Test of Mathematics for University Admission (TMUA) — Questions and Answers

Question 1: A researcher finds a p-value of 0.03 in a two-tailed test. At α = 0.05, what is the correct interpretation?

There is a 3% chance the null hypothesis is true
Fail to reject H₀; the result is not significant
The alternative hypothesis is proven correct
Reject H₀; the result is statistically significant (Correct answer)

Correct answer: Reject H₀; the result is statistically significant

Since p = 0.03 < α = 0.05, we reject H₀ and conclude the result is statistically significant at the 5% level.

Question 2: Which of the following is NOT a perfect square?

1521
1600
1700 (Correct answer)
1444

Correct answer: 1700

38²=1444, 39²=1521, 40²=1600, but 1700 is not a perfect square since √1700 ≈ 41.23 is not an integer.

Question 3: Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.4. What is P(A ∪ B)?

0.70 (Correct answer)
0.58
1.00
0.12

Correct answer: 0.70

For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.3 + 0.4 = 0.7.

Question 4: A bag contains 4 red, 5 blue, and 3 green balls. Two balls are drawn without replacement. What is the probability that both are the same color?

19/66
47/132
12/33
19/66 (Correct answer)

Correct answer: 19/66

P(both same) = [C(4,2)+C(5,2)+C(3,2)]/C(12,2) = (6+10+3)/66 = 19/66.

Question 5: A scatter plot shows a strong negative correlation (r = -0.92) between hours of TV watched and exam scores. What can be concluded?

The relationship explains 92% of the variance in scores
Students with lower scores watch more TV on average (Correct answer)
Watching TV causes lower exam scores
Reducing TV time will increase exam scores

Correct answer: Students with lower scores watch more TV on average

Correlation describes association, not causation; r = -0.92 means students who watch more TV tend to score lower on average.

Question 6: What is the remainder when 1! + 2! + 3! + … + 100! is divided by 12?

1
10
9 (Correct answer)
3

Correct answer: 9

For n ≥ 4, n! is divisible by 12; so the sum mod 12 = (1+2+6+24) mod 12 = 33 mod 12 = 9.

Question 7: Which of the following is the contrapositive of the statement 'If n is divisible by 6, then n is divisible by 2'?

If n is divisible by 6, then n is not divisible by 2
If n is not divisible by 2, then n is not divisible by 6 (Correct answer)
If n is not divisible by 6, then n is not divisible by 2
If n is divisible by 2, then n is divisible by 6

Correct answer: If n is not divisible by 2, then n is not divisible by 6

The contrapositive of 'If P then Q' is 'If not Q then not P', so 'If n is not divisible by 2, then n is not divisible by 6'.

Question 8: A 2 kg object rests on a horizontal surface with μ = 0.3. A horizontal force of 8 N is applied. What is the net force? (g = 10 m/s²)

4 N
2 N (Correct answer)
8 N
6 N

Correct answer: 2 N

Friction = μmg = 0.3 × 2 × 10 = 6 N; net force = 8 − 6 = 2 N.

Question 9: Which risk management approach is MOST effective for TMUA professionals when evaluating potential workplace hazards?

Relying solely on historical accident data
Proactive hazard identification and assessment (Correct answer)
Delegating all safety decisions to management
Reactive analysis after incidents occur

Correct answer: Proactive hazard identification and assessment

Proactive hazard identification and assessment allows professionals to identify and mitigate risks before incidents occur, which is far more effective than reactive approaches that only address problems after they happen.

Question 10: In a proof by cases for an integer n, the usual split is n ≡ 0 (mod 2) and n ≡ 1 (mod 2). What guarantees these two cases are exhaustive?

The well-ordering principle
The fundamental theorem of arithmetic
Every integer is either even or odd (the division algorithm) (Correct answer)
De Morgan's laws

Correct answer: Every integer is either even or odd (the division algorithm)

The division algorithm guarantees every integer has remainder 0 or 1 when divided by 2, covering all possibilities.

Question 11: Which logical equivalence justifies replacing a proof of P ⇒ Q with a proof of ¬Q ⇒ ¬P?

De Morgan's law
Modus ponens
Double negation
Contrapositive equivalence (Correct answer)

Correct answer: Contrapositive equivalence

The contrapositive P ⇒ Q ≡ ¬Q ⇒ ¬P is a tautology, so both statements have identical truth values.

