PSAT Math Master — 20 Essential Practice Questions

Domain 1

Algebra

📐 Core Concept

Linear Equations & Systems

Slope-intercept: y = mx + b Standard form: Ax + By = C Slope: m = (y₂ - y₁) / (x₂ - x₁) Systems — substitution / elimination Two equations, two unknowns → unique, infinite, or no solution

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
If two linear equations are equivalent (same line) → infinitely many solutions.
No solution → lines are parallel (same slope, different y-intercept).
Always check your solution by substituting back into both equations.

Worked Example

Solve the system: 2x + y = 7 and x - y = 2

Step-by-Step Solution

Add both equations: 3x = 9, so x = 3

Substitute into 2nd equation: 3 - y = 2 → y = 1

Answer: (3, 1). Check: 2(3)+1=7 ✓ and 3-1=2 ✓

Q1 Easy

No calculator · Linear Equations

If 3x + 5 = 20, what is the value of 6x - 4?

A16

B26

C30

D34

✅ Correct!

From 3x+5=20, we get 3x=15, so x=5. Then 6x-4 = 6(5)-4 = 30-4 = 26. Tip: notice 6x-4 = 2(3x)-4 = 2(15)-4 = 26 — a shortcut without solving for x!

❌ Incorrect

From 3x+5=20, subtract 5: 3x=15, so x=5. Then 6x-4 = 30-4 = 26. The answer is B.

Q2 Medium

No calculator · Systems of Linear Equations

The system kx + 3y = 12 and 2x + y = 4 has infinitely many solutions. What is the value of k?

✅ Correct!

For infinitely many solutions, the equations must be multiples of each other. Multiply the 2nd equation by 3: 6x+3y=12. Comparing with kx+3y=12, we get k=6.

❌ Incorrect

Infinitely many solutions means the equations are identical. Multiply 2x+y=4 by 3 to match the right side (12): 6x+3y=12. So k=6. Answer: C.

Q3 Medium

Linear Inequalities

Which of the following is the solution to -2x + 6 < 14?

Ax < -4

Bx > -4

Cx < 4

Dx > 4

✅ Correct!

-2x+6<14 → -2x<8 → divide by -2 (flip the inequality!) → x>-4. Key rule: when you divide or multiply by a negative number, always reverse the inequality sign.

❌ Incorrect

-2x+6<14 → -2x<8 → divide both sides by −2 and flip the sign → x>-4. Don't forget to reverse the inequality when dividing by a negative! Answer: B.

Q4 Hard

Linear Functions · Slope & Intercepts

A line passes through (2, 7) and (-1, 1). What is the y-intercept of this line?

✅ Correct!

Slope: m = (7-1)/(2-(-1)) = 6/3 = 2. Using point-slope with (2,7): y-7 = 2(x-2) → y = 2x+3. Y-intercept = 3.

❌ Incorrect

Step 1: slope = (7-1)/(2-(-1)) = 6/3 = 2. Step 2: y-7=2(x-2) → y=2x+3. Y-intercept is 3. Answer: C.

Domain 2

Advanced Math

🔢 Core Concept

Quadratics, Polynomials & Functions

Quadratic formula: x = (-b ± √(b²-4ac)) / 2a Discriminant: D = b²-4ac D > 0 → two real roots D = 0 → one real root (repeated) D < 0 → no real roots Vertex form: f(x) = a(x-h)² + k [vertex at (h, k)] Factored form: f(x) = a(x-r₁)(x-r₂) Sum of roots: r₁+r₂ = -b/a Product of roots: r₁·r₂ = c/a

Parabola opens upward if a > 0 (minimum at vertex); downward if a < 0 (maximum).
Axis of symmetry: x = -b / (2a)
FOIL: (a+b)(c+d) = ac + ad + bc + bd
Difference of squares: a² - b² = (a+b)(a-b)
Perfect square trinomial: (a+b)² = a² + 2ab + b²

Worked Example

For the function f(x) = x² - 6x + 8, find the roots and vertex.

Step-by-Step Solution

Factor: (x-2)(x-4) = 0 → roots: x=2 and x=4

Axis of symmetry: x = (2+4)/2 = 3

Vertex: f(3) = 9-18+8 = -1 → vertex (3,-1)

Q5Easy

Quadratic Equations · Factoring

Which values of x satisfy x² + x - 6 = 0?

