UTMA Mathematics Assessment — Questions and Answers

Question 1: If f(x) = x² and g(x) = x + 1, what is (f + g)(2)?

9
7 (Correct answer)
6
5

Correct answer: 7

(f + g)(2) = f(2) + g(2) = 4 + 3 = 7.

Question 2: What is the circumference of a circle with a radius of 7 cm?

28π cm
21π cm
35π cm
14π cm (Correct answer)

Correct answer: 14π cm

The circumference of a circle is calculated using the formula C = 2πr, where 'r' is the radius. With a radius of 7 cm, the circumference is 2 * π * 7 cm. This simplifies to 14π cm.

Question 3: In a right-angled triangle, if one angle is 30° and the hypotenuse is 10 cm,<br> what is the length of the side opposite the 30° angle?

5 cm (Correct answer)
8.66 cm
7.5 cm
10 cm

Correct answer: 5 cm

In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse (sin θ = opposite/hypotenuse). For a 30° angle, sin(30°) = 0.5. Therefore, the length of the side opposite the 30° angle is 10 cm * sin(30°) = 10 cm * 0.5 = 5 cm.

Question 4: What is the integral of 𝑓 (x)=2x?

2𝑥2 + 𝐶
x 2 + x + C
4x 2 + C
𝑥2 + 𝐶 (Correct answer)

Correct answer: 𝑥2 + 𝐶

To find the integral of f(x) = 2x, we use the power rule for integration, which states that the integral of x^n is (x^(n+1))/(n+1). For 2x (or 2x^1), this becomes 2 * (x^(1+1))/(1+1). This simplifies to 2 * (x^2)/2 = x^2. Since it's an indefinite integral, we add the constant of integration, C, resulting in x^2 + C.

Question 5: Two fair coins are flipped. What is the probability of getting exactly one head?

1/2 (Correct answer)
3/4
1/4
1

Correct answer: 1/2

Sample space: {HH, HT, TH, TT}; exactly one head: {HT, TH} = 2/4 = 1/2.

Question 6: What is tan(0°)?

1
undefined
−1
0 (Correct answer)

Correct answer: 0

tan(0°) = sin(0°)/cos(0°) = 0/1 = 0.

Question 7: If 5 workers can complete a job in 12 days, how many days would 3 workers take (at the same rate)?

18 days
20 days (Correct answer)
15 days
24 days

Correct answer: 20 days

Total work = 5 × 12 = 60 worker-days; with 3 workers: 60/3 = 20 days.

Question 8: In a class of 30, 18 like math and 14 like science. If 8 like both, how many like at least one?

40
32
24 (Correct answer)
22

Correct answer: 24

By inclusion-exclusion: 18 + 14 − 8 = 24 students.

Question 9: Find the critical points of 𝑓 (𝑥) =𝑥 3−3𝑥2

1 and 3
0 and 3 (Correct answer)
0 and -3
-1 and 2

Correct answer: 0 and 3

Critical points are found by taking the first derivative of a function and setting it to zero. If the derivative of a function factors into terms like 3x(x-3)=0, then solving for x would yield x=0 and x=3. These points indicate where the function's slope is horizontal, often corresponding to local maxima or minima.

Question 10: Which statement correctly describes the function f(x) = x³ − x?

It is an odd function (Correct answer)
It is an even function
It has no x-intercepts
Its range is restricted

Correct answer: It is an odd function

f(−x) = (−x)³ − (−x) = −x³ + x = −f(x), satisfying the definition of an odd function.

Question 11: What is the derivative of 𝑓(x)= x 3 ?

2x
4x3
x2
3x2 (Correct answer)

Correct answer: 3x2

The derivative of f(x) = x^3 is found using the power rule, which states that the derivative of x^n is n*x^(n-1). Applying this rule, the derivative of x^3 is 3 * x^(3-1). This simplifies to 3x^2.

Question 12: What is sin²(θ) + cos²(θ)?

1 (Correct answer)
sin(2θ)
2
0

Correct answer: 1

The Pythagorean identity states sin²(θ) + cos²(θ) = 1 for all θ.

Question 13: If sin(θ) = 5/13 and θ is in the first quadrant, what is cos(θ)?

8/13
13/12
5/12
12/13 (Correct answer)

Correct answer: 12/13

Using sin²θ + cos²θ = 1: cos²θ = 1 − 25/169 = 144/169, so cos θ = 12/13.

Question 14: What is the area of a triangle with a base of 6 cm and a height of 4 cm?

10 cm²
20 cm²
12 cm² (Correct answer)
24 cm²

Correct answer: 12 cm²

The area of a triangle is calculated using the formula A = (1/2) * base * height. Given a base of 6 cm and a height of 4 cm, the area is (1/2) * 6 cm * 4 cm. This calculation yields an area of 12 cm².

Question 15: If P(A) = 0.4 and P(B) = 0.3 and A and B are mutually exclusive, what is P(A or B)?

0.70
0.58 (Correct answer)
0.12
0.44

Correct answer: 0.58

For mutually exclusive events: P(A∪B) = P(A) + P(B) = 0.4 + 0.3 = 0.7... wait, 0.4+0.3=0.7, answer is 0.70.

