Digital SAT Hard Math Practice Flashcards

The equation x² + y² + 4x − 10y + 20 = 0 represents a circle. What is the area of this circle?

Answer: 9π

Complete the square: (x+2)² + (y−5)² = −20 + 4 + 25 = 9; radius = 3, area = 9π.

If f(x) = 2^x and g(x) = f(f(x)), what is g(3)?

Answer: 256

f(3) = 2³ = 8; g(3) = f(f(3)) = f(8) = 2⁸ = 256.

In a right triangle, the two acute angles satisfy α + β = 90°. If tan α = 5/12, what is sin β?

Answer: 12/13

Since β = 90°−α, sin β = cos α. With tan α = 5/12, hypotenuse = 13 and cos α = 12/13.

The function f(x) = (x² − 1)/(x + 1) has a removable discontinuity at x = −1. What value would make f continuous there?

Answer: −2

Factor: (x²−1)/(x+1) = (x−1)(x+1)/(x+1) = x − 1 for x ≠ −1; at x = −1, the limit is −2.

A survey of 200 students found that 120 passed Math, 90 passed English, and 40 passed both. How many passed neither subject?

Answer: 30

By inclusion-exclusion: passed at least one = 120 + 90 − 40 = 170; passed neither = 200 − 170 = 30.

If p(x) = x⁴ − 5x² + 4, how many real roots does p(x) have?

Answer: 4

Substituting u = x²: u² − 5u + 4 = (u−1)(u−4) = 0; u = 1 → x = ±1 and u = 4 → x = ±2; total 4 real roots.

A population grows from 2,000 to 8,000 in 6 years following exponential growth. What is the annual growth rate, rounded to the nearest percent?

Answer: 26%

8000 = 2000·(1+r)⁶ → (1+r)⁶ = 4 → 1+r = 4^(1/6) ≈ 1.2599 → r ≈ 26%.