₱20,000 invested at 6% per year for 3 years?
    Answer: C. ₱3,600    Word Problems
  35. 35₱5,000 is invested at 10% compounded annually. What is its value after 2 years?
    Answer: D. ₱6,050    Word Problems
  36. 36Six people at a meeting each shake hands once with every other person. How many handshakes occur in total?
    Answer: A. 15    Combinatorics
  37. 37How many distinguishable arrangements can be made from the letters of the word LEVEL?
    Answer: A. 30    Combinatorics
  38. 38A box contains 5 red marbles and 3 blue marbles. If one marble is drawn at random, what is the probability it is red?
    Answer: B. 5/8    Probability
  39. 39What is the slope of any line perpendicular to y = 2x + 1?
    Answer: B. −1/2    Coordinate Geometry
  40. 40What is the vertex of the parabola y = x² − 6x + 5?
    Answer: C. (3, −4)    Algebra
  41. 41What is the domain of f(x) = 1 / (x − 3)?
    Answer: D. All real numbers except 3    Algebra
  42. 42Evaluate: log₂ 32
    Answer: B. 5    Algebra
  43. 43Solve for x: 2^(x + 1) = 32
    Answer: A. x = 4    Algebra
  44. 44What is the sum of the interior angles of a hexagon?
    Answer: B. 720°    Geometry
  45. 45What is the measure of each interior angle of a regular pentagon?
    Answer: A. 108°    Geometry
  46. 46In a circle of radius 6 cm, what is the length of an arc subtending a central angle of 60°?
    Answer: C. 2π cm    Geometry
  47. 47Three consecutive integers have a sum of 51. What is the largest of the three?
    Answer: D. 18    Word Problems
  48. 48Maria is three times as old as her son. In 12 years she will be twice as old as he is then. How old is Maria now?
    Answer: D. 36    Word Problems
  49. 49Rationalize the denominator and simplify: 6 / √3
    Answer: D. 2√3    Algebra
  50. 50A trapezoid has parallel sides measuring 8 cm and 12 cm, with a height of 5 cm. What is its area?
    Answer: C. 50 cm²    Geometry

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