Paper Set

2002

2002 AMC12 Paper & Solutions Pick Paper A

25 questions with solutions, early exam, good for beginners.

Sample Problems Download PDF

Scan to get free 2002 PDF + solutions

25 Questions

75 Minutes

Max Score 150

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Exam Overview

This exam emphasizes multiple math topics. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry30%

Number Theory15%

Combinatorics20%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

87+

Honor Roll

Honor Roll (Top 5%)

77+

Achievement Roll

Grade 10 and below

62+

Sample Problems

2002 AMC12 Sample Problems (12 questions)

Sample reference problems by difficulty — click an option to check your answer

Q1EasyPolynomial

If f(x) = 2x² - 5x + 3, find f(4).

A) 11

B) 13

C) 15

D) 17

E) 19

Steps

1f(4) = 2×4² - 5×4 + 3

2= 2×16 - 20 + 3 = 15

Answer: CSubstitute, then exponentiate, multiply/divide, add/subtract

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=5, d=4, find the 12-th term.

A) 39

B) 44

C) 54

D) 49

E) 59

Steps

1aₙ = a₁ + (n-1)d

2a_12 = 5 + 11×4 = 49

Answer: DGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 1024, find x.

A) 3

B) 4

C) 6

D) 7

E) 5

Steps

11024 = 4^5

2so x = 5

Answer: EConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 7x + 6 = 0?

A) 7

B) 3

C) 5

D) 9

E) 11

Steps

1Vieta's:sum of roots = -(-7)/1 = 7

2product of roots = 6

Answer: AFor x²-px+q=0, sum of roots = p

Q10MediumDivisibility

How many integers from 1 to 32 are divisible by 5?

A) 2

B) 6

C) 4

D) 8

E) 10

Steps

1⌊32 / 5⌋ = 6

Answer: BCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 9-gon have?

A) 21

B) 24

C) 27

D) 30

E) 33

Steps

1diagonals = n(n-3)/2

2= 9×6/2 = 27

Answer: Cn-gon diagonals: n(n-3)/2

Q15MediumCombination

Compute C(8, 3).

A) 44

B) 50

C) 62

D) 56

E) 68

Steps

1C(n,3) = n(n-1)(n-2)/6

2= 8×7×6/6 = 56

Answer: DCombination: C(n,k)=n!/(k!(n-k)!)

Q17MediumPythagorean

Right triangle legs 6 and 8, find the hypotenuse.

A) 6

B) 8

C) 12

D) 14

E) 10

Steps

1c² = 6² + 8² = 36 + 64 = 100

2c = √100 = 10

Answer: EPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 5^6 divided by 7.

A) 1

B) 0

C) 2

D) 3

E) 4

Steps

1compute 5^6 mod 7

2simplify step by step (modular arithmetic)

3=1

Answer: AModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 8x + 21.

A) 1

B) 5

C) 3

D) 7

E) 9

Steps

1complete the square:f(x) = (x - 4)² + 5

2when x = 4 , min at 5

Answer: BCompleting the square for quadratic extrema

Q24HardInclusion-Excl

6 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?

A) 430

B) 485

C) 540

D) 595

E) 650

Steps

1total 3^6 = 729

2subtract empty boxes: -C(3,1)×2^6 = -192

3add back 2 empty boxes: +C(3,2)×1 = +3

4total 729 - 192 + 3 = 540

Answer: CInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=7, b=8, cos C=1/2, find c².

A) 45

B) 51

C) 63

D) 57

E) 69

Steps

1Law of cosines c² = a² + b² - 2ab·cos C

2= 49 + 64 - 56 = 57

Answer: DLaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 7x + 3, find f(4).

A) 3

B) 5

C) 7

D) 9

E) 11

Steps

1f(4) = 2×4² - 7×4 + 3

2= 2×16 - 28 + 3 = 7

Answer: CSubstitute, then exponentiate, multiply/divide, add/subtract

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=7, d=4, find the 12-th term.

A) 39

B) 45

C) 57

D) 51

E) 63

Steps

1aₙ = a₁ + (n-1)d

2a_12 = 7 + 11×4 = 51

Answer: DGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 1024, find x.

A) 3

B) 4

C) 6

D) 7

E) 5

Steps

11024 = 4^5

2so x = 5

Answer: EConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 9x + 6 = 0?

A) 9

B) 5

C) 7

D) 11

E) 13

Steps

1Vieta's:sum of roots = -(-9)/1 = 9

2product of roots = 6

Answer: AFor x²-px+q=0, sum of roots = p

Q10MediumDivisibility

How many integers from 1 to 34 are divisible by 5?

A) 2

B) 6

C) 4

D) 8

E) 10

Steps

1⌊34 / 5⌋ = 6

Answer: BCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 11-gon have?

A) 38

B) 41

C) 44

D) 47

E) 50

Steps

1diagonals = n(n-3)/2

2= 11×8/2 = 44

Answer: Cn-gon diagonals: n(n-3)/2

Q15MediumCombination

Compute C(10, 3).

A) 94

B) 107

C) 133

D) 120

E) 146

Steps

1C(n,3) = n(n-1)(n-2)/6

2= 10×9×8/6 = 120

Answer: DCombination: C(n,k)=n!/(k!(n-k)!)

Q17MediumPythagorean

Right triangle legs 7 and 24, find the hypotenuse.

A) 21

B) 23

C) 27

D) 29

E) 25

Steps

1c² = 7² + 24² = 49 + 576 = 625

2c = √625 = 25

Answer: EPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 5^6 divided by 7.

A) 1

B) 0

C) 2

D) 3

E) 4

Steps

1compute 5^6 mod 7

2simplify step by step (modular arithmetic)

3=1

Answer: AModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 8x + 23.

A) 3

B) 7

C) 5

D) 9

E) 11

Steps

1complete the square:f(x) = (x - 4)² + 7

2when x = 4 , min at 7

Answer: BCompleting the square for quadratic extrema

Q24HardInclusion-Excl

6 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?

A) 430

B) 485

C) 540

D) 595

E) 650

Steps

1total 3^6 = 729

2subtract empty boxes: -C(3,1)×2^6 = -192

3add back 2 empty boxes: +C(3,2)×1 = +3

4total 729 - 192 + 3 = 540

Answer: CInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=9, b=8, cos C=1/2, find c².

A) 57

B) 65

C) 81

D) 73

E) 89

Steps

1Law of cosines c² = a² + b² - 2ab·cos C

2= 81 + 64 - 72 = 73

Answer: DLaw of cosines is key for solving triangles

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2002 AMC12 Sample Problems (12 questions)

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