Ax = 2 and x = 3

Bx = -2 and x = 3

Cx = 2 and x = -3

Dx = -2 and x = -3

✅ Correct!

Factor: (x+3)(x-2) = 0. So x=-3 or x=2. Check: multiply the two roots: (-3)(2)=-6 ✓, sum: (-3)+(2)=-1=coefficient of x with sign ✓

❌ Incorrect

We need two numbers that multiply to −6 and add to +1. That's +3 and −2. So (x+3)(x-2)=0 → x=-3 or x=2. Answer: C.

Q6Medium

Quadratic Functions · Vertex Form

The function f(x) = 2(x-3)² + 5 has a minimum value of:

D11

✅ Correct!

In vertex form f(x)=a(x-h)²+k, the vertex is (h,k)=(3,5). Since a=2>0, the parabola opens upward, so the minimum value is k=5.

❌ Incorrect

Vertex form: vertex at (3,5). Since a=2>0 the parabola opens up → minimum at the vertex = 5. Answer: C.

Q7Hard

Discriminant · Number of Solutions

For what value of k does x² - 4x + k = 0 have exactly one real solution?

✅ Correct!

For exactly one solution, the discriminant = 0. b²-4ac=(-4)²-4(1)(k)=16-4k=0 → k=4.

❌ Incorrect

Set the discriminant to 0: (-4)²-4(1)(k)=0 → 16=4k → k=4. Answer: B.

Q8Medium

Polynomial Operations

If (x+3)(x-3) = x² + mx - 9, what is the value of m?

A-6

✅ Correct!

(x+3)(x-3) = x²-9 (difference of squares). So x²+mx-9 = x²+0x-9, meaning m=0.

❌ Incorrect

Use the difference of squares rule: (x+3)(x-3) = x²-9. There is no x-term, so m=0. Answer: B.

Q9Hard

Function Composition & Notation

If f(x) = 2x - 1 and g(x) = x² + 3, what is g(f(2))?

B11

C12

D15

✅ Correct!

First find f(2) = 2(2)-1 = 3. Then g(f(2)) = g(3) = 3²+3 = 12. Always work from the inside out!

❌ Incorrect

Work inside out: f(2)=2(2)-1=3. Then g(3)=3²+3=12. Answer: C.

Domain 3

Problem Solving & Data Analysis

📊 Core Concept

Ratios, Percents, Statistics & Probability

Percent change: ((New - Old) / Old) × 100 Percent of: Part = Percent × Whole Mean: Sum of values / Number of values Median: Middle value (sorted list) Mode: Most frequent value Range: Max - Min Probability: P(E) = favorable outcomes / total outcomes

When comparing ratios, convert both to common form or decimals.
In a two-way table, read row/column totals carefully for conditional probability.
Outliers affect the mean more than the median.
Standard deviation measures the spread; higher SD = more spread out.
For "at least one" probability: P(at least 1) = 1 - P(none)

Worked Example

A store raised a price from $80 to $100. What is the percent increase?

Step-by-Step Solution

Change = 100 - 80 = $20

Percent increase = 20/80 × 100 = 25%

Always divide by the ORIGINAL value (80), not the new value.

Q10Easy

Ratios & Proportions

A recipe uses 2 cups of flour for every 3 cups of sugar. If a baker uses 8 cups of flour, how many cups of sugar are needed?

A10

B12

C14

D16

✅ Correct!

Set up the proportion: 2/3 = 8/x. Cross multiply: 2x=24 → x=12. Or: 8÷2=4, so multiply 3×4=12.

❌ Incorrect

Ratio is 2:3. Scale factor: 8÷2=4. Sugar = 3×4=12 cups. Answer: B.

Q11Medium

Percent Change

A laptop's price decreased from $1,200 to $900. By approximately what percent did the price decrease?

A20%

B25%

C30%

D33%

✅ Correct!

Decrease = 1200-900=300. Percent decrease = 300/1200 × 100 = 25%. Always divide by the original price.

❌ Incorrect

Decrease = $300. Divide by original: 300/1200 = 0.25 = 25%. Answer: B.

Q12Medium

Statistics · Mean & Median

The five data values are: 4, 7, 9, 9, 11. Which of the following is true?

