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20 Questions · 35 min
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College Board · Digital SAT · Math Module 2

20 Hard Questions
You Must Master

Each question is reverse-engineered from the solution up — every answer choice targets a specific common error. Master these patterns and you master the SAT Math section.

Linear & Systems Quadratics Polynomials Functions Statistics Geometry Trigonometry Probability Exponentials Complex Numbers

Core Concepts Must Memorize

Unit 01
Linear Equations
Key Formula
y = mx + b
m = (y₂−y₁)/(x₂−x₁)
Ax + By = C
Parallel lines: equal slopes (m₁ = m₂). Perpendicular: slopes are negative reciprocals (m₁ · m₂ = −1).
Unit 02
Systems of Equations
Key Formula
Substitution or Elimination
No solution: parallel lines
∞ solutions: same line
Multiply one equation to make coefficients equal, then add or subtract to eliminate a variable.
Unit 03
Quadratic Functions
Key Formula
f(x) = ax² + bx + c
Vertex: x = −b/(2a)
Vertex form: a(x−h)² + k
The vertex is the minimum (a>0) or maximum (a<0). Axis of symmetry: x = h.
Unit 04
Quadratic Formula
Key Formula
x = (−b ± √(b²−4ac)) / 2a
Discriminant D = b²−4ac
D>0: 2 real roots
D = 0 → one real root (tangent). D < 0 → no real roots (two complex roots).
Unit 05
Polynomials & Factor Theorem
Key Rule
If f(a) = 0, then (x−a) is a factor
Remainder Theorem:
f(a) = remainder when f(x)÷(x−a)
To find remainders quickly, plug the zero of the divisor directly into f(x).
Unit 06
Rational Expressions
Key Rule
Undefined when denominator = 0
a/b ÷ c/d = a/b × d/c
LCD to add/subtract
Always factor the numerator and denominator before simplifying — look for common factors to cancel.
Unit 07
Exponents & Radicals
Key Rules
aᵐ·aⁿ = aᵐ⁺ⁿ
aᵐ/aⁿ = aᵐ⁻ⁿ
√a = a^(1/2)
∛a = a^(1/3)
Negative exponent: a⁻ⁿ = 1/aⁿ. Fractional exponent: aᵐ/ⁿ = (ⁿ√a)ᵐ.
Unit 08
Function Composition
Key Formula
(f∘g)(x) = f(g(x))
Domain: x valid in g AND
g(x) valid in f
Work from the inside out: compute g(x) first, then substitute the result into f.
Unit 09
Statistics & Data
Key Formulas
Mean = Σx / n
Median = middle value
Range = max − min
On SAT, adding a constant k to all values shifts mean by k but does NOT change standard deviation.
Unit 10
Geometry & Trig
Key Formulas
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
Arc length = rθ (radians)
sin²θ + cos²θ = 1. Complementary angles: sin(θ) = cos(90°−θ).

Worked Examples Study Before Practicing

Example A — Quadratic Vertex
The function f(x) = 2x² − 8x + 5 has a minimum value. What is that minimum value?
Step 1: Find the x-coordinate of the vertex: x = −b/(2a) = −(−8)/(2·2) = 8/4 = 2
Step 2: Substitute back: f(2) = 2(4) − 8(2) + 5 = 8 − 16 + 5 = −3
Answer: −3
Example B — Systems of Equations
If 3x + 2y = 16 and x − y = 1, what is the value of x + y?
Step 1: From x − y = 1, write x = y + 1.
Step 2: Substitute: 3(y+1) + 2y = 16 → 5y = 13 → y = 13/5. Then x = 1 + 13/5 = 18/5.
Step 3: x + y = 18/5 + 13/5 = 31/5 = 6.2
Example C — Exponential Growth
A population doubles every 3 years. If the initial population is 500, what is the population after 9 years?
Formula: P = 500 · 2^(t/3), where t = 9 years.
P = 500 · 2^(9/3) = 500 · 2³ = 500 · 8 = 4,000

