SAT Math Master

Advanced Practice Problems

20 Hard SAT-Style Questions  ·  All Major Topics  ·  Full Explanations

TIME 45:00 0 / 20 answered
Heart of Algebra

Linear Equations & Systems

Solve for variables, interpret slope/intercept, use substitution or elimination for systems.

y = mx + b  |  ax + by = c
Elimination: multiply & add/subtract
Passport to Adv. Math

Quadratics

Factor, use quadratic formula, complete the square. Know vertex form and discriminant.

x = (−b ± √(b²−4ac)) / 2a
Vertex: x = −b/(2a)
Passport to Adv. Math

Polynomials & Functions

Remainder theorem, factor theorem, end behavior, zeros, and function composition/inverse.

f(a) = r  →  (x−a) is factor if r=0
f(g(x)) = composition
Passport to Adv. Math

Exponential Functions

Growth/decay models, properties of exponents, converting between forms.

y = a·bˣ  |  y = a·eˣ
bˣ·bʸ = bˣ⁺ʸ  |  (bˣ)ʸ = bˣʸ
Additional Topics

Circles & Geometry

Circle equation, arc length, sector area, inscribed angles, tangent properties.

(x−h)²+(y−k)²=r²
Arc = (θ/360)·2πr
Additional Topics

Trigonometry

SOH-CAH-TOA, unit circle, radian measure, sine/cosine graphs, special triangles.

sin²θ + cos²θ = 1
30-60-90: 1 : √3 : 2
Problem Solving

Ratios, Rates & Proportions

Set up proportions carefully. Check units. Percent change = (new−old)/old × 100.

a/b = c/d  →  ad = bc
% change = Δ/original × 100
Data Analysis

Statistics & Probability

Mean, median, standard deviation, normal distribution, conditional probability, two-way tables.

P(A|B) = P(A∩B)/P(B)
Mean = Σx / n
Additional Topics

Complex Numbers

i² = −1. Add/subtract real and imaginary parts separately. Multiply using FOIL.

i² = −1, i³ = −i, i⁴ = 1
(a+bi)(c+di) = (ac−bd)+(ad+bc)i
Heart of Algebra

Absolute Value & Inequalities

|x| = a → x = a or x = −a. For inequalities, flip sign when dividing by negative.

|x − a| < b → −b < x−a < b
|x − a| > b → x<a−b or x>a+b
SECTION 01

Heart of Algebra

Q 01 Linear Systems Hard

If the system of equations below has infinitely many solutions, what is the value of k?

3x − 6y = 12
kx − 2ky = 4k
Q 02 Linear Equations Hard

Line ℓ passes through (−3, 7) and is perpendicular to the line 4x − 2y = 10. What is the y-intercept of line ℓ?

Q 03 Absolute Value / Inequalities Very Hard

How many integer solutions does the inequality |2x − 5| < 7 have?

SECTION 02

Passport to Advanced Math

Q 04 Quadratics Hard

The parabola y = x² − 6x + k is tangent to the x-axis (touches at exactly one point). What is the value of k?

Q 05 Polynomials Very Hard

If (x − 3) is a factor of p(x) = x³ − 2x² + cx − 12, what is the remainder when p(x) is divided by (x + 1)?

Q 06 Exponential Functions Hard

A bacteria culture doubles every 3 hours. It starts with 200 bacteria at t = 0. At what time t (in hours) will the population first exceed 25,600?

Q 07 Rational & Radical Equations Very Hard

Which value of x satisfies the equation below? (Check for extraneous solutions.)

√(3x + 13) = x + 3
Q 08 Function Composition Hard

Let f(x) = 2x + 1 and g(x) = x² − 3. What is the value of f(g(f(2)))?

SECTION 03

Problem Solving & Data Analysis

Q 09 Statistics — Mean & Median Hard

A data set of 7 values has a mean of 12 and a median of 10. If the largest value in the set is removed, the new mean becomes 10. What was the largest value?

Q 10 Probability Very Hard

A class has 30 students. 18 play soccer, 14 play basketball, and 6 play both. A student is chosen at random. What is the probability (as a fraction) that the student plays neither sport?

Q 11 Ratios & Percent Change Hard

A store first increases the price of a jacket by 25%, then decreases the new price by 20%. What is the overall percent change from the original price?

Q 12 Scatterplot & Line of Best Fit Hard

A line of best fit for a data set is modeled by ŷ = 3.2x + 4.8. If the actual y-value when x = 5 is 20, what is the residual?

SECTION 04

Additional Topics in Math

Q 13 Circles Hard

A circle has center (3, −2) and passes through the point (7, 1). What is the area of the circle?

Q 14 Trigonometry Very Hard

In a right triangle, sin θ = 5/13. What is the value of tan θ? (Assume θ is an acute angle.)

Q 15 Complex Numbers Very Hard

What is the real part of the expression (3 + 2i)² − (1 − i)?

Q 16 3D Geometry — Volume Hard

A cone has a radius of 6 and a height of 8. A sphere has the same volume as the cone. What is the radius of the sphere? (Vcone = πr²h/3, Vsphere = 4πr³/3)

Q 17 Coordinate Geometry Hard

Points A(1, 2) and B(7, 10) are the endpoints of a diameter of a circle. What is the equation of the circle?

SECTION 05

Mixed Hard & Very Hard

Q 18 Quadratic & Linear System Very Hard

The line y = 2x + k and the parabola y = x² − 4x + 7 intersect at exactly one point. What is the value of k?

Q 19 Sequences & Series Very Hard

In a geometric sequence, the second term is 6 and the fifth term is 162. What is the first term of the sequence?

Q 20 Modeling & Optimization Very Hard

A farmer wants to fence a rectangular garden using 120 meters of fencing along three sides (one side is against a barn wall and needs no fence). What is the maximum area of the garden?

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