AMC12 Algebra 2

Question 1

If a+b=5 and ab=4, find a^3+b^3.

Accepted Answer

65

Answer

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.

Accepted Answer

6

Answer

By Vieta's formulas, the sum of roots equals 6.

Question 3

Solve 3^(2x) - 4*3^x + 3 = 0. What are the solutions?

Accepted Answer

x=0 or x=1

Answer

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?

Accepted Answer

23

Answer

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).

Accepted Answer

24

Answer

f(2)=6, f(4)=12, f(8)=24.

AMC12 - American Mathematics Competition 12 Practice Test