SAT Math Level 2 — Practice Test A
25 multiple-choice questions · Worked solutions included Topics: arc length · distance · probability · circles · functions · summation · sequences · trigonometry · domain & range · vectors · matrices · statistics
Questions 1–10
1.
If f(x) = (3x + 7) / (6x + 4), what value does f(x) approach as x gets infinitely larger?
a) 0 | b) 1/2 |
c) 3/4 | d) 7/4 |
e) infinity |
✅ Answer: B Since the expression is neither "top heavy" nor "bottom heavy," compare lead coefficients: 3/6 = 1/2
2.
O is the center of the circle, and the diameter is 12. What is the arc length AB?
A
|
|
O------+
\ B
\ /
\__/
∠AOB = 90°
a) 3π | b) 6π |
c) 9π | d) 18π |
e) 36π |
✅ Answer: A Circumference = π · diameter = 12π. Since ∠AOB = 90°, arc AB is 1/4 of the circumference: (90/360) · 12π = 3π
3.
What is the distance in space between (1, 0, 5) and (−3, 6, 3)?
a) 4 | b) 6 |
c) 2√11 | d) 2√14 |
e) 12 |
✅ Answer: D distance = √[(x₁−x₂)² + (y₁−y₂)² + (z₁−z₂)²] = √[(1−(−3))² + (0−6)² + (5−3)²] = √(16+36+4) = √56 = 2√14
4.
|5 − 7| − |7 − 5| =
a) −4 | b) 0 |
c) 2 | d) 4 |
e) 12 |
✅ Answer: B |−2| − |2| = 2 − 2 = 0
5.
A line has parametric equations x = t + 5 and y = t + 10. What is the slope of the line?
a) 1 | b) 2 |
c) 5 | d) 10 |
e) 50 |
✅ Answer: A From x = t + 5, t = x − 5. Substitute: y = (x − 5) + 10 → y = x + 5 (slope = 1)
6.
Two dice are tossed. What is the probability that neither die is a 4?
a) 1/6 | b) 1/3 |
c) 2/3 | d) 25/36 |
e) 5/6 |
✅ Answer: D P(1st not 4) = 5/6, P(2nd not 4) = 5/6. Since independent: P(neither is 4) = (5/6)(5/6) = 25/36
7.
(1, 6), (3, −2), and (−2, K) are collinear points. What is K?
a) −6 | b) 2 |
c) 8 | d) 10 |
e) 18 |
✅ Answer: E Collinear points share the same slope. Slope of (1,6) and (3,−2) = (−2−6)/(3−1) = −4. So (K−6)/(−2−1) = −4 → K = 18
8.
Vectors u and v are given by u = (3, 0) and v = (1, −4). What is the length of vector w, given w = 2u − v?
a) 2√10 | b) √41 |
c) 6 − √17 | d) 3 |
e) √23 |
✅ Answer: B w = 2(3,0) − (1,−4) = (5, 4). ‖w‖ = √(5² + 4²) = √41
9.
The domain of g(x) = 3 / √(4 − x²) is:
a) [−2, 2] | b) (−2, 2) |
c) (0, 2) | d) (−∞, −2) |
e) (−∞, 2) |
✅ Answer: B Cannot have a negative under the radical, and cannot have zero in the denominator — so x must be strictly between −2 and 2
10.
The radius of circle x² − 4x + y² + 6y = 3 is:
a) 3 | b) √3 |
c) 4 | d) 8 |
e) 16 |
✅ Answer: C Complete the square: x² − 4x + 4 + y² + 6y + 9 = 3 + 4 + 9 → (x−2)² + (y+3)² = 16 → radius = 4
Questions 11–20
11.
f(x) = 2x + 1, g(x) = x² − 1. What is (f∘g)(x)?
a) x² + 2x | b) 2x³ + x² − 2x − 1 |
c) 2x² − 1 | d) 4x² + 4x |
e) 2(x² + x + 1) |
✅ Answer: C f(g(x)) = 2(x² − 1) + 1 = 2x² − 2 + 1 = 2x² − 1
12.
The intersection of line x + y = 2 and circle x² + y² = 4 occurs when x =
a) −2, 2 | b) −2, 0, 2 |
c) −2, 4 | d) 0, 2 |
e) 0, 4 |
✅ Answer: D y = −x + 2. Substitute: x² + (−x+2)² = 4 → 2x² − 4x = 0 → 2x(x−2) = 0 → x = 0, 2
13.
