AMC 12 Algebra & Functions 5

Question 1

If r, s, t are the roots of x^3 - 3x^2 + 5x - 7 = 0, what is r^2 + s^2 + t^2?

Accepted Answer

-1

Answer

r^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + st + tr) = 9 - 10 = -1.

Question 2

What is the sum of the coefficients of the expansion of (2x - 1)^5?

Accepted Answer

1

Answer

The sum of coefficients equals P(1) = (2 - 1)^5 = 1.

Question 3

Which of the following functions is even?

Accepted Answer

f(x) = x^4 - 3x^2 + cos x

Answer

Every term of x^4 - 3x^2 + cos x is unchanged when x is replaced by -x.

Question 4

What is log_3(81) + log_2(1/8)?

Accepted Answer

1

Answer

log_3(81) = 4 and log_2(1/8) = -3, so the sum is 1.

Question 5

What is i^2026, where i^2 = -1?

Accepted Answer

-1

Answer

Since 2026 ≡ 2 (mod 4), i^2026 = i^2 = -1.

AMC 12 - - American Mathematics Contest 12 Practice Test