Digital SAT Hard Math Practice Flashcards

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

Answer: x ≥ 3

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

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

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.

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

Answer: 7/25

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

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

Answer: 108

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

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

Answer: −3

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

If log₂(x + 1) + log₂(x − 3) = 5, what is the value of x?

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).

In the xy-plane, the circle x² + y² − 6x + 8y = 0 has center (h, k). What is h + k?

Answer: −1

Completing the square: (x−3)² + (y+4)² = 25; center is (3, −4), so h + k = 3 + (−4) = −1.