Digital SAT — Math, Module 2 (Harder Form · Practice Test A)

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 $f(x) = 3x^2 - 12x + 7$, for what value of $x$ does $f$ attain its minimum? A) $-2$ B) 2 C) 4 D) 7

2. The equation $2x^2 - 8x + k = 0$ has exactly one real solution. What is $k$? A) 2 B) 4 C) 8 D) 16

3. If $\dfrac{3}{x} + \dfrac{2}{x} = \dfrac{1}{4}$, what is $x$? A) 10 B) 16 C) 20 D) 24

4. A function is defined by $g(x) = 2^x$. If $g(a + 3) = 8,g(a)$, the equation holds for: A) only $a = 0$ B) only $a = 3$ C) all values of $a$ D) no value of $a$

5. The line $y = mx + b$ passes through $(2, 5)$ and $(6, 17)$. What is $b$? A) $-1$ B) 1 C) 3 D) 5

6. If $x^2 - y^2 = 24$ and $x - y = 4$, what is $x + y$? A) 4 B) 6 C) 8 D) 12

7. A quantity decays so that it halves every 5 years. Which models the amount $A$ after $t$ years if $A_0 = 80$? A) $A = 80(2)^{t/5}$ B) $A = 80(0.5)^{t/5}$ C) $A = 80(0.5)^{5t}$ D) $A = 80 - 0.5t$

8. In the xy-plane, the parabola $y = x^2 + bx + 9$ is tangent to the x-axis. What is a possible value of $b$? A) 3 B) 6 C) 9 D) 18

9. If $\sqrt{2x + 3} = x$, what is the value of $x$? (Consider only valid solutions.) A) 1 B) 3 C) $-1$ D) 9

10. The expression $\dfrac{x^2 + x - 6}{x^2 - 9}$ simplifies to (for $x \ne \pm 3$): A) $\dfrac{x+2}{x+3}$ B) $\dfrac{x-2}{x-3}$ C) $\dfrac{x-2}{x+3}$ D) $\dfrac{x-3}{x+3}$

11. A circle has equation $x^2 + y^2 - 6x + 8y = 0$. What is its radius? A) 5 B) 10 C) $\sqrt{14}$ D) 25

12. If $3^{2x} = 27^{x-1}$, what is $x$? A) 1 B) 2 C) 3 D) 4

13. Data set: the values 4, 4, 5, 9, 13. If 13 is replaced by 33, which statistic changes the most? A) median B) mode C) mean D) range stays equal

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) $\frac{1}{2}$ B) $\frac{1}{3}$ C) $\frac{2}{3}$ D) equal

15. If $f(x) = ax + b$, $f(1) = 5$, and $f(4) = 17$, what is $a$? A) 3 B) 4 C) 5 D) 12

16. In a right triangle, one acute angle $\theta$ has $\cos\theta = \frac{3}{5}$. What is $\sin\theta$? A) $\frac{3}{5}$ B) $\frac{4}{5}$ C) $\frac{5}{4}$ D) $\frac{5}{3}$

17. The sum of two numbers is 20 and their product is 96. What is the larger number? A) 8 B) 10 C) 12 D) 16

18. A population of bacteria triples every hour. If it starts at 200, how many are there after 3 hours? A) 600 B) 1,800 C) 5,400 D) 1,200

19. The graph of $y = |x - 3| - 2$ has its minimum value at $y =$ A) $-2$ B) 0 C) 1 D) 3

Questions 20–22 are student-produced responses (grid-in).

20. If $x^2 + 6x + 5 = 0$, what is the sum of the solutions?

21. A line perpendicular to $y = \frac{1}{2}x + 4$ 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

Worked solutions

1. (B) Vertex at $x = -\dfrac{b}{2a} = -\dfrac{-12}{2(3)} = \dfrac{12}{6} = 2$.

2. (C) One real solution ⇒ discriminant $= 0$: $(-8)^2 - 4(2)k = 0 \Rightarrow 64 = 8k \Rightarrow k = 8$.

3. (C) $\dfrac{5}{x} = \dfrac{1}{4} \Rightarrow x = 20$.

4. (C) $g(a+3) = 2^{a+3} = 2^a \cdot 2^3 = 8 \cdot 2^a = 8,g(a)$ — true for all $a$.

5. (A) Slope $= \dfrac{17-5}{6-2} = 3$. Using $(2,5)$: $5 = 3(2) + b \Rightarrow b = -1$.

6. (B) $x^2 - y^2 = (x-y)(x+y) = 4(x+y) = 24 \Rightarrow x+y = 6$.

7. (B) Halving every 5 years ⇒ factor $0.5$ raised to $t/5$: $A = 80(0.5)^{t/5}$.

8. (B) Tangent to x-axis ⇒ discriminant $0$: $b^2 - 4(1)(9) = 0 \Rightarrow b^2 = 36 \Rightarrow b = \pm 6$. So $6$ works.

9. (B) Square: $2x + 3 = x^2 \Rightarrow x^2 - 2x - 3 = 0 \Rightarrow (x-3)(x+1)=0$. Only $x = 3$ satisfies the original (the root must be non-negative and check: $\sqrt{9}=3$ ✓; $x=-1$ fails).

10. (B) Factor: numerator $x^2 + x - 6 = (x+3)(x-2)$, denominator $x^2 - 9 = (x+3)(x-3)$. Cancel $(x+3)$: $\dfrac{x-2}{x-3}$.

11. (A) Complete the square: $(x-3)^2 + (y+4)^2 = 9 + 16 = 25 \Rightarrow r = 5$.

12. (C) $27^{x-1} = 3^{3(x-1)}$, so $2x = 3x - 3 \Rightarrow x = 3$.

13. (C) Replacing 13 with 33 adds 20 to the sum, shifting the mean by $20/5 = 4$; the median (5) and mode (4) are unchanged. Mean changes most.

14. (B) Cone volume $= \frac{1}{3}\pi r^2 h$ vs cylinder $\pi r^2 h$: ratio $\frac{1}{3}$.

15. (B) Slope $a = \dfrac{17 - 5}{4 - 1} = \dfrac{12}{3} = 4$.

16. (B) $\sin^2\theta = 1 - (3/5)^2 = 1 - 9/25 = 16/25 \Rightarrow \sin\theta = 4/5$ (3-4-5 triangle).

17. (C) Numbers solve $t^2 - 20t + 96 = 0 \Rightarrow (t-8)(t-12)=0$; larger is 12.

18. (C) $200 \cdot 3^3 = 200 \cdot 27 = 5400$.

19. (A) $|x-3| \ge 0$, minimum 0 at $x=3$, so least $y = 0 - 2 = -2$.

20. (-6) Sum of roots $= -\dfrac{b}{a} = -\dfrac{6}{1} = -6$ (roots $-1, -5$).

21. (-2) Perpendicular slope is the negative reciprocal of $\frac{1}{2}$, i.e. $-2$.

22. (12) Let reds $= 2k$, blues $= 3k$. After adding 4 reds: $2k + 4 = 3k \Rightarrow k = 4$. Blues $= 3(4) = 12$.

Scoring note

This harder module weights more toward your final score band. 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.