AMC12 Algebra 2 — Questions and Answers
Question 1: If a+b=5 and ab=4, find a^3+b^3.
- 25
- 65 (Correct answer)
- 45
- 80
Correct answer: 65
a^3+b^3=(a+b)^3-3ab(a+b)=125-60=65.
Question 2: The polynomial x^3-6x^2+11x-6 has roots r, s, t. Find r+s+t.
- 11
- 3
- 6 (Correct answer)
- -6
Correct answer: 6
By Vieta's formulas, the sum of roots equals 6.
Question 3: Solve 3^(2x) - 4*3^x + 3 = 0. What are the solutions?
- x=1 or x=2
- x=-1 or x=0
- x=0 or x=3
- x=0 or x=1 (Correct answer)
Correct answer: x=0 or x=1
Let u=3^x: u^2-4u+3=0 gives u=1 (x=0) or u=3 (x=1).
Question 4: If x + 1/x = 5, what is x^2 + 1/x^2?
- 23 (Correct answer)
- 25
- 27
- 21
Correct answer: 23
Squaring x+1/x=5 gives x^2+2+1/x^2=25, so x^2+1/x^2=23.
Question 5: A function satisfies f(2x)=2f(x) and f(1)=3. Find f(8).
- 6
- 12
- 24 (Correct answer)
- 48
Correct answer: 24
f(2)=6, f(4)=12, f(8)=24.
Question 6: For how many integers n is (n-1)/(n+1) also an integer?
- 2
- 1
- 3
- 4 (Correct answer)
Correct answer: 4
Writing n-1=(n+1)-2, n+1 must divide 2, so n+1 in {-2,-1,1,2} giving 4 integers.
AMC12 Algebra 2
If a+b=5 and ab=4, find a^3+b^3.