Which is a zero of the function f(x) = sin(2x) − 1/2?
a) π/12 | b) π/3 |
c) π/2 | d) 0 |
e) π |
✅ Answer: A sin(2x) − 1/2 = 0 → sin(2x) = 1/2 → 2x = π/6 → x = π/12
14.
A circle is inscribed in a square. If the area of the square is 36 sq. units, what is the area of the circle?
┌───────────┐
│ ╭───╮ │
│ ╱ ╲ │
│ │ r │ │ side = 6
│ ╲ ╱ │
│ ╰───╯ │
└───────────┘
a) 6π | b) 9π |
c) 12π | d) 18π |
e) 36π |
✅ Answer: B Area of square = 36, so each side = 6. The inscribed circle's radius = 3. Area = πr² = π(3)² = 9π
15.
Find the sum: Σ (n = 1 to 100) of (n − 2)
a) −100 | b) 2250 |
c) 4500 | d) 4550 |
e) 4850 |
✅ Answer: E Σ(n−2) = Σn − Σ2 = [(100+1)(50)] − 2(100) = 5050 − 200 = 4850
16.
In the xy-plane, the vertices of a triangle are (−1, 3), (6, 3), and (−1, −4). The area of the triangle is:
a) 10 | b) 17.5 |
c) 24.5 | d) 35 |
e) 42 |
✅ Answer: C The right angle is at (−1,3): legs measure 7 (horizontal) and 7 (vertical). Area = 1/2(7)(7) = 49/2 = 24.5
17.
A right cylinder has radius 5 and height 5. If A and B are points on the surface, what is the maximum possible (line) distance between them?
______________
/ \ A
|________________| /|
| | / |
| 5 | / | 5
| | / |
\______________/ /____|
B
10 (diameter)
a) 5 | b) 5√2 |
c) 10 | d) 12.5 |
e) 5√5 |
✅ Answer: E Radius is 5, so diameter is 10. Max distance is the hypotenuse of a right triangle with legs 5 and 10: 5² + 10² = AB² → AB = √125 = 5√5
18.
In the figure, AB = BC and ∠A = 80°. What is x?
A
/\
/80\
/ \
/ \
B /________\ C
x
a) 8 | b) 10.3 |
c) 11.4 | d) 12 |
e) 12.7 |
✅ Answer: B Since AB = BC, angles B and C are equal; angles sum to 180°, so B = C = 50°. Law of sines: x/sinA = 8/sinC → x = 8sin(80°)/sin(50°) = 8(0.985)/(0.766) ≈ 10.3
19.
f(x) = 2x² is translated 2 units to the right and 1 unit up. If the resulting graph is g(x), what is g(3)?
a) 3 | b) 11 |
c) 15 | d) 26 |
e) 31 |
✅ Answer: A Shift right 2: 2(x−2)². Shift up 1: g(x) = 2(x−2)² + 1. g(3) = 2(1)² + 1 = 3
20.
A sequence is recursively defined as a₁ = 0, a₂ = 1, and for n > 2, aₙ = aₙ₋₁ − 3aₙ₋₂. What is a₆?
a) −8 | b) 1 |
c) 18 | d) 72 |
e) 109 |
✅ Answer: B a₁=0, a₂=1, a₃=1−3(0)=1, a₄=1−3(1)=−2, a₅=−2−3(1)=−5, a₆=−5−3(−2)=1
Questions 21–25
21.
If sin(θ) = 0.47, then sin(π − θ) =
a) −0.47 | b) −0.43 |
c) 0 | d) 0.43 |
e) 0.47 |
✅ Answer: E sin(π−θ) = sinπ·cosθ − cosπ·sinθ = 0(cosθ) − (−1)(sinθ) = sinθ = 0.47
22.
What is the range of the function g(x) = 1/x + 3?
a) all real numbers | b) all real numbers except 0 |
c) all real numbers > 3 | d) all real numbers except 3 |
e) all real numbers > 1 |
✅ Answer: D The range is all possible outputs. Since 1/x never equals zero, the output 3 is never reached: all real numbers except 3
23.
For matrices A = [[1, 3], [0, −1]] and B = [[2, 2], [−1, 4]], what is BA?
a) [[3, 5], [−1, 3]] | b) [[1, −1], [−1, 5]] |
c) [[−1, 14], [1, −4]] | d) [[2, 4], [−1, −7]] |
e) none of the above |
✅ Answer: D Note BA ≠ AB. BA = [[2(1)+2(0), 2(3)+2(−1)], [−1(1)+4(0), −1(3)+4(−1)]] = [[2, 4], [−1, −7]]
24.
