AMC 12 Algebra & Functions 3

Question 1

If a + b = 5 and ab = 3, what is a^2 + b^2?

Accepted Answer

19

Answer

a^2 + b^2 = (a + b)^2 - 2ab = 25 - 6 = 19.

Question 2

What is the minimum value of f(x) = |x - 2| + |x + 4|?

Accepted Answer

6

Answer

The sum of distances from x to 2 and to -4 is minimized at 6 for any x between them.

Question 3

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

Accepted Answer

4

Answer

It factors as (x^2 - 1)(x^2 - 4) = 0, giving x = ±1, ±2.

Question 4

If f(x) + 2f(1 - x) = x for all real x, what is f(0)?

Accepted Answer

2/3

Answer

Plugging x = 0 and x = 1 gives f(0) + 2f(1) = 0 and f(1) + 2f(0) = 1, so f(0) = 2/3.

Question 5

What is the sum of the infinite series 1 + 1/3 + 1/9 + 1/27 + ...?

Accepted Answer

3/2

Answer

An infinite geometric series sums to a/(1 - r) = 1/(2/3) = 3/2.

AMC 12 - - American Mathematics Contest 12 Practice Test