TMUA 2025 Mathematics Tmua.fyi
10 questions10 marks20Updated August 2026
The TMUA 2025 Mathematics Tmua.fyi paper in full: all 10 questions, each with its answer. TMUA is the Test of Mathematics for University Admission. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.
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Question 1
1 markEvaluate the integral of (3x⁵ − 4x³ + 5x + 2) with respect to x, from x = −2 to x = 2.
- A.0
- B.4
- C.8
- D.16
- E.40
- F.88
Answer: C
Question 2
1 markThe line y = 2x + c does not meet the curve y = x² − 3x + 7. What is the complete range of values of c?
- A.c < 3/4
- B.c > 3/4
- C.c < −3/4
- D.c > −3/4
- E.−3/4 < c < 3/4
- F.c < 7
Answer: A
Question 3
1 markFind the sum of all real solutions of 2^(2x) − 5(2^x) + 4 = 0.
- A.0
- B.1
- C.2
- D.3
- E.4
- F.log₂5
Answer: C
Question 4
1 markHow many solutions does 2(sin x)² = sin x + 1 have in the interval 0 ≤ x < 2π?
- A.1
- B.2
- C.3
- D.4
- E.5
- F.6
Answer: C
Question 5
1 markIn an arithmetic series, the sum of the first 10 terms equals the sum of the first 20 terms. What is the sum of the first 30 terms?
- A.0
- B.15d
- C.−300
- D.30a
- E.it depends on a and d
- F.300
Answer: A
Question 6
1 markThe curve y = x³ − 6x² + 9x + 1 has two stationary points. By how much does the y-coordinate of the local maximum exceed that of the local minimum?
- A.2
- B.4
- C.5
- D.6
- E.8
- F.27
Answer: B
Question 7
1 marka and b are real numbers. Consider two statements. R: a + b > 0. S: a > 0 and b > 0. Which is true?
- A.R is necessary and sufficient for S
- B.R is necessary but not sufficient for S
- C.R is sufficient but not necessary for S
- D.R is neither necessary nor sufficient for S
Answer: B
Question 8
1 markA student tries to prove: 'For all real numbers a and b, if a² = b² then a = b.' The attempt is: Step I: suppose a² = b². Step II: then a² − b² = 0. Step III: so (a − b)(a + b) = 0. Step IV: therefore a − b = 0. Step V: hence a = b. Which best describes the attempt?
- A.it is completely correct
- B.first error is at step I
- C.first error is at step II
- D.first error is at step III
- E.first error is at step IV
- F.first error is at step V
Answer: E
Question 9
1 markConsider the claim: 'For every integer n, if n² is even then n is even.' Which of the following would, on its own, be a valid proof of this claim? I: checking that it holds for n = 2, 4 and 6. II: assuming n is odd and deducing that n² is odd (a contradiction). III: proving the contrapositive, that if n is odd then n² is odd. Which is/are valid?
- A.none
- B.I only
- C.II only
- D.III only
- E.I and II
- F.I and III
- G.II and III
- H.I, II and III
Answer: G
Question 10
1 markWhich of the following statements about positive integers is/are true? 1: if n is prime and n > 3, then n = 6k + 1 or n = 6k − 1 for some integer k. 2: if n = 6k + 1 or n = 6k − 1 for some integer k, then n is prime. 3: if n and n + 2 are both prime and n > 3, then n = 6k − 1 for some integer k.
- A.none
- B.1 only
- C.2 only
- D.3 only
- E.1 and 2
- F.1 and 3
- G.2 and 3
- H.1, 2 and 3
Answer: F