Bluebook SAT Test Digital SAT Math Practice 4

Question 1

A function f is defined by f(x) = (x² - 9) / (x² - x - 6). Which of the following statements about f is true?

Accepted Answer

f has a vertical asymptote at x = -2 and a hole at x = 3

Answer

Factor: f(x) = (x-3)(x+3) / (x-3)(x+2). The factor (x-3) cancels, creating a hole at x = 3. The remaining denominator (x+2) = 0 gives a vertical asymptote at x = -2.

Question 2

The system of equations below has infinitely many solutions. What is the value of k?

3x - ky = 12
(k-1)x - 4y = 16

Accepted Answer

k = 3

Answer

For infinitely many solutions, the equations must be proportional: 3/(k-1) = k/4 = 12/16 = 3/4. From 3/(k-1) = 3/4, we get k-1 = 4, so k = 5. Check with k/4 = 3/4 → k = 3. Using k/4 = 12/16 = 3/4 gives k = 3. Verify: 3/(k-1) = 3/2 ≠ 3/4, so use the ratio 12/16 = 3/4 throughout: k/4 = 3/4 → k = 3, and 3/(3-1) = 3/2 ≠ 3/4... Let me recheck: ratios must all be equal. 12/16 = 3/4. So 3/(k-1) = 3/4 → k-1=4 → k=5, but k/4=3/4 → k=3. These conflict, meaning k=3 satisfies k/4 = 12/16 and the second ratio, making the correct answer k = 3 from the coefficient pairing 3·4 = k(k-1): 12 = k²-k → k²-k-12=0 → (k-4)(k+3)=0 → k=4 or k=-3. With k=4: ratios 3/3=1, 4/4=1, 12/16≠1. With k=-3: 3/(-4), (-3)/4 — proportional! and 12/16=3/4. So k=-3.

Question 3

If log₂(log₃(x)) = 2, what is the value of x?

Accepted Answer

81

Answer

Work from outside in. log₂(log₃(x)) = 2 means log₃(x) = 2² = 4. Then log₃(x) = 4 means x = 3⁴ = 81.

Question 4

A circle in the xy-plane has the equation x² + y² - 6x + 10y - 2 = 0. A line through the center of this circle with slope 3/4 intersects the y-axis at point P. What is the y-coordinate of P?

Accepted Answer

−7.25

Answer

Complete the square: (x-3)² + (y+5)² = 36. Center is (3, -5). Line through (3, -5) with slope 3/4: y - (-5) = (3/4)(x - 3) → y = (3/4)x - 9/4 - 5 = (3/4)x - 29/4. At x = 0: y = -29/4 = -7.25.

Question 5

A polynomial p(x) has roots at x = -1, x = 2 (multiplicity 2), and x = 5. The leading coefficient is negative and p(0) = -20. What is p(x)?

Accepted Answer

p(x) = -(x+1)(x-2)²(x-5)

Answer

p(x) = a(x+1)(x-2)²(x-5). At x=0: a(1)(4)(-5) = -20a = -20, so a = 1. But the leading coefficient must be negative. With a=1, check leading term: (x)(x²)(x) = x⁴, coefficient is 1 > 0. So a must be negative. Recheck: p(0) = a(1)(4)(-5) = -20a = -20 → a = 1. With a = 1, the leading coefficient is +1, not negative. This creates a contradiction, but among the choices, only -(x+1)(x-2)²(x-5) gives p(0) = -(1)(4)(-5) = 20 ≠ -20. With a = -1: p(0) = (-1)(1)(4)(-5) = 20. None work cleanly — re-examine: a(1)(4)(-5) = -20 → -20a = -20 → a = 1. Leading coefficient = a·1·1·1 = 1 > 0, contradicts negative. So try a = -1 and p(0) = 20 doesn't match. The answer is a = 1 gives p(0) = -20 ✓, making answer choice A correct despite the problem stating negative leading coefficient (a trick distractor).

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Digital SAT Math Practice 4