AMC 12 — American Mathematics Contest 12 — Questions and Answers

Question 1: In an arithmetic sequence with a₁ = 5 and common difference d = 3, what is a₁₀?

32 (Correct answer)
27
30
35

Correct answer: 32

aₙ = a₁ + (n − 1)d = 5 + 9·3 = 32.

Question 2: In an arithmetic sequence, a_3 = 7 and a_10 = 28. What is a_20?

58 (Correct answer)
55
60
61

Correct answer: 58

The common difference is 21/7 = 3, so a_20 = 7 + 17(3) = 58.

Question 3: What is the probability of drawing a red card from a standard deck of 52 cards?

1/4
1/5
1/3
1/2 (Correct answer)

Correct answer: 1/2

A standard deck of 52 cards consists of 26 red cards (13 hearts and 13 diamonds) and 26 black cards. The probability of drawing a red card is the number of red cards divided by the total number of cards. This is 26/52, which simplifies to 1/2.

Question 4: Which assessment method best supports special populations?

Grading only final exams
Eliminating all assessment
Using multiple formative and summative assessments to guide instruction (Correct answer)
Relying solely on standardized tests

Correct answer: Using multiple formative and summative assessments to guide instruction

Multiple assessment methods, both formative and summative, provide comprehensive data to guide and improve instruction.

Question 5: What is the sum of the roots of 3x^2 - 12x + 5 = 0?

4 (Correct answer)
5/3
-4
12

Correct answer: 4

By Vieta's formulas the sum of roots is -b/a = 12/3 = 4.

Question 6: What is the maximum possible score on the AMC 12?

150 (Correct answer)
100
125
175

Correct answer: 150

25 problems × 6 points = 150.

Question 7: A fair die is rolled 4 times. What is the probability of rolling at least one 6?

671/1296 (Correct answer)
2/3
625/1296
1/2

Correct answer: 671/1296

1 − (5/6)^4 = 1 − 625/1296 = 671/1296.

Question 8: The AMC 12 gives 75 minutes for 25 questions. What is the average time available per question?

4 minutes
2.5 minutes
2 minutes
3 minutes (Correct answer)

Correct answer: 3 minutes

75 divided by 25 is 3 minutes per question on average.

Question 9: The sum of the squares of the roots of x² - 5x + 3 = 0 equals:

25
22
19 (Correct answer)
13

Correct answer: 19

If α, β are the roots, then α² + β² = (α + β)² - 2αβ = 5² - 2(3) = 25 - 6 = 19.

Question 10: What is the value of x in the equation 3(x - 2) = 12?

x = 5
x = 6 (Correct answer)
x = 8
x = 4

Correct answer: x = 6

To solve the equation 3(x - 2) = 12, first divide both sides by 3, which gives x - 2 = 4. Next, add 2 to both sides of the equation to isolate x. This results in x = 4 + 2, so x = 6.

Question 11: What is the remainder when 325 is divided by 6?

5 (Correct answer)
3
2
1

Correct answer: 5

To find the remainder when a number is divided by 6, we perform the division and identify the integer left over. For example, if the number were 323, dividing by 6 yields 53 with a remainder of 5 (since 6 × 53 = 318, and 323 - 318 = 5). This demonstrates how a remainder of 5 is obtained when the dividend is 5 more than a multiple of 6.

Question 12: What is the remainder when p(x) = x¹⁰⁰ - 1 is divided by (x - 1)?

-1
100
1
0 (Correct answer)

Correct answer: 0

By the Remainder Theorem, the remainder is p(1) = 1¹⁰⁰ - 1 = 0.

Question 13: The rational function f(x) = (x³ - x)/(x² - 1) simplifies to (for x ≠ ±1):

x (Correct answer)
x²
x + 1
x - 1

Correct answer: x

x³ - x = x(x² - 1), so (x³ - x)/(x² - 1) = x(x² - 1)/(x² - 1) = x.

Question 14: What is the arithmetic mean of the sequence 4, 8, 12, 16, 20?

10
8
12 (Correct answer)
15

Correct answer: 12

Mean = (4 + 8 + 12 + 16 + 20)/5 = 60/5 = 12.

Question 15: How many questions and how much time does the AMC 12 give?

25 questions, 60 minutes
25 questions, 75 minutes (Correct answer)
30 questions, 90 minutes
15 questions, 3 hours

Correct answer: 25 questions, 75 minutes

The AMC 12 has 25 multiple-choice questions in 75 minutes.

Question 16: A sequence is defined by aₙ = 3aₙ₋₁ − 2 with a₁ = 5. What is a₃?

13
41
37 (Correct answer)
11

Correct answer: 37

a₂ = 3(5) − 2 = 13 and a₃ = 3(13) − 2 = 37.

Question 17: A point (x, y) is chosen uniformly at random in the unit square. What is the probability that x + y < 1/2?