AMean = Median = 9

BMean < Median

CMean = 8, Median = 9

DMode = Mean = 9

✅ Correct!

Mean = (4+7+9+9+11)/5 = 40/5 = 8. Median (middle of sorted list) = 9. So mean=8 < median=9. Answer C correctly states mean=8 and median=9.

❌ Incorrect

Mean = (4+7+9+9+11)/5 = 40/5 = 8. Median = 9 (middle value). So C is correct: mean=8, median=9. Answer: C.

Q13Hard

Probability

A bag contains 5 red marbles and 3 blue marbles. If two marbles are drawn without replacement, what is the probability that both are red?

A5/16

B25/64

C5/14

D5/12

✅ Correct!

P(1st red) = 5/8. After drawing one red, 4 red remain out of 7 total: P(2nd red | 1st red) = 4/7. So P(both red) = (5/8)(4/7) = 20/56 = 5/14.

❌ Incorrect

Without replacement: P = (5/8) × (4/7) = 20/56 = 5/14. Answer: C.

Q14Hard

Scatterplots & Linear Models

A scatterplot shows a study time (hours) vs. test score (%) linear relationship. The line of best fit is y = 8x + 40. Based on this model, how many hours of study are needed to predict a score of 88%?

A4 hours

B6 hours

C8 hours

D10 hours

✅ Correct!

Set 88 = 8x+40. Subtract 40: 48=8x → x=6 hours. The slope (8) means each additional hour adds 8 points.

❌ Incorrect

Set y=88: 88=8x+40 → 8x=48 → x=6. Answer: B.

Domain 4

Geometry & Trigonometry

📏 Core Concept

Geometry Formulas & Trig Ratios

Area: Rectangle = lw | Triangle = ½bh | Circle = πr² Trapezoid = ½(b₁+b₂)h Perimeter: Rectangle = 2(l+w) | Circle (circumference) = 2πr Volume: Rectangular prism = lwh | Cylinder = πr²h Cone = ⅓πr²h | Sphere = (4/3)πr³ Pythagorean theorem: a² + b² = c² (c = hypotenuse) TRIGONOMETRY (SOH-CAH-TOA): sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent Special triangles: 30-60-90: sides = 1 : √3 : 2 45-45-90: sides = 1 : 1 : √2

The sum of interior angles in a triangle is always 180°.
Supplementary angles sum to 180°; complementary angles sum to 90°.
Similar triangles have proportional sides and equal angles.
Inscribed angle = half the central angle intercepting the same arc.

Worked Example

A right triangle has legs of length 6 and 8. Find the hypotenuse and sin of the smaller acute angle.

Step-by-Step Solution

Hypotenuse: c = √(6²+8²) = √(36+64) = √100 = 10

Smaller angle (opposite side = 6): sin θ = 6/10 = 0.6

This is a 3-4-5 triangle scaled by 2: (6,8,10) = 2×(3,4,5)

Q15Easy

Geometry · Area & Perimeter

A rectangle has a length of 12 and a width of 5. What is the length of its diagonal?

A13

B14

C15

D17

✅ Correct!

Use Pythagorean theorem: d = √(12²+5²) = √(144+25) = √169 = 13. Recognize this as a 5-12-13 Pythagorean triple!

❌ Incorrect

Pythagorean theorem: √(12²+5²) = √169 = 13. This is the 5-12-13 triple. Answer: A.

Q16Medium

Circles · Arc Length & Area

A circle has a radius of 6. A central angle of 90° forms a sector. What is the area of this sector? (Use π ≈ 3.14)

A9π

B12π

C18π

D36π

✅ Correct!

Area of full circle = π(6)²=36π. A 90° sector is 90/360 = 1/4 of the circle. Sector area = 36π/4 = 9π.

❌ Incorrect

Full circle area = π(6²)=36π. 90° is 1/4 of 360°, so sector = 36π × (1/4) = 9π. Answer: A.

Q17Medium

Trigonometry · SOH-CAH-TOA

In a right triangle, the angle θ has an opposite side of 5 and a hypotenuse of 10. What is cos θ?

A1/2

B√3/2

C√2/2

D5/10

✅ Correct!

sin θ = 5/10 = 1/2, so θ = 30°. For a 30-60-90 triangle with hypotenuse 10: adjacent = 10·cos(30°) = 10·(√3/2) = 5√3. So cos θ = 5√3/10 = √3/2.

