AMC12 Algebra 1 — Questions and Answers
Question 1: If 2^x = 3, what is the value of 4^x?
- 9 (Correct answer)
- 6
- 12
- 8
Correct answer: 9
4^x = (2^2)^x = (2^x)^2 = 3^2 = 9.
Question 2: The roots of x^2 - 5x + 6 = 0 are r and s. What is r^2 + s^2?
- 25
- 13 (Correct answer)
- 12
- 7
Correct answer: 13
By Vieta's, r+s=5 and rs=6, so r^2+s^2=(r+s)^2-2rs=25-12=13.
Question 3: If log2(x) + log2(x-2) = 3, find x.
- 2
- 8
- 4 (Correct answer)
- 6
Correct answer: 4
x(x-2)=8 gives x^2-2x-8=0, so (x-4)(x+2)=0 and x=4.
Question 4: For what positive k does 3x^2 + kx + 12 = 0 have exactly one real solution?
- 6
- 9
- 4
- 12 (Correct answer)
Correct answer: 12
The discriminant k^2-144=0 gives k=12.
Question 5: If f(x) = 2x+1 and g(x) = x^2, what is f(g(3))?
- 37
- 49
- 19 (Correct answer)
- 10
Correct answer: 19
g(3)=9 and f(9)=2(9)+1=19.
Question 6: Solve |2x-3|=7. What is the sum of all solutions?
- 8
- 6
- 0
- 3 (Correct answer)
Correct answer: 3
2x-3=7 gives x=5 and 2x-3=-7 gives x=-2; sum=3.
AMC12 Algebra 1
If 2^x = 3, what is the value of 4^x?