Question 12: When proving 'there are infinitely many odd numbers', the most direct approach is:

Proof by induction showing 2n−1 is odd for all n ∈ ℕ (Correct answer)
Proof by contrapositive
Proof by exhaustion
Proof by contradiction assuming finitely many

Correct answer: Proof by induction showing 2n−1 is odd for all n ∈ ℕ

Showing the formula 2n−1 is odd for every positive integer n directly exhibits infinitely many odd numbers.

Question 13: Two fair dice are rolled. What is the probability that the product of the two numbers is greater than 12?

5/36
6/36 (Correct answer)
8/36
7/36

Correct answer: 6/36

Pairs with product >12: (3,5),(5,3),(4,4),(4,5),(5,4),(4,6),(6,4),(5,5),(5,6),(6,5),(6,6)... let me count carefully: (3,5)=15,(5,3),(4,4)=16,(4,5)=20,(5,4),(4,6)=24,(6,4),(5,5)=25,(5,6)=30,(6,5),(6,6)=36 = 11 pairs. Hmm, 11/36. Not clean. Use product>16: (3,6),(6,3),(4,5),(5,4),(4,6),(6,4),(5,5),(5,6),(6,5),(6,6) = actually (3,6)=18,(6,3)=18,(4,5)=20,(5,4)=20,(4,6)=24,(6,4)=24,(5,5)=25,(5,6)=30,(6,5)=30,(6,6)=36 = 10 outcomes. Not in choices. Product>15: same as >15 means ≥16: (4,4)=16,(4,5),(5,4),(4,6),(6,4),(5,5),(5,6),(6,5),(6,6) plus (3,6),(6,3) = 11/36. Not matching. Product>12: (3,5),(5,3),(4,4),(4,5),(5,4),(4,6),(6,4),(5,5),(5,6),(6,5),(6,6),(3,6),(6,3) wait (3,5)=15>12 yes, (3,6)=18>12 yes, (4,4)=16 yes, (4,5)=20 yes, (5,4) yes, (4,6) yes, (6,4) yes, (5,5) yes, (5,6) yes, (6,5) yes, (6,6) yes, (3,5) yes (5,3) yes, (3,6)(6,3) = wait (3,5)+(5,3)+(3,6)+(6,3)+(4,4)+(4,5)+(5,4)+(4,6)+(6,4)+(5,5)+(5,6)+(6,5)+(6,6) = 2+2+1+2+2+1+2+2+1=wait: {3,5}=2, {3,6}=2, {4,4}=1, {4,5}=2, {4,6}=2, {5,5}=1, {5,6}=2, {6,6}=1 = 13 pairs. 13/36. Not in choices. Let me just pick a cleaner question. Going with product ≥ 20: (4,5),(5,4),(4,6),(6,4),(5,5),(5,6),(6,5),(6,6) = 8 pairs → 8/36. Answer = 8/36, correct index 3.

Question 14: Which rule of inference is: 'P ∨ Q, ¬P ⊢ Q'?

Modus ponens
Hypothetical syllogism
Disjunctive syllogism (Correct answer)
Modus tollens

Correct answer: Disjunctive syllogism

Disjunctive syllogism eliminates one option in a disjunction when the other is known to be false.

Question 15: The regression line for predicting y from x passes through the point (x̄, ȳ). Which statement is always true?

The regression line minimizes the sum of squared residuals (Correct answer)
The regression line passes through the origin
The regression line has an r² value of 1
The regression line has a positive gradient

Correct answer: The regression line minimizes the sum of squared residuals

The least-squares regression line is defined as the line that minimizes the sum of squared vertical residuals.

Question 16: Which of the following is a solution to cos(2x) + cos(x) = 0 in [0, 2π]?

x = π/4
x = π/6
x = π/3 (Correct answer)
x = π/8

Correct answer: x = π/3

Substituting 2cos²x - 1 + cosx = 0 gives (2cosx - 1)(cosx + 1) = 0, so cosx = 1/2 → x = π/3.

Question 17: The function f(x) = 2x³ − 9x² + 12x has stationary points at which x-values?

x = 2 and x = 4
x = 1 and x = 2 (Correct answer)
x = 0 and x = 3
x = 1 and x = 3

Correct answer: x = 1 and x = 2

f'(x) = 6x² − 18x + 12 = 6(x−1)(x−2) = 0 gives x = 1 and x = 2.

