AMC12 Algebra

Question 1

If 2^x = 3, what is the value of 4^x?

Accepted Answer

9

Answer

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?

Accepted Answer

13

Answer

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.

Accepted Answer

4

Answer

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?

Accepted Answer

12

Answer

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))?

Accepted Answer

19

Answer

g(3)=9 and f(9)=2(9)+1=19.

AMC12 - American Mathematics Competition 12 Practice Test