Practice Questions Hard Level · SAT Style

1
Linear Equations · Slope & Intercepts
Hard
Line \(\ell\) passes through the points \((2,\,1)\) and \((6,\,9)\). Line \(m\) is perpendicular to \(\ell\) and passes through \((4,\,3)\). What is the \(y\)-intercept of line \(m\)?
✦ Full Explanation
1
Slope of \(\ell\): \(m_\ell = \dfrac{9-1}{6-2} = \dfrac{8}{4} = 2\)
2
Slope of \(m\) (perpendicular): \(m_m = -\dfrac{1}{2}\)
3
Line \(m\) through \((4,3)\): \(y - 3 = -\dfrac{1}{2}(x-4)\) → \(y = -\dfrac{1}{2}x + 2 + 3 = -\dfrac{1}{2}x + 5\)
4
Set \(x = 0\): \(y\)-intercept \(= \mathbf{5}\)
Correct Answer: C — 5
2
Systems of Equations · Infinite Solutions
Hard
The system below has infinitely many solutions. What is the value of \(k\)? \[4x - 6y = 10\] \[2x - ky = 5\]
✦ Full Explanation
1
For infinite solutions, the equations must be multiples of each other.
2
Multiply the second equation by 2: \(4x - 2ky = 10\).
3
Compare with \(4x - 6y = 10\): we need \(2k = 6\), so \(k = 3\).
Correct Answer: B — 3
3
Quadratic Functions · Vertex & Max/Min
Hard
The function \(f(x) = -3x^2 + 12x - 7\) has a maximum value. What is that maximum value?
✦ Full Explanation
1
Since \(a = -3 < 0\), the parabola opens downward, confirming a maximum.
2
Vertex x-coordinate: \(x = -\dfrac{b}{2a} = -\dfrac{12}{2(-3)} = -\dfrac{12}{-6} = 2\)
3
Compute \(f(2) = -3(4) + 12(2) - 7 = -12 + 24 - 7 = \mathbf{5}\)
Correct Answer: C — 5
4
Quadratic Formula · Discriminant
Hard
For what value of \(c\) does the equation \(x^2 - 6x + c = 0\) have exactly one real solution?
✦ Full Explanation
1
For exactly one real solution, the discriminant \(D = b^2 - 4ac = 0\).
2
Here \(a=1,\, b=-6,\, c=c\): \((-6)^2 - 4(1)(c) = 0\)
3
\(36 - 4c = 0 \Rightarrow c = 9\)
Correct Answer: D — 9
5
Polynomials · Remainder Theorem
Hard
When the polynomial \(p(x) = x^3 - 4x^2 + 3x + k\) is divided by \((x - 2)\), the remainder is 5. What is the value of \(k\)?
✦ Full Explanation
1
By the Remainder Theorem, the remainder when dividing by \((x-2)\) equals \(p(2)\).
2
\(p(2) = 8 - 16 + 6 + k = -2 + k\)
3
Set equal to remainder: \(-2 + k = 5 \Rightarrow k = 7\)
Correct Answer: C — 7
6
Rational Expressions · Undefined Values
Hard
For what value(s) of \(x\) is the expression \(\dfrac{x^2 - 9}{x^2 - x - 6}\) undefined?
✦ Full Explanation
1
The expression is undefined when the denominator equals zero.
2
Factor denominator: \(x^2 - x - 6 = (x-3)(x+2)\)
3
Set each factor to zero: \(x = 3\) or \(x = -2\).
4
Note: even though the numerator also has a factor \((x-3)\), which cancels, the original expression is still undefined at \(x=3\) because we cannot divide by zero in the original form.
Correct Answer: B — x = 3 and x = −2
7
Exponents & Radicals · Simplification
Hard
If \(x > 0\), which expression is equivalent to \(\dfrac{x^{5/3}}{x^{2/3}}\)?
✦ Full Explanation
1
When dividing with the same base, subtract the exponents: \(\dfrac{x^{5/3}}{x^{2/3}} = x^{5/3 - 2/3}\)
2
\(x^{5/3 - 2/3} = x^{3/3} = x^1 = x\)
Correct Answer: A — x
8
Functions · Composition
Hard
If \(f(x) = 2x + 1\) and \(g(x) = x^2 - 3\), what is the value of \(f(g(3))\)?
✦ Full Explanation
1
Compute the inside first: \(g(3) = 3^2 - 3 = 9 - 3 = 6\)
2
Now compute \(f(6) = 2(6) + 1 = 12 + 1 = \mathbf{13}\)
Correct Answer: B — 13
9
Absolute Value · Equations
Hard
How many distinct real solutions does the equation \(|2x - 4| = x + 1\) have?
✦ Full Explanation
1
Case 1: \(2x - 4 = x + 1 \Rightarrow x = 5\). Check: \(|2(5)-4| = 6 = 5+1 = 6\) ✓
2
Case 2: \(2x - 4 = -(x+1) \Rightarrow 3x = 3 \Rightarrow x = 1\). Check: \(|2(1)-4| = 2 = 1+1 = 2\) ✓
3
Both solutions are valid: \(x = 5\) and \(x = 1\). That is 2 distinct solutions.
Correct Answer: C — 2
10
Inequalities · Linear Systems
Hard
Which of the following points \((x, y)\) satisfies the system of inequalities below? \[y < 2x + 3\] \[y \geq -x + 1\]
✦ Full Explanation
1
Test \((2,1)\): Is \(1 < 2(2)+3 = 7\)? Yes ✓. Is \(1 \geq -2+1 = -1\)? Yes ✓. Both satisfied.
2
Verify others fail: \((0,4)\): \(4 < 3\)? No ✗. \((-1,2)\): \(2 < 1\)? No ✗. \((1,5)\): \(5 < 5\)? No ✗.
Correct Answer: C — (2, 1)
11
Statistics · Mean & Median
Hard
The five values 3, 7, 12, \(x\), and 18 have a mean of 10. What is the median of the five values?
✦ Full Explanation