There is a regular hexagon with sides of length 6. If the coordinate of A is (9, 0), what is the y-coordinate of B?
B ___
/ \
| |
\____/ A (9,0)
─────●──────────
│
side length = 6
a) 0 | b) 3 |
c) 3√2 | d) 3√3 |
e) 4½ |
✅ Answer: D Interior angles of a regular hexagon are 120°. Using the resulting 30-60-90 triangle with hypotenuse 6, the short leg is 3 and the long leg is 3√3 — the y-coordinate of B.
25.
A sample of 25 scores has a mean 75, median 79, and standard deviation of 8. If you increase every score by 10, which is true?
I. The new mean is 85 II. The new median is 89 III. The new standard deviation is 18
a) I only | b) II only |
c) I and II only | d) none |
e) I, II, and III |
✅ Answer: C Adding 10 to every score shifts the mean to 85 and the median to 89 (I, II true). Each score's distance from the mean is unchanged, since all scores shifted together — so standard deviation stays 8, not 18 (III false).
SAT Math Level 2 Practice Test A
Improve your problem-solving skills with SAT Math Level 2 Practice Test A, a comprehensive practice exam designed to help students prepare for advanced mathematics questions commonly found on standardized tests. This practice test includes 25 multiple-choice questions with detailed answer explanations, allowing you to understand not only the correct answer but also the reasoning behind it.
The questions cover a wide range of important math concepts, including functions, circles, arc length, coordinate geometry, probability, trigonometry, sequences, summation, vectors, matrices, statistics, domain and range, and three-dimensional geometry. Each problem is carefully selected to strengthen analytical thinking, mathematical reasoning, and test-taking confidence.
Whether you're reviewing key concepts, preparing for an upcoming exam, or looking to improve your overall math performance, this SAT Math Level 2 practice test provides an effective way to measure your progress and identify areas for improvement.
What You'll Learn
By completing this practice test, you'll reinforce your understanding of:
Rational and polynomial functions
Limits and function behavior
Circle geometry and arc length
Coordinate and three-dimensional geometry
Distance formula applications
Probability and counting principles
Absolute value expressions
Parametric equations
Trigonometric identities
Sequences and series
Sigma notation and summation
Matrix operations
Vector calculations
Statistics and data interpretation
Domain and range of functions
Why Practice SAT Math Regularly?
Consistent practice is one of the most effective ways to improve your SAT Math performance. Working through realistic multiple-choice questions helps you become familiar with common question formats, strengthens your conceptual understanding, and increases your speed and accuracy under timed conditions.
Practice tests also help you identify weak areas, making it easier to focus your study time on the topics that need the most attention before exam day.
Who Should Use This Practice Test?
This SAT Math Level 2 practice test is ideal for:
High school students preparing for college entrance exams
Students reviewing advanced algebra and geometry concepts
Learners looking to improve mathematical reasoning skills
Teachers seeking classroom practice materials
Parents supporting students with exam preparation
Anyone wanting additional SAT-style math practice
How to Get the Most from This Practice Test
For the best results:
Complete all questions without looking at the answers.
Time yourself to simulate real exam conditions.
Review every explanation carefully, even for questions you answered correctly.
Make note of recurring mistakes and revisit those topics.
Repeat similar practice tests regularly to track your improvement.
Skills Assessed
This practice test evaluates several essential mathematical skills, including:
Problem-solving
Logical reasoning
Algebraic manipulation
Geometric reasoning
Data analysis
Mathematical modeling
Formula application
Quantitative comparison
Multi-step calculations
Why Choose This Practice Test?
Unlike simple question banks, this practice test is designed to help you learn through detailed explanations. Each solution breaks down the reasoning process step by step, helping you build confidence and develop effective strategies for solving similar problems independently.
Frequently Asked Questions
Is this practice test suitable for self-study?
Yes. The included answer explanations make it an excellent resource for independent learning and exam preparation.
Are the questions multiple choice?
Yes. All questions follow a multiple-choice format similar to standardized math exams.
Does this practice test include explanations?
Yes. Every question is accompanied by a worked solution to help you understand the concepts and methods used.
What topics are covered?
Topics include functions, circles, probability, trigonometry, vectors, matrices, statistics, sequences, summation, coordinate geometry, domain and range, and more.
Can I use this practice test for revision?
Absolutely. It is suitable for both first-time learning and final revision before an exam.
Continue Practicing
After completing this practice test, explore additional SAT Math practice sets covering algebra, geometry, trigonometry, statistics, and advanced problem-solving to further improve your accuracy, speed, and confidence.