Bluebook SAT Test Digital SAT Hard Math 1 — Questions and Answers
Question 1: If f(x) = x² - 6x + 9 and g(x) = x - 3, for what value of x does f(x)/g(x) = 7?
- x = 10 (Correct answer)
- x = 4
- x = 7
- x = 3
Correct answer: x = 10
f(x)/g(x) = (x-3)²/(x-3) = x-3 for x ≠ 3. Setting x - 3 = 7 gives x = 10.
f(x) = x² - 6x + 9 = (x-3)². So f(x)/g(x) = (x-3)²/(x-3) = x - 3 for x ≠ 3. Setting x - 3 = 7 yields x = 10. This tests knowledge of factoring perfect square trinomials and simplifying rational expressions.
Question 2: A population of bacteria triples every 4 hours. If there are 200 bacteria initially, which expression gives the number after t hours?
- 200 · 3^(t/4) (Correct answer)
- 200 · 3^(4t)
- 200 · (t/4)^3
- 3 · 200^(t/4)
Correct answer: 200 · 3^(t/4)
Every 4 hours the count multiplies by 3, so after t hours there are t/4 intervals of tripling, giving 200 · 3^(t/4).
Exponential growth with period p and growth factor b is modeled as A₀ · b^(t/p). Here the initial amount A₀ = 200, b = 3 (tripling), and p = 4 hours. Substituting gives 200 · 3^(t/4). After 4 hours (t=4), this equals 200 · 3¹ = 600, confirming the tripling behavior.
Question 3: The system of equations 3x + ky = 12 and 6x + 4y = 24 has infinitely many solutions. What is the value of k?
- 2 (Correct answer)
- 6
- 3
- 4
Correct answer: 2
For infinitely many solutions, the equations must be proportional. Multiplying the first by 2 gives 6x + 2ky = 24. For this to match 6x + 4y = 24, we need 2k = 4, so k = 2.
Two linear equations have infinitely many solutions when they represent the same line — one is a constant multiple of the other. The second equation is exactly 2 times the first: 2(3x + ky) = 6x + 2ky = 24. Comparing 2ky with 4y, we get 2k = 4, so k = 2. Always check all coefficients and the constant when verifying proportionality.
Question 4: A quadratic equation has roots at x = 2 + √5 and x = 2 - √5. Which of the following could be the equation?
- x² - 4x - 1 = 0 (Correct answer)
- x² + 4x - 1 = 0
- x² - 4x + 1 = 0
- x² - 4x + 9 = 0
Correct answer: x² - 4x - 1 = 0
Sum of roots = 4 (coefficient of x is -4), product of roots = (2)² - (√5)² = 4 - 5 = -1 (constant term is -1).
For roots r₁ = 2+√5 and r₂ = 2-√5: sum = r₁+r₂ = 4, product = r₁·r₂ = (2+√5)(2-√5) = 4-5 = -1. A monic quadratic with these roots is x² - (sum)x + product = x² - 4x + (-1) = x² - 4x - 1 = 0. This tests the Vieta's formulas relationship.
Question 5: If 2^(x+1) = 5^(x-1), what is the value of x? (Round to the nearest hundredth.)
- x ≈ 4.31 (Correct answer)
- x ≈ 2.15
- x ≈ 1.76
- x ≈ 3.87
Correct answer: x ≈ 4.31
Taking log of both sides: (x+1)ln2 = (x-1)ln5. Solving: x(ln2-ln5) = -ln5-ln2, so x = (ln5+ln2)/(ln5-ln2) ≈ 4.31.
Applying ln to both sides: (x+1)ln2 = (x-1)ln5. Expanding: x·ln2 + ln2 = x·ln5 - ln5. Rearranging: ln2 + ln5 = x·ln5 - x·ln2 = x(ln5-ln2). So x = (ln2+ln5)/(ln5-ln2) = ln10/(ln5-ln2) ≈ 2.3026/0.5346 ≈ 4.31.
Question 6: A circle has center (3, -2) and passes through the point (7, 1). What is the area of the circle?
- 25π (Correct answer)
- 16π
- 5π
- √(25)π
Correct answer: 25π
Radius = √((7-3)² + (1-(-2))²) = √(16+9) = √25 = 5. Area = π·5² = 25π.
The radius of a circle is the distance from the center to any point on the circle. Using the distance formula: r = √((7-3)² + (1-(-2))²) = √(4² + 3²) = √(16+9) = √25 = 5. The area is then πr² = π(5²) = 25π. This is a classic application of the Pythagorean theorem in coordinate geometry.
Question 7: The function h(t) = -16t² + 64t + 6 models the height (in feet) of a projectile t seconds after launch. What is the maximum height reached?
- 70 feet (Correct answer)
- 64 feet
- 80 feet
- 86 feet
Correct answer: 70 feet
Maximum occurs at t = -64/(2·(-16)) = 2. h(2) = -16(4) + 64(2) + 6 = -64 + 128 + 6 = 70.
For h(t) = -16t² + 64t + 6, the vertex (maximum) occurs at t = -b/(2a) = -64/(2·(-16)) = -64/(-32) = 2 seconds. Substituting back: h(2) = -16(2²) + 64(2) + 6 = -16(4) + 128 + 6 = -64 + 128 + 6 = 70 feet. The downward parabola confirms this is a maximum.
Question 8: If |3x - 7| = 2x + 1, which of the following gives all solutions for x?
- x = 8 or x = 6/5 (Correct answer)
- x = 8 only
- x = 6/5 only
- No solution
Correct answer: x = 8 or x = 6/5
Case 1: 3x-7 = 2x+1 → x = 8 (check: |17| = 17 ✓). Case 2: 3x-7 = -(2x+1) → 5x = 6 → x = 6/5 (check: |18/5-7| = 17/5, 2(6/5)+1 = 17/5 ✓).
For |3x-7| = 2x+1, note 2x+1 must be ≥ 0, so x ≥ -1/2. Case 1: 3x-7 = 2x+1 gives x = 8. Check: |3(8)-7| = |17| = 17 = 2(8)+1 ✓. Case 2: 3x-7 = -(2x+1) = -2x-1 gives 5x = 6, x = 6/5. Check: |3(6/5)-7| = |18/5-35/5| = 17/5 = 2(6/5)+1 = 17/5 ✓. Both solutions are valid.
Bluebook SAT Test Digital SAT Hard Math 1
If f(x) = x² - 6x + 9 and g(x) = x - 3, for what value of x does f(x)/g(x) = 7?