❌ Incorrect

sin θ = 5/10 → θ = 30°. Then cos 30° = √3/2. Alternatively: adjacent = √(10²-5²)=√75=5√3, cos θ = 5√3/10 = √3/2. Answer: B.

Q18Hard

Volume · 3D Geometry

A cylinder has a radius of 3 and a height of 8. A cone with the same base and height is removed from inside. What is the remaining volume? (Leave answer in terms of π)

A48π

B60π

C72π

D96π

✅ Correct!

Cylinder: π(3²)(8)=72π. Cone: (1/3)π(3²)(8)=24π. Remaining = 72π-24π=48π. The cone is always 1/3 of the enclosing cylinder.

❌ Incorrect

Cylinder = 72π. Cone = (1/3)(72π)=24π. Remaining = 72π-24π=48π. Answer: A.

Q19Medium

Word Problem · Real-World Algebra

Two trains start at the same station at the same time, traveling in opposite directions. Train A travels at 60 mph and Train B travels at 80 mph. After how many hours will they be 350 miles apart?

A2 hours

B2.5 hours

C3 hours

D3.5 hours

✅ Correct!

Combined speed = 60+80 = 140 mph. Time = Distance ÷ Speed = 350/140 = 2.5 hours. When objects move in opposite directions, add their speeds.

❌ Incorrect

Combined speed = 140 mph. Time = 350/140 = 2.5 hours. Answer: B.

Q20Hard

Mixed · Exponential Growth

A population of bacteria doubles every 3 hours. If the initial population is 500, which expression represents the population after t hours?

A500 · 2^(t/3)

B500 · 3^(t/2)

C500 · 2^(3t)

D500 · t^2

✅ Correct!

The exponential growth formula is P = P₀ · b^(t/T) where T is the doubling time. Here: P = 500 · 2^(t/3). Check: at t=3: 500·2¹=1000 ✓ (doubled). At t=6: 500·2²=2000 ✓ (doubled again).

❌ Incorrect

Doubles every 3 hours: P = 500 · 2^(t/3). The exponent t/3 counts how many doubling periods have passed. Answer: A.

Answer Key & Explanations

Q1 — Answer: B (26)

3x+5=20 → x=5. Then 6x-4=30-4=26. Shortcut: 6x-4=2(3x)-4=2(15)-4=26.

Q2 — Answer: C (6)

Infinite solutions: multiply 2x+y=4 by 3 → 6x+3y=12. So k=6.

Q3 — Answer: B (x > -4)

-2x+6<14 → -2x<8 → x>-4. Flip inequality when dividing by negative.

Q4 — Answer: C (3)

Slope=(7-1)/(2-(-1))=2. Point-slope: y-7=2(x-2) → y=2x+3. Y-intercept=3.

Q5 — Answer: C (x=2 and x=-3)

Factor: (x+3)(x-2)=0 → x=-3 or x=2.

Q6 — Answer: C (5)

Vertex form: vertex=(3,5). a=2>0 opens up → minimum=5.

Q7 — Answer: B (4)

Discriminant=0: 16-4k=0 → k=4.

Q8 — Answer: B (0)

(x+3)(x-3)=x²-9. No x-term, so m=0.

Q9 — Answer: C (12)

f(2)=3. g(3)=9+3=12.

Q10 — Answer: B (12)

Ratio 2:3. Scale=8÷2=4. Sugar=3×4=12.

Q11 — Answer: B (25%)

300/1200×100=25%.

Q12 — Answer: C (Mean=8, Median=9)

Sum=40, mean=8. Middle value=9.

Q13 — Answer: C (5/14)

(5/8)×(4/7)=20/56=5/14.

Q14 — Answer: B (6 hours)

88=8x+40 → x=6.

Q15 — Answer: A (13)

√(144+25)=√169=13. (5-12-13 triple)

Q16 — Answer: A (9π)

36π×(90/360)=9π.

Q17 — Answer: B (√3/2)

sinθ=1/2 → θ=30°. cos30°=√3/2.

Q18 — Answer: A (48π)

72π-24π=48π.

Q19 — Answer: B (2.5 hours)

350/140=2.5 hours.

Q20 — Answer: A (500·2^(t/3))

Doubling time T=3: P=500·2^(t/3).

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