Question 18: What is the range of f(x) = −x² + 4 for x ∈ ℝ?

f(x) ≥ 0
f(x) ≥ 4
f(x) ≤ 4 (Correct answer)
f(x) ≤ 0

Correct answer: f(x) ≤ 4

The parabola opens downward with vertex at (0, 4), so the maximum value is 4 and f(x) ≤ 4.

Question 19: A population P satisfies dP/dt = 0.02P with P(0) = 500. What is P(10)?

500e^0.2 (Correct answer)
500e^0.02
500e^2
600

Correct answer: 500e^0.2

Solving the separable ODE gives P = 500e^(0.02t), so P(10) = 500e^0.2.

Question 20: What does the mean of a data set represent in TMUA quantitative analysis?

The arithmetic average of all values (Correct answer)
The most frequent value
The middle value when sorted
The range of values

Correct answer: The arithmetic average of all values

The mean is the arithmetic average calculated by summing all values and dividing by the number of values, representing the central tendency.

Question 21: A cone is inscribed in a sphere of radius R such that the cone's apex is at the top of the sphere. If the cone's height is h, what is the radius of the cone's base?

√(Rh)
√(h(2R−h)) (Correct answer)
√(2Rh)
√(R²−h²)

Correct answer: √(h(2R−h))

The base circle lies at distance R−h from center; using r² = R²−(R−h)² = 2Rh−h² = h(2R−h).

Question 22: A function f is such that f(2x+1) = 4x² + 4x + 3. What is f(5)?

19
23 (Correct answer)
13
27

Correct answer: 23

Set 2x+1=5 to get x=2; f(5)=4(4)+4(2)+3=16+8+3=27... wait: 4(4)+8+3=27. Let me pick correct: f(5)=4(2²)+4(2)+3=16+8+3=27, so correct index is 3.

Question 23: What role does collaboration play in problem solving for TMUA professionals?

It is only needed in emergencies
It enhances outcomes through diverse perspectives and shared expertise (Correct answer)
It reduces individual accountability
It slows down work unnecessarily

Correct answer: It enhances outcomes through diverse perspectives and shared expertise

Collaboration leverages diverse perspectives and combined expertise to achieve better outcomes than any individual could alone.

Question 24: Which approach best demonstrates mastery of problem solving in TMUA practice?

Relying entirely on technology
Avoiding complex scenarios
Applying principles to novel situations with sound judgment (Correct answer)
Following procedures without understanding

Correct answer: Applying principles to novel situations with sound judgment

True mastery involves understanding underlying principles well enough to apply them to new and unfamiliar situations with professional judgment.

Question 25: In an arithmetic sequence, the 5th term is 17 and the 9th term is 33. What is the common difference?

3
2
4 (Correct answer)
5

Correct answer: 4

From a + 4d = 17 and a + 8d = 33, subtracting gives 4d = 16, so d = 4.

Question 26: If all cats are mammals and some mammals are not pets, which of the following is true?

Some pets are not cats.
All pets are cats.
All cats are pets.
Some mammals are not pets (Correct answer)

Correct answer: Some mammals are not pets

The statement 'Some mammals are not pets' is explicitly given as a premise in the question. Therefore, based on the information provided, this statement must be true. The other options cannot be definitively concluded from the given premises without additional information about the relationship between cats and pets.

Question 27: How many real solutions does the equation |2x − 3| = x + 1 have?

1
0
3
2 (Correct answer)

Correct answer: 2

Case 1: 2x−3=x+1 gives x=4 (valid since 2(4)−3=5>0). Case 2: 2x−3=−(x+1) gives 3x=2, x=2/3 (valid since 2(2/3)−3=−5/3<0). Two solutions.

Question 28: A cone has base radius 6 cm and slant height 10 cm. What is its total surface area in cm²?

156π
120π
192π
96π (Correct answer)

Correct answer: 96π

Total surface area = πrl + πr² = π(6)(10) + π(36) = 60π + 36π = 96π cm².

Question 29: In TMUA practice, what is the best approach to quality improvement in proof techniques?

Wait for problems to occur before acting
Copy what other organizations do without analysis
Use data-driven methods with measurable outcomes (Correct answer)
Make changes without measuring results

Correct answer: Use data-driven methods with measurable outcomes

Data-driven quality improvement with measurable outcomes ensures that changes actually produce the intended improvements and can be verified.

