AMC12 Algebra 3

Question 1

If log(xy)=5 and log(x/y)=1, what is log x?

Accepted Answer

3

Answer

Adding the two equations gives 2log(x)=6, so log(x)=3.

Question 2

How many real solutions does x^4 - 5x^2 + 4 = 0 have?

Accepted Answer

4

Answer

Substituting u=x^2 gives (u-1)(u-4)=0, yielding x=+-1 or x=+-2, so 4 real solutions.

Question 3

If a and b are roots of 2x^2-3x+1=0, what is a/b + b/a?

Accepted Answer

5/2

Answer

a/b+b/a=(a^2+b^2)/ab=((a+b)^2-2ab)/ab=(9/4-1)/(1/2)=5/2.

Question 4

What is the range of f(x) = 3/(x^2+1)?

Accepted Answer

(0,3]

Answer

Since x^2+1>=1, we have 0<3/(x^2+1)<=3, with maximum 3 at x=0.

Question 5

If 2^(x+1) = 3^x, what is x?

Accepted Answer

log(2)/log(3/2)

Answer

Taking logs: (x+1)log2=x*log3, so x(log3-log2)=log2, giving x=log2/log(3/2).

AMC12 - American Mathematics Competition 12 Practice Test