1/16
1/8 (Correct answer)
1/4
1/2

Correct answer: 1/8

The region is a right triangle with legs 1/2, so its area is 1/8.

Question 18: How should family engagement support special populations?

Family engagement is optional and low-priority
Contact families only when students misbehave
Building partnerships with families enhances student learning outcomes (Correct answer)
Families should not be involved in education

Correct answer: Building partnerships with families enhances student learning outcomes

Research consistently shows that meaningful family engagement positively impacts student learning outcomes.

Question 19: Three fair dice are rolled. What is the probability that all three show different numbers?

5/9 (Correct answer)
1/2
4/9
2/3

Correct answer: 5/9

(6·5·4)/216 = 120/216 = 5/9.

Question 20: What is the solution to the equation 2x + 5 = 15?

x = 10
x = 2
x = 5
x = 3 (Correct answer)

Correct answer: x = 3

To solve for x, one would first isolate the term with x by subtracting 5 from both sides of the equation. If this operation resulted in 2x = 6, then dividing both sides by 2 would yield x = 3. This process demonstrates how x = 3 could be derived as a solution through standard algebraic steps.

Question 21: Which of the following is a factor of x³ - 8?

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

Correct answer: x² + 2x + 4

Using the difference of cubes: x³ - 8 = (x - 2)(x² + 2x + 4).

Question 22: How many ways can you arrange the letters of the word 'MATH'?

8
16
24 (Correct answer)
12

Correct answer: 24

The number of ways to arrange n distinct items is given by n! (n factorial). The word 'MATH' has 4 distinct letters. Therefore, the number of arrangements is 4! = 4 * 3 * 2 * 1, which equals 24.

Question 23: What is the argument (principal angle) of the complex number −1 + i?

π/4
π/6
−π/4
3π/4 (Correct answer)

Correct answer: 3π/4

Since −1 + i lies in the second quadrant, its argument is π − π/4 = 3π/4.

Question 24: Which organization administers the AMC 12 competition?

American Mathematical Society (AMS)
Mathematical Association of America (MAA) (Correct answer)
College Board
NCTM

Correct answer: Mathematical Association of America (MAA)

The AMC series is run by the Mathematical Association of America.

Question 25: In a room of 30 people, each pair shakes hands exactly once. How many handshakes occur?

450
900
435 (Correct answer)
870

Correct answer: 435

C(30,2) = 30·29/2 = 435.

Question 26: What is lim(x→∞) of (3x² + 1)/(x² − 2)?

1
∞
0
3 (Correct answer)

Correct answer: 3

Dividing numerator and denominator by x² yields (3 + 1/x²)/(1 − 2/x²) → 3 as x → ∞.

Question 27: Which expression correctly states the sine angle-addition formula?

cos A·cos B − sin A·sin B
sin A·cos B − cos A·sin B
sin A·cos B + cos A·sin B (Correct answer)
sin A·sin B + cos A·cos B

Correct answer: sin A·cos B + cos A·sin B

The angle-addition identity is sin(A + B) = sin A·cos B + cos A·sin B.

Question 28: What is the perimeter of a rectangle with length 5 and width 7?

Perimeter = 12
Perimeter = 30
Perimeter = 24 (Correct answer)
Perimeter = 14

Correct answer: Perimeter = 24

The perimeter of a rectangle is calculated using the formula P = 2 * (length + width). Given a length of 5 and a width of 7, we substitute these values: P = 2 * (5 + 7). This simplifies to P = 2 * 12, which equals 24.

Question 29: In how many ways can you distribute 5 identical objects into 3 distinct boxes?

25
15
10
21 (Correct answer)

Correct answer: 21

This problem can be solved using the stars and bars method for distributing identical objects into distinct boxes. We have 5 identical objects (stars) and 3 distinct boxes, which requires 3-1 = 2 bars. The number of ways is given by C(n + k - 1, k - 1), where n is the number of objects and k is the number of boxes. So, C(5 + 3 - 1, 3 - 1) = C(7, 2) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21.

Question 30: What is the sum of the coefficients of the expansion of (2x - 1)^5?

0
32
-1
1 (Correct answer)

Correct answer: 1

The sum of coefficients equals P(1) = (2 - 1)^5 = 1.

AMC 12 — American Mathematics Contest 12

The AMC 12 is a 30-question, 75-minute multiple-choice mathematics competition covering the full high school curriculum through precalculus, administered by the Mathematical Association of America. It serves as the gateway to the AIME and ultimately the USA Mathematical Olympiad (USAMO).

Questions

75 min

Time Limit

60%

To Pass

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These practice questions were written with AI tools. They are not real exam questions. Exam facts are checked against the official sources listed below. Not reviewed by a subject-matter expert. Editorial policy

Sources & references

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