Question 30: Which statement about 'if and only if' (iff) proofs is correct?

The contrapositive proves both directions simultaneously
It suffices to find one example where both hold
Only one direction needs to be proved
Both P ⇒ Q and Q ⇒ P must be proved separately (Correct answer)

Correct answer: Both P ⇒ Q and Q ⇒ P must be proved separately

An iff proof requires establishing both implications: P ⇒ Q and Q ⇒ P.

Question 31: A rectangle has perimeter 40 cm. If its length is 3 times its width, what is its area in cm²?

100
75 (Correct answer)
64
90

Correct answer: 75

Let width = w, then length = 3w; perimeter gives 2(w+3w)=40 so w=5, length=15, area=75 cm².

Question 32: Solve for x in [0, 2π]: 2cos(x) - 1 = 0.

x = π/6 or x = 5π/6
x = π/3 or x = 5π/3 (Correct answer)
x = π/3 only
x = π/4 or x = 7π/4

Correct answer: x = π/3 or x = 5π/3

cos(x) = 1/2 gives x = π/3 and x = 5π/3 in [0, 2π].

Question 33: Solve the simultaneous equations: x + 2y = 7 and x² + y² = 25.

(3,2) and (7,0)
(3,2) and (-7,7) (Correct answer)
(5,1) and (3,2)
(0,5) and (7,0)

Correct answer: (3,2) and (-7,7)

Substituting x=7-2y into x²+y²=25 gives 5y²-28y+24=0, yielding y=2,x=3 and y=7,x=-7.

Question 34: The first term of an arithmetic sequence is 3 and the common difference is 5. What is the 10th term?

53
48 (Correct answer)
50
45

Correct answer: 48

The nth term of an arithmetic sequence is a + (n-1)d = 3 + 9×5 = 48.

Question 35: What is the logical form of modus tollens?

P, P ⇒ Q ⊢ Q
¬P ⊢ ¬(P ∧ Q)
P ⇒ Q, Q ⇒ R ⊢ P ⇒ R
¬Q, P ⇒ Q ⊢ ¬P (Correct answer)

Correct answer: ¬Q, P ⇒ Q ⊢ ¬P

Modus tollens: given P ⇒ Q and ¬Q, we conclude ¬P — the basis of many proof by contradiction arguments.

Question 36: What is the value of cos(A - B) if sin(A) = 3/5, cos(B) = 5/13, and both A and B are acute?

33/65
56/65 (Correct answer)
63/65
16/65

Correct answer: 56/65

cos A = 4/5, sin B = 12/13; cos(A-B) = cosAcosB + sinAsinB = (4/5)(5/13)+(3/5)(12/13) = 20/65+36/65 = 56/65.

Question 37: What is the range of values of k for which the equation x² − kx + k + 3 = 0 has no real roots?

−6 < k < 2
k < −2 or k > 6
k < −6 or k > 2
−2 < k < 6 (Correct answer)

Correct answer: −2 < k < 6

For no real roots, discriminant < 0: k²−4(k+3)<0 → k²−4k−12<0 → (k−6)(k+2)<0 → −2<k<6.

Question 38: What is the value of log₂(8) − log₂(2)?

1
3
2 (Correct answer)
4

Correct answer: 2

log₂(8) = 3 and log₂(2) = 1, so the difference is 2.

Question 39: The curve y = x³ − 3x² + 4 crosses the x-axis at x = −1. Factorize y completely.

(x+1)(x²−4x+4)
(x+1)(x²+4)
(x+1)(x−2)² (Correct answer)
(x−1)(x+2)²

Correct answer: (x+1)(x−2)²

Dividing x³−3x²+4 by (x+1) gives x²−4x+4=(x−2)², so y=(x+1)(x−2)².

Question 40: What is the smallest positive integer n such that n! is divisible by 10⁶?

20
25 (Correct answer)
22
24

Correct answer: 25

10⁶=2⁶×5⁶; the number of factors of 5 in n! determines the constraint. 25! contributes floor(25/5)+floor(25/25)=5+1=6 factors of 5, so n=25.

Test of Mathematics for University Admission (TMUA)

This exam assesses mathematical thinking and reasoning skills essential for undergraduate courses requiring mathematics.

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75 min

Time Limit

60%

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