1
Mean = 10, so sum = \(10 \times 5 = 50\).
2
\(3 + 7 + 12 + x + 18 = 50 \Rightarrow 40 + x = 50 \Rightarrow x = 10\)
3
Sorted values: \(3, 7, 10, 12, 18\). The middle (3rd) value is the median: \(\mathbf{10}\).
Correct Answer: B — 10
12
Data Analysis · Percentages & Ratios
Hard
A store reduces the price of a jacket by 20%, then later reduces the new price by 25%. What is the total percentage reduction from the original price?
✦ Full Explanation
1
Let original price = \(P\). After 20% off: \(0.80P\).
2
After 25% off the new price: \(0.80P \times 0.75 = 0.60P\).
3
Final price is 60% of original, so the total reduction is \(100\% - 60\% = \mathbf{40\%}\).
Correct Answer: B — 40%
13
Geometry · Similar Triangles
Hard
Triangle \(ABC\) is similar to triangle \(DEF\). The sides of \(\triangle ABC\) are 5, 12, and 13. If the shortest side of \(\triangle DEF\) is 15, what is the perimeter of \(\triangle DEF\)?
✦ Full Explanation
1
The perimeter of \(\triangle ABC = 5 + 12 + 13 = 30\).
2
Scale factor: \(\dfrac{15}{5} = 3\) (shortest side of DEF ÷ shortest side of ABC).
3
Perimeter of \(\triangle DEF = 30 \times 3 = \mathbf{90}\).
Correct Answer: C — 90
14
Circles · Arc Length & Sector
Hard
A circle has a radius of 6. A central angle of 60° is formed by two radii. What is the length of the arc intercepted by this angle? (Use \(\pi\) in your answer.)
✦ Full Explanation
1
Arc length formula: \(s = \dfrac{\theta}{360°} \times 2\pi r\)
2
\(s = \dfrac{60°}{360°} \times 2\pi(6) = \dfrac{1}{6} \times 12\pi = 2\pi\)
Correct Answer: B — 2π
15
Trigonometry · Right Triangles
Hard
In a right triangle, the side opposite to angle \(\theta\) has length 8 and the hypotenuse has length 10. What is \(\cos\theta\)?
✦ Full Explanation
1
\(\sin\theta = \dfrac{\text{opp}}{\text{hyp}} = \dfrac{8}{10} = \dfrac{4}{5}\)
2
Adjacent side: \(\sqrt{10^2 - 8^2} = \sqrt{36} = 6\)
3
\(\cos\theta = \dfrac{\text{adj}}{\text{hyp}} = \dfrac{6}{10} = \mathbf{\dfrac{3}{5}}\)
Correct Answer: C — 3/5
16
Word Problems · Rate, Distance, Time
Hard
Car A travels at 60 mph. Car B travels at 45 mph in the same direction. If Car A starts 30 miles behind Car B, how many hours will it take Car A to catch Car B?
✦ Full Explanation
1
Relative speed of Car A closing the gap: \(60 - 45 = 15\) mph.
2
Time to close 30-mile gap: \(t = \dfrac{30}{15} = \mathbf{2}\) hours.
Correct Answer: C — 2 hours
17
Exponential Functions · Growth & Decay
Hard
The value of a car depreciates at 15% per year. If the car originally costs \(\$20{,}000\), which expression represents the value after \(t\) years?
✦ Full Explanation
1
Depreciation of 15% per year means the car retains \(100\% - 15\% = 85\% = 0.85\) of its value each year.
2
Exponential decay formula: \(V(t) = 20000 \times (0.85)^t\)
3
Note: \(20000(0.15)^t\) would be wrong because 0.15 is just the rate lost, not the fraction remaining.
Correct Answer: B — 20000(0.85)ᵗ
18
Scatterplot · Line of Best Fit
Hard
A line of best fit for a data set is given by \(\hat{y} = 3.2x + 14.5\). According to this model, by approximately how much does \(y\) increase when \(x\) increases by 4?
✦ Full Explanation
1
In \(\hat{y} = 3.2x + 14.5\), the slope 3.2 tells us: for every 1-unit increase in \(x\), \(y\) increases by 3.2.
2
For a 4-unit increase in \(x\): \(\Delta y = 3.2 \times 4 = \mathbf{12.8}\)
Correct Answer: C — 12.8
19
Probability · Conditional Probability
Hard
In a class of 30 students, 18 play soccer, 12 play basketball, and 6 play both. If a student is selected at random from those who play soccer, what is the probability that the student also plays basketball?
✦ Full Explanation
1
Conditional probability: \(P(\text{basketball} \mid \text{soccer}) = \dfrac{P(\text{both})}{P(\text{soccer})}\)
2
Number who play soccer = 18; number who play both = 6.
3
\(P = \dfrac{6}{18} = \dfrac{1}{3}\)
Correct Answer: B — 1/3
20
Complex Numbers · Operations
Hard
What is the result of \((3 + 2i)(1 - 4i)\), where \(i = \sqrt{-1}\)?
✦ Full Explanation
1
Use FOIL: \((3+2i)(1-4i) = 3(1) + 3(-4i) + 2i(1) + 2i(-4i)\)
2
\(= 3 - 12i + 2i - 8i^2\)
3
Since \(i^2 = -1\): \(= 3 - 10i - 8(-1) = 3 - 10i + 8 = \mathbf{11 - 10i}\)
Correct Answer: D — 11 − 10i
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Answer Key

