Format: 22 questions · 35 minutes · calculator permitted throughout
Coverage: Algebra · Advanced Math · Problem-Solving & Data Analysis · Geometry & Trigonometry
When to use this module: Attempt this after scoring 16+ on Math Module 1. On the real adaptive test, strong Module 1 performance routes you to this harder second module, where the questions carry more weight. Full worked solutions and a scoring note follow.
Questions
1. If f(x) = 3x² − 12x + 7, for what value of x does f attain its minimum?
A) 7 B) 4 C) −2 D) 2
2. The equation 2x² − 8x + k = 0 has exactly one real solution. What is k?
A) 8 B) 2 C) 4 D) 16
3. If 3/x + 2/x = 1/4, what is x?
A) 24 B) 10 C) 16 D) 20
4. A function is defined by g(x) = 2ˣ. If g(a + 3) = 8 g(a), the equation holds for:
A) only a = 0 B) no value of a C) only a = 3 D) all values of a
5. The line y = mx + b passes through (2, 5) and (6, 17). What is b?
A) 1 B) −1 C) 5 D) 3
6. If x² − y² = 24 and x − y = 4, what is x + y?
A) 4 B) 12 C) 6 D) 8
7. A quantity decays so that it halves every 5 years. Which models the amount A after t years if A₀ = 80?
A) A = 80(2)^(t/5) B) A = 80(0.5)^(t/5) C) A = 80 − 0.5t D) A = 80(0.5)⁵ᵗ
8. In the xy-plane, the parabola y = x² + bx + 9 is tangent to the x-axis. What is a possible value of b?
A) 9 B) 3 C) 6 D) 18
9. If √(2x + 3) = x, what is the value of x? (Consider only valid solutions.)
A) 1 B) −1 C) 3 D) 9
10. The expression (x² + x − 6)/(x² − 9) simplifies to (for x ≠ ±3):
A) (x − 3)/(x + 3) B) (x − 2)/(x + 3) C) (x − 2)/(x − 3) D) (x + 2)/(x + 3)
11. A circle has equation x² + y² − 6x + 8y = 0. What is its radius?
A) 10 B) 25 C) √14 D) 5
12. If 3²ˣ = 27^(x − 1), what is x?
A) 2 B) 3 C) 4 D) 1
13. A data set consists of the values 4, 4, 5, 9 and 13. If the value 13 is replaced by 33, by how much does the mean of the data set increase?
A) 20 B) 0 C) 6.6 D) 4
14. A right circular cone and a cylinder share the same radius and height. The cone's volume is what fraction of the cylinder's?
A) 2/3 B) 1/3 C) 1/2 D) equal
15. If f(x) = ax + b, f(1) = 5, and f(4) = 17, what is a?
A) 4 B) 12 C) 3 D) 5
16. In a right triangle, one acute angle θ has cos θ = 3/5. What is sin θ?
A) 4/5 B) 3/5 C) 5/3 D) 5/4
17. The sum of two numbers is 20 and their product is 96. What is the larger number?
A) 12 B) 10 C) 16 D) 8
18. A population of bacteria triples every hour. If it starts at 200, how many are there after 3 hours?
A) 5,400 B) 1,200 C) 1,800 D) 600
19. The graph of y = |x − 3| − 2 has its minimum value at y =
A) 3 B) −2 C) 0 D) 1
Questions 20–22 are student-produced responses (grid-in).
20. If x² + 6x + 5 = 0, what is the sum of the solutions?
21. A line perpendicular to y = (1/2)x + 4 passes through (0, 3). What is its slope?
22. A jar holds red and blue marbles in the ratio 2:3. If 4 red marbles are added, the ratio becomes 1:1. How many blue marbles are in the jar?