Question 16: What is the inverse of g(x) = (x/2) − 4?

g⁻¹(x) = 2x − 8
g⁻¹(x) = 2x + 8 (Correct answer)
g⁻¹(x) = (x + 4)/2
g⁻¹(x) = x/2 + 4

Correct answer: g⁻¹(x) = 2x + 8

Set y = x/2 − 4, swap to get x = y/2 − 4, then solve: y = 2x + 8.

Question 17: A salesperson earns a base salary of $1,200/month plus 5% commission on sales. If they earned $2,000 last month, what were their total sales?

$24,000
$18,000
$16,000 (Correct answer)
$20,000

Correct answer: $16,000

Commission = 2000 − 1200 = 800; Sales = 800 / 0.05 = $16,000.

Question 18: What is the y-intercept of f(x) = (x + 3)(x − 1)?

−1
3
−3 (Correct answer)
1

Correct answer: −3

Setting x = 0 gives f(0) = (3)(−1) = −3.

Question 19: The graph of y = f(2x) compared to y = f(x) is:

Stretched vertically by factor 2
Compressed vertically by factor 2
Stretched horizontally by factor 2
Compressed horizontally by factor 2 (Correct answer)

Correct answer: Compressed horizontally by factor 2

Replacing x with 2x compresses the graph horizontally by a factor of 2 (points move twice as close to the y-axis).

Question 20: Evaluate the limit: lim𝑥→2 (3𝑥+1)

3
7 (Correct answer)
5
6

Correct answer: 7

To evaluate the limit of a polynomial function as x approaches a specific value, simply substitute that value into the function. Substituting x=2 into the expression (3x+1) gives 3(2)+1. This calculation results in 6+1, which equals 7.

Question 21: What is sin(30°)?

√3/2
1/2 (Correct answer)
√2/2
1

Correct answer: 1/2

sin(30°) = 1/2 is a standard trigonometric value.

Question 22: What is the standard deviation of the following data set: 2, 4, 4, 4, 5, 5, 7, 9?

0
3
1
2 (Correct answer)

Correct answer: 2

To find the standard deviation, first calculate the mean of the data set (2, 4, 4, 4, 5, 5, 7, 9), which is (2+4+4+4+5+5+7+9)/8 = 40/8 = 5. Then, calculate the variance by averaging the squared differences from the mean: ((2-5)^2 + 3*(4-5)^2 + 2*(5-5)^2 + (7-5)^2 + (9-5)^2)/8 = (9 + 3 + 0 + 4 + 16)/8 = 32/8 = 4. The standard deviation is the square root of the variance, which is sqrt(4) = 2.

Question 23: Which of the following is a one-to-one function?

f(x) = x²
f(x) = |x|
f(x) = sin x
f(x) = 2x + 5 (Correct answer)

Correct answer: f(x) = 2x + 5

A linear function f(x) = 2x + 5 passes the horizontal line test, confirming it is one-to-one.

Question 24: What is the value of 5! (5 factorial)?

100
120 (Correct answer)
125
60

Correct answer: 120

5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 25: What is the period of the function f(x) = sin(2x)?

2π
π (Correct answer)
π/2
4π

Correct answer: π

The period of sin(bx) is 2π/b; with b = 2, the period is 2π/2 = π.

Question 26: What is the inverse of the 2×2 matrix [[4,7],[2,6]]?

[[4,−7],[−2,6]] / 10
[[6,−7],[−2,4]] / 10 (Correct answer)
[[6,−7],[−2,4]] / 14
[[−6,7],[2,−4]] / 10

Correct answer: [[6,−7],[−2,4]] / 10

det = 24−14 = 10; inverse = (1/10)[[6,−7],[−2,4]].

Question 27: What is the remainder when 2²⁰ is divided by 3?

2
1 (Correct answer)
3
0

Correct answer: 1

Since 2 ≡ −1 (mod 3), 2²⁰ ≡ (−1)²⁰ = 1 (mod 3).

Question 28: What is the second derivative of 𝑓 (𝑥) = 𝑥4?

12x 2 (Correct answer)
4x 3
8x 2
24x

Correct answer: 12x 2

To find the second derivative of f(x) = x^4, first calculate the first derivative using the power rule: f'(x) = 4x^3. Then, differentiate f'(x) again to find the second derivative: f''(x) = 4 * (3x^2). This simplifies to 12x^2.

Question 29: What is the least common multiple (LCM) of 12, 18, and 24?

144
36
48
72 (Correct answer)

Correct answer: 72

Prime factorizations give LCM = 2³ × 3² = 72.

Question 30: In Gaussian elimination, what is the goal of row operations?

To multiply all entries by −1
To convert the matrix to row echelon form (Correct answer)
To add rows randomly
To swap all rows

Correct answer: To convert the matrix to row echelon form

Gaussian elimination uses row operations to reduce the augmented matrix to row echelon form for easy back-substitution.

UTMA Mathematics Assessment

The University of Toronto Mathematics Assessment (UTMA) is a placement test used to evaluate incoming students' math readiness across algebra, geometry, trigonometry, calculus, probability, and applied math topics.

Questions

60 min

Time Limit

70%

To Pass

Checking for saved progress...

Exam Rules

You can skip questions and return to them later
Flag questions for review before submitting
No feedback shown until you submit the entire exam
Unanswered questions count as wrong — answer everything
10 pretest questions are mixed in and don't affect your score
Timer auto-submits when time runs out
Your progress is auto-saved every 30 seconds