Bluebook SAT Test Digital SAT Hard Math Practice 6

Question 1

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

Accepted Answer

9π

Answer

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

Question 2

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

Accepted Answer

256

Answer

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

Question 3

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

Accepted Answer

12/13

Answer

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

Question 4

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

Accepted Answer

−2

Answer

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

Question 5

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

Accepted Answer

30

Answer

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

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