Bluebook SAT Test Digital SAT Hard Math Practice 2

Question 1

Let f(x) = √(3x − 9). What is the domain of f?

Accepted Answer

x ≥ 3

Answer

The radicand must be non-negative: 3x − 9 ≥ 0 → x ≥ 3.

Question 2

The graphs of y = ax² + bx + c and y = dx + e intersect at exactly one point. Which condition must be true?

Accepted Answer

The discriminant of ax² + (b−d)x + (c−e) = 0 equals zero

Answer

Substituting y = dx + e into the quadratic gives ax² + (b−d)x + (c−e) = 0; exactly one intersection means the discriminant (b−d)² − 4a(c−e) = 0.

Question 3

If cos(2θ) = 1 − 2sin²θ, and sin θ = 3/5, what is cos(2θ)?

Accepted Answer

7/25

Answer

sin²θ = 9/25, so cos(2θ) = 1 − 2(9/25) = 1 − 18/25 = 7/25.

Question 4

A function f satisfies f(x + 2) = 3f(x) for all real x, and f(0) = 4. What is f(6)?

Accepted Answer

108

Answer

f(2) = 3(4) = 12; f(4) = 3(12) = 36; f(6) = 3(36) = 108.

Question 5

The polynomial p(x) = x³ + ax² + bx − 8 has (x − 2) as a factor, and p(−1) = −18. What is the value of a?

Accepted Answer

−3

Answer

p(2)=0: 4a+2b=0 → b=−2a. p(−1)=−18: a−b=−9 → 3a=−9 → a=−3.

Bluebook SAT Practice Test