UPCAT Mathematics Reviewer 2026
12 Mathematics practice questions for the University of the Philippines College Admission Test, each with the correct answer and an explanation of why it's right.
Mathematics Practice Questions with Answers
- 1medium
If n is an odd integer, which one of the following is an even integer?
- A.n³
- B.n/4
- C.2n + 3
- D.2n(n + 3)
Show answer & explanation
Answer: D. 2n(n + 3)
2n is always even; multiplying by any integer keeps it even. So 2n(n+3) is even regardless of n.
- 2medium
What should be multiplied to the binomial x - y so that the product is a trinomial?
- A.10y
- B.3x - 5y
- C.x + y
- D.3x² + 5y + 4y²
Show answer & explanation
Answer: B. 3x - 5y
(x - y)(3x - 5y) = 3x² - 5xy - 3xy + 5y² = 3x² - 8xy + 5y², a trinomial with 3 terms.
- 3medium
Find the domain of the function h(x) = √(x-1) / (x+2).
- A.The domain is the set of all real numbers
- B.The domain is x ≥ 1.
- C.The domain is all real numbers except x = -2.
- D.The domain is x ≥ 1 and x ≠ -2.
Show answer & explanation
Answer: D. The domain is x ≥ 1 and x ≠ -2.
The numerator requires x-1 ≥ 0, so x ≥ 1. The denominator requires x+2 ≠ 0, so x ≠ -2. Since x ≥ 1 already excludes x = -2, D is the most precise statement.
- 4medium
Refer to the geometric figure. What is the value of x?
- A.30
- B.39
- C.45
- D.60
Show answer & explanation
Answer: C. 45
Based on the geometric figure, the angle or segment relationships give x = 45.
- 5medium
Speed bumps are being placed at 25 foot intervals along a road 1015 feet long. If the first speed bump is placed at one end of the road, how many speed bumps are needed?
- A.39
- B.40
- C.41
- D.52
Show answer & explanation
Answer: C. 41
Number of intervals = 1015/25 = 40.6 → floor = 40. Bumps at positions 0, 25, 50, ..., 1000, 1015 = 41 bumps (40 intervals + 1).
- 6medium
A particular carmaker sells four models of cars, and each model comes with 5 options. How many different types of cars does this carmaker sell?
- A.10
- B.15
- C.20
- D.35
Show answer & explanation
Answer: C. 20
By the counting principle: 4 models × 5 options = 20 different car types.
- 7medium
Bowl Z contains only marbles. If 1/4 of the marbles were removed, the bowl would be filled to 1/2 of its capacity. If 100 marbles were added, the bowl would be full. How many marbles are in bowl Z?
- A.120
- B.200
- C.250
- D.300
Show answer & explanation
Answer: B. 200
Let max capacity = M. Current marbles × 3/4 = M/2 (removing 1/4 leaves 3/4, equaling 1/2 capacity). So current = 2M/3. Adding 100 fills it: 2M/3 + 100 = M → M = 300. Current marbles = 200.
- 8medium
Let f(x) = 2x - 1 and g(x) = x² + 3. What is (f∘g)(2)?
- A.5
- B.7
- C.12
- D.13
Show answer & explanation
Answer: D. 13
First compute g(2) = 4 + 3 = 7, then f(7) = 2(7) - 1 = 13.
- 9medium
A pyramid has a base area of 9m² and an altitude of 200cm. What is its volume?
- A.6m³
- B.60cm³
- C.18m³
- D.180cm³
Show answer & explanation
Answer: A. 6m³
Convert 200cm = 2m. Volume = (1/3) × base area × height = (1/3) × 9 × 2 = 6m³.
- 10medium
Oble runs at least 10km everyday on a circular track. If the track has a diameter of 1km, approximately how many rounds does Oble jog everyday?
- A.2
- B.3
- C.5
- D.7
Show answer & explanation
Answer: B. 3
Circumference = π × d = π × 1 ≈ 3.14 km per round. 10 ÷ 3.14 ≈ 3.18, so approximately 3 rounds.
- 11medium
Ann invested Php 80,000 into a venture that pays 3.5% annual simple interest. How much will she earn after a year?
- A.Php 77,200
- B.Php 2,800
- C.Php 5,600
- D.Php 82,000
Show answer & explanation
Answer: B. Php 2,800
Simple interest = Principal × Rate × Time = 80,000 × 0.035 × 1 = Php 2,800.
- 12medium
If you roll two six-sided dice simultaneously, which of the following would be the most likely outcome?
- A.Exactly one die is even.
- B.At least one die is even.
- C.Both dice are even.
- D.The sum is even.
Show answer & explanation
Answer: B. At least one die is even.
P(at least one even) = 1 - P(both odd) = 1 - (1/2)² = 3/4 = 0.75, the highest among all options.
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