Answer key
| Q | Ans | Q | Ans | Q | Ans |
|---|---|---|---|---|---|
| 1 | D | 9 | C | 17 | A |
| 2 | A | 10 | C | 18 | A |
| 3 | D | 11 | D | 19 | B |
| 4 | D | 12 | B | 20 | −6 |
| 5 | B | 13 | D | 21 | −2 |
| 6 | C | 14 | B | 22 | 12 |
| 7 | B | 15 | A | ||
| 8 | C | 16 | A |
Worked solutions
1. (D) Vertex at x = −b/2a = −(−12)/(2 × 3) = 12/6 = 2.
2. (A) One real solution ⇒ discriminant = 0: (−8)² − 4(2)k = 0 ⇒ 64 = 8k ⇒ k = 8.
3. (D) 5/x = 1/4 ⇒ x = 20.
4. (D) g(a+3) = 2ᵃ⁺³ = 2ᵃ · 2³ = 8 · 2ᵃ = 8 g(a) — true for all a.
5. (B) Slope = (17 − 5)/(6 − 2) = 3. Using (2,5): 5 = 3(2) + b ⇒ b = −1.
6. (C) x² − y² = (x − y)(x+y) = 4(x+y) = 24 ⇒ x+y = 6.
7. (B) Halving every 5 years ⇒ factor 0.5 raised to t/5: A = 80(0.5)^(t/5).
8. (C) Tangent to x-axis ⇒ discriminant 0: b² − 4(1)(9) = 0 ⇒ b² = 36 ⇒ b = ±6. So 6 works.
9. (C) Square: 2x + 3 = x² ⇒ x² − 2x − 3 = 0 ⇒ (x − 3)(x+1)=0. Only x = 3 satisfies the original (the root must be non-negative and check: √9=3 ✓; x=−1 fails).
10. (C) Factor: numerator x² + x − 6 = (x+3)(x − 2), denominator x² − 9 = (x+3)(x − 3). Cancel (x+3): (x − 2)/(x − 3).
11. (D) Complete the square: (x − 3)² + (y+4)² = 9 + 16 = 25 ⇒ r = 5.
12. (B) 27^(x − 1) = 3^(3(x − 1)), so 2x = 3x − 3 ⇒ x = 3.
13. (D) Replacing 13 with 33 adds 20 to the sum of the five values, so the mean rises by 20/5 = 4 (from 7 to 11). (20 is the change in the sum, not the mean; 0 is the change in the median; 6.6 is 33/5.)
14. (B) Cone volume = (1/3)πr²h and cylinder volume = πr²h, so the ratio is 1/3.
15. (A) Slope a = (17 − 5)/(4 − 1) = 12/3 = 4.
16. (A) sin²θ = 1 − (3/5)² = 1 − 9/25 = 16/25 ⇒ sin θ = 4/5 (3-4-5 triangle).
17. (A) Numbers solve t² − 20t + 96 = 0 ⇒ (t − 8)(t − 12)=0; larger is 12.
18. (A) 200 · 3³ = 200 · 27 = 5400.
19. (B) |x − 3| ≥ 0, minimum 0 at x=3, so least y = 0 − 2 = −2.
20. (−6) Sum of roots = −b/a = −6/1 = −6 (roots −1, −5).
21. (−2) Perpendicular slope is the negative reciprocal of 1/2, i.e. −2.
22. (12) Let reds = 2k, blues = 3k. After adding 4 reds: 2k + 4 = 3k ⇒ k = 4. Blues = 3(4) = 12.
Scoring note
Only the harder second module gives access to the top of the score range. Aim for 15+ correct here to land in the upper score range. If you missed:
- Quadratics/discriminant (Q2, Q8, Q17, Q20): revise the Advanced Math guide — discriminant and Vieta's relationships.
- Exponential models (Q4, Q7, Q12, Q18): revise growth/decay factors.
- Coordinate geometry (Q5, Q11, Q21): slope, perpendicularity, circle completing-the-square.
- Ratios & statistics (Q13, Q22): the Problem-Solving & Data Analysis guide.
Pedagogy: Harder SAT items reward structural moves — factoring to reveal (x − y)(x+y) in Q6, recognizing a 3-4-5 triangle in Q16, using sum/product of roots in Q17/Q20 — rather than brute force. Train the pattern, not just the answer.