Bluebook SAT Test Digital SAT Hard Math Practice 2 — Questions and Answers
Question 1: Let f(x) = √(3x − 9). What is the domain of f?
- x ≥ 3 (Correct answer)
- x > 3
- x ≥ 0
- All real numbers
Correct answer: x ≥ 3
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?
- b² − 4ac = 0 after substitution
- The discriminant of ax² + (b−d)x + (c−e) = 0 equals zero (Correct answer)
- a = d
- c = e
Correct answer: The discriminant of ax² + (b−d)x + (c−e) = 0 equals zero
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)?
- 7/25 (Correct answer)
- −7/25
- 1/25
- 24/25
Correct answer: 7/25
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)?
- 108 (Correct answer)
- 36
- 324
- 12
Correct answer: 108
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?
- −3 (Correct answer)
- 3
- −1
- 2
Correct answer: −3
p(2)=0: 4a+2b=0 → b=−2a. p(−1)=−18: a−b=−9 → 3a=−9 → a=−3.
Question 6: If log₂(x + 1) + log₂(x − 3) = 5, what is the value of x?
- 7 (Correct answer)
- 5
- 9
- 3
Correct answer: 7
Combining: log₂[(x+1)(x−3)] = 5 → (x+1)(x−3) = 32 → x² − 2x − 3 = 32 → x² − 2x − 35 = 0 → (x−7)(x+5) = 0; x = 7 (x = −5 rejected since x − 3 must be positive).
Question 7: In the xy-plane, the circle x² + y² − 6x + 8y = 0 has center (h, k). What is h + k?
- −1 (Correct answer)
- 1
- 7
- −7
Correct answer: −1
Completing the square: (x−3)² + (y+4)² = 25; center is (3, −4), so h + k = 3 + (−4) = −1.
Bluebook SAT Test Digital SAT Hard Math Practice 2
Let f(x) = √(3x − 9).
What is the domain of f?