Digital SAT — Math, Module 2 (Harder Form · Practice Test B)
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 $f(x) = 2x^2 - 8x + 5$, at what $x$ is $f$ minimized? A) 2 B) 4 C) 8 D) 5
2. The equation $x^2 - 6x + k = 0$ has exactly one real solution. What is $k$? A) 3 B) 6 C) 9 D) 12
3. If $\dfrac{4}{x} + \dfrac{2}{x} = \dfrac{3}{2}$, what is $x$? A) 2 B) 3 C) 4 D) 6
4. If $x^2 - y^2 = 40$ and $x - y = 5$, what is $x + y$? A) 5 B) 8 C) 10 D) 35
5. A quantity halves every 3 hours. Which models the amount $A$ after $t$ hours if $A_0 = 64$? A) $64(2)^{t/3}$ B) $64(0.5)^{t/3}$ C) $64(0.5)^{3t}$ D) $64 - 0.5t$
6. The parabola $y = x^2 + bx + 16$ is tangent to the $x$-axis. A possible value of $b$ is: A) 4 B) 8 C) 16 D) 32
7. If $\sqrt{3x + 4} = x$, what is the valid solution? A) $-1$ B) 1 C) 4 D) 9
8. Simplify $\dfrac{x^2 + 2x - 15}{x^2 - 9}$ (for $x \ne \pm 3$). A) $\dfrac{x+5}{x+3}$ B) $\dfrac{x-5}{x-3}$ C) $\dfrac{x+5}{x-3}$ D) $\dfrac{x-3}{x+3}$
9. A circle has equation $x^2 + y^2 - 4x + 6y = 12$. What is its radius? A) 5 B) 12 C) $\sqrt{12}$ D) 25
10. If $4^{x} = 8^{x-1}$, what is $x$? A) 1 B) 2 C) 3 D) 4
11. In a right triangle, $\cos\theta = \tfrac{5}{13}$. What is $\sin\theta$? A) $\tfrac{5}{13}$ B) $\tfrac{12}{13}$ C) $\tfrac{13}{12}$ D) $\tfrac{13}{5}$
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 $g(x) = a x + b$ has $g(2) = 7$ and $g(5) = 16$. What is $a$? A) 2 B) 3 C) 4 D) 9
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) $\tfrac{1}{3}$ D) equal
16. If $x^2 + 8x + 7 = 0$, what is the sum of the solutions? A) $-8$ B) 8 C) $-7$ D) 7
17. A line perpendicular to $y = \tfrac{1}{3}x + 2$ has slope: A) $\tfrac{1}{3}$ B) $-\tfrac{1}{3}$ 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 $x^2 - 10x + 21 = 0$, 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 | C | 11 | B | 19 | A |
| 4 | B | 12 | B | 20 | 7 |
| 5 | B | 13 | B | 21 | -1 |
| 6 | B | 14 | C | 22 | 8 |
| 7 | C | 15 | B | ||
| 8 | A | 16 | A |
Key distribution (MC): A×4, B×7, C×5, D×1.
Worked solutions
1. (A) Vertex $x = -\frac{-8}{2(2)} = \frac{8}{4} = 2$. 2. (C) Discriminant 0: $36 - 4k = 0 \Rightarrow k = 9$. 3. (C) $\frac{6}{x} = \frac{3}{2} \Rightarrow x = 4$. 4. (B) $(x-y)(x+y) = 40$; $5(x+y) = 40 \Rightarrow x+y = 8$. 5. (B) Halving every 3 hours → $64(0.5)^{t/3}$. 6. (B) Tangent ⇒ discriminant 0: $b^2 - 64 = 0 \Rightarrow b = \pm 8$. 7. (C) $3x + 4 = x^2 \Rightarrow x^2 - 3x - 4 = 0 \Rightarrow (x-4)(x+1)=0$; valid root $x = 4$. 8. (A) $x^2+2x-15=(x+5)(x-3)$ and $x^2-9=(x+3)(x-3)$; cancel $(x-3)$ → $\frac{x+5}{x+3}$. 9. (A) Complete the square: $(x-2)^2 + (y+3)^2 = 12 + 4 + 9 = 25 \Rightarrow r = 5$. 10. (C) $4^x = 2^{2x}$, $8^{x-1} = 2^{3(x-1)}$; $2x = 3x - 3 \Rightarrow x = 3$. 11. (B) $\sin\theta = \sqrt{1 - (5/13)^2} = \sqrt{144/169} = 12/13$. 12. (B) Roots of $t^2 - 15t + 56 = 0$: $(t-7)(t-8)$; larger is 8. 13. (B) Slope $a = \frac{16-7}{5-2} = 3$. 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 $\tfrac13$ = $-3$. 18. (C) $100 \cdot 3^{6/2} = 100 \cdot 27 = 2700$. 19. (A) $|x+1| \ge 0$; minimum $0 - 3 = -3$. 20. (7) $(x-3)(x-7) = 0$; larger root 7. 21. (-1) $y = 3x + b$; $5 = 6 + b \Rightarrow b = -1$. 22. (8) Acid: $0.3(20) + a = 0.5(20 + a) \Rightarrow 6 + a = 10 + 0.5a \Rightarrow 0.5a = 4 \Rightarrow a = 8$ 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).