AMC 12 Algebra & Functions 4

Question 1

What is the remainder when x^3 - 2x^2 + 4x - 5 is divided by x - 2?

Accepted Answer

3

Answer

By the Remainder Theorem, P(2) = 8 - 8 + 8 - 5 = 3.

Question 2

A polynomial P satisfies P(1) = 3 and P(-1) = 5. What is the remainder when P(x) is divided by x^2 - 1?

Accepted Answer

-x + 4

Answer

Writing the remainder as ax + b gives a + b = 3 and -a + b = 5, so a = -1 and b = 4.

Question 3

What is the value of ⌊√1⌋ + ⌊√2⌋ + ... + ⌊√10⌋?

Accepted Answer

19

Answer

Three 1s, five 2s, and two 3s give 3 + 10 + 6 = 19.

Question 4

In an arithmetic sequence, a_3 = 7 and a_10 = 28. What is a_20?

Accepted Answer

58

Answer

The common difference is 21/7 = 3, so a_20 = 7 + 17(3) = 58.

Question 5

If f(x) = 1/(1 - x), what is f(f(x)) for x ≠ 0, 1?

Accepted Answer

(x - 1)/x

Answer

f(f(x)) = 1/(1 - 1/(1 - x)) = (1 - x)/(-x) = (x - 1)/x.

AMC 12 - - American Mathematics Contest 12 Practice Test