QAnsQAnsQAnsQAnsQAns
1C2B3C4D5C
6B7A8B9C10C
11B12B13C14B15C
16C17B18C19B20D

Explanations

Q1. Slope of line l = (9-1)/(6-2) = 2. Perpendicular slope = -1/2. Through (4,3): y = -1/2x + 5. y-intercept = 5. Answer: C
Q2. Multiply eq2 by 2: 4x - 2ky = 10. Compare to 4x - 6y = 10: 2k = 6, k = 3. Answer: B
Q3. Vertex x = -12/(2x-3) = 2. f(2) = -12+24-7 = 5. Answer: C
Q4. D = 36 - 4c = 0 → c = 9. Answer: D
Q5. p(2) = 8-16+6+k = -2+k = 5 → k = 7. Answer: C
Q6. Denominator: (x-3)(x+2) = 0 → x=3 or x=-2. Answer: B
Q7. x^(5/3 - 2/3) = x^(3/3) = x. Answer: A
Q8. g(3)=6, f(6)=13. Answer: B
Q9. x=5 and x=1 both valid (2 solutions). Answer: C
Q10. (2,1): 1<7 and 1≥-1. Both satisfied. Answer: C
Q11. x=10; sorted: 3,7,10,12,18. Median=10. Answer: B
Q12. 0.80 x 0.75 = 0.60; reduction = 40%. Answer: B
Q13. Scale = 15/5=3; perimeter = 30x3=90. Answer: C
Q14. (60/360)x2pi(6)=2pi. Answer: B
Q15. adj=6, cos=6/10=3/5. Answer: C
Q16. Relative speed=15 mph; time=30/15=2 hours. Answer: C
Q17. Retain 85% per year: 20000(0.85)^t. Answer: B
Q18. Slope=3.2; for 4 units: 3.2x4=12.8. Answer: C
Q19. P = 6/18 = 1/3. Answer: B
Q20. FOIL: 3-12i+2i+8=11-10i. Answer: D