Bluebook SAT Test Digital SAT Hard Math 4

Question 1

The function f is defined by f(x) = (x² − 9) / (x² − x − 6). At which value of x is f(x) undefined but does NOT have a vertical asymptote?

Accepted Answer

x = 3

Answer

Factor both the numerator and denominator: numerator = (x − 3)(x + 3), denominator = (x − 3)(x + 2). The function is undefined wherever the denominator equals zero: x = 3 and x = −2. At x = 3, the factor (x − 3) cancels from both, producing a removable discontinuity (a hole in the graph) — not a vertical asymptote. At x = −2, only the denominator is zero, so there is a true vertical asymptote. Therefore x = 3 is the answer.

Question 2

Positive integers a and b satisfy a² − b² = 105 and a − b = 3. What is the value of a + b?

Accepted Answer

35

Answer

Apply the difference of squares identity: a² − b² = (a + b)(a − b). Substituting the known value a − b = 3 gives (a + b)(3) = 105, so a + b = 35. Verification: a + b = 35 and a − b = 3 yields a = 19, b = 16. Then 19² − 16² = 361 − 256 = 105. ✓

Question 3

A parabola is defined by y = 2x² + bx + c. Its axis of symmetry is the line x = 3, and it passes through the point (0, 4). What is the value of c − b?

Accepted Answer

16

Answer

The axis of symmetry of y = ax² + bx + c is x = −b/(2a). With a = 2 and axis x = 3: 3 = −b/4, so b = −12. Substituting the point (0, 4): 4 = 2(0)² + (−12)(0) + c, giving c = 4. Therefore c − b = 4 − (−12) = 16.

Question 4

In the xy-plane, line ℓ passes through the origin and is perpendicular to the line 3x − 5y = 15. Which of the following points lies on line ℓ?

Accepted Answer

(3, −5)

Answer

Rewrite 3x − 5y = 15 as y = (3/5)x − 3, giving a slope of 3/5. The perpendicular slope is the negative reciprocal: −5/3. Since line ℓ passes through the origin, its equation is y = (−5/3)x. Testing (3, −5): y = (−5/3)(3) = −5 ✓. The other choices fail: (5, −3) gives −25/3; (−3, −5) gives 5; (−5, 3) gives 25/3.

Question 5

If 4^x = 8^(x − 1), what is the value of x?

Accepted Answer

3

Answer

Rewrite both sides as powers of 2: 4^x = (2²)^x = 2^(2x) and 8^(x−1) = (2³)^(x−1) = 2^(3x−3). Since the bases are equal, set the exponents equal: 2x = 3x − 3, giving x = 3. Verification: 4³ = 64 and 8^(3−1) = 8² = 64. ✓

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Digital SAT Hard Math 4