Format: 22 questions · 35 minutes · calculator permitted
When to use: after scoring 16+ on Practice B Module 1. Harder second module; full worked solutions follow. (Answer key balanced across A–D.)
Questions
1. If f(x) = 2x² − 8x + 5, at what x is f minimized?
A) 2 B) 4 C) 8 D) 5
2. The equation x² − 6x + k = 0 has exactly one real solution. What is k?
A) 3 B) 6 C) 9 D) 12
3. If 4/x + 2/x = 3/2, what is x?
A) 2 B) 3 C) 6 D) 4
4. If x² − y² = 40 and x − y = 5, what is x + y?
A) 5 B) 10 C) 35 D) 8
5. A quantity halves every 3 hours. Which models the amount A after t hours if A₀ = 64?
A) 64(2)^(t/3) B) 64(0.5)^(t/3) C) 64(0.5)³ᵗ D) 64 − 0.5t
6. The parabola y = x² + bx + 16 is tangent to the x-axis. A possible value of b is:
A) 4 B) 16 C) 32 D) 8
7. If √(3x + 4) = x, what is the valid solution?
A) −1 B) 1 C) 4 D) 9
8. Simplify (x² + 2x − 15)/(x² − 9) (for x ≠ ±3).
A) (x+5)/(x+3) B) (x − 5)/(x − 3) C) (x+5)/(x − 3) D) (x − 3)/(x+3)
9. A circle has equation x² + y² − 4x + 6y = 12. What is its radius?
A) 5 B) 12 C) √12 D) 25
10. If 4ˣ = 8^(x − 1), what is x?
A) 1 B) 2 C) 3 D) 4
11. In a right triangle, cos θ = 5/13. What is sin θ?
A) 5/13 B) 12/13 C) 13/12 D) 13/5
12. The sum of two numbers is 15 and their product is 56. What is the larger number?
A) 7 B) 8 C) 9 D) 14
13. A function g(x) = ax + b has g(2) = 7 and g(5) = 16. What is a?
A) 2 B) 4 C) 9 D) 3
14. The data 3, 3, 4, 8, 12 has 12 replaced by 42. Which changes most?
A) median B) mode C) mean D) they stay equal
15. A cone and a cylinder share radius and height. The cylinder's volume is how many times the cone's?
A) 2 B) 3 C) 1/3 D) equal
16. If x² + 8x + 7 = 0, what is the sum of the solutions?
A) −8 B) 8 C) −7 D) 7
17. A line perpendicular to y = (1/3)x + 2 has slope:
A) 1/3 B) −1/3 C) 3 D) −3
18. A population triples every 2 days. Starting at 100, after 6 days it is:
A) 300 B) 900 C) 2,700 D) 600
19. The graph of y = |x + 1| − 3 has a minimum value of:
A) −3 B) −1 C) 1 D) 3
Questions 20–22 are student-produced responses.
20. If x² − 10x + 21 = 0, what is the larger solution?
21. A line passes through (2, 5) with slope 3. What is its y-intercept?
22. A mixture is 30% acid. How many liters of pure acid must be added to 20 L to make it 50% acid?
Answer key
| Q | Ans | Q | Ans | Q | Ans |
|---|---|---|---|---|---|
| 1 | A | 9 | A | 17 | D |
| 2 | C | 10 | C | 18 | C |
| 3 | D | 11 | B | 19 | A |
| 4 | D | 12 | B | 20 | 7 |
| 5 | B | 13 | D | 21 | −1 |
| 6 | D | 14 | C | 22 | 8 |
| 7 | C | 15 | B | ||
| 8 | A | 16 | A |
Key distribution (MC): A×5, B×4, C×5, D×5.
Worked solutions
1. (A) Vertex x = −b/(2a) = −(−8)/(2 × 2) = 8/4 = 2.
2. (C) Discriminant 0: 36 − 4k = 0 ⇒ k = 9.
3. (D) 6/x = 3/2 ⇒ x = 4.
4. (D) (x − y)(x+y) = 40; 5(x+y) = 40 ⇒ x+y = 8.
5. (B) Halving every 3 hours → 64(0.5)^(t/3).
6. (D) Tangent ⇒ discriminant 0: b² − 64 = 0 ⇒ b = ±8, so 8 is a possible value.
7. (C) 3x + 4 = x² ⇒ x² − 3x − 4 = 0 ⇒ (x − 4)(x+1)=0; valid root x = 4.
8. (A) x²+2x − 15=(x+5)(x − 3) and x² − 9=(x+3)(x − 3); cancel (x − 3) → (x+5)/(x+3).
9. (A) Complete the square: (x − 2)² + (y+3)² = 12 + 4 + 9 = 25 ⇒ r = 5.
10. (C) 4ˣ = 2²ˣ, 8^(x − 1) = 2^(3(x − 1)); 2x = 3x − 3 ⇒ x = 3.
11. (B) sin θ = √(1 − (5/13)²) = √(144/169) = 12/13.
12. (B) Roots of t² − 15t + 56 = 0: (t − 7)(t − 8); larger is 8.
13. (D) Slope a = (16 − 7)/(5 − 2) = 3.
14. (C) Replacing 12 with 42 adds 30 to the sum → mean shifts by 6; median/mode unchanged.
15. (B) Cylinder = 3 × cone (cone is one-third).
16. (A) Sum of roots = −b/a = −8.
17. (D) Perpendicular slope = negative reciprocal of 1/3 = −3.
18. (C) 100 · 3^(6/2) = 100 · 27 = 2700.
19. (A) |x+1| ≥ 0; minimum 0 − 3 = −3.
20. (7) (x − 3)(x − 7) = 0; larger root 7.
21. (−1) y = 3x + b; 5 = 6 + b ⇒ b = −1.
22. (8) Acid: 0.3(20) + a = 0.5(20 + a) ⇒ 6 + a = 10 + 0.5a ⇒ 0.5a = 4 ⇒ a = 8 L.
Scoring note
Aim for 15+ on this weighted module. Review by topic: discriminant/quadratics (Q1, Q2, Q6, Q16, Q20), exponential models (Q5, Q10, Q18), coordinate geometry (Q9, Q17, Q21), and mixture/rate setups (Q22).