This is a hard SAT Math practice test: 5 questions from our question bank, all at the top of our difficulty scale. Give yourself 8 minutes, about the pace of a harder second module, and keep Desmos open. The answer key and worked solutions are at the bottom.
Scoring yourself: 5 right means you are ready for the hardest questions on the test. 3 or 4 means review the misses carefully. 2 or fewer means go back to the linked notes first.
Question 1 (Hard)
For how many values of K does |k-3|+2 is equal to one?
- A. None
- B. one
- C. two
- D. more than two
Question 2 (Hard)
Light travels through a vacuum at a speed of approximately 186,000 miles per second. Approximately how many miles will a ray of light travel through a vacuum in one day?
- A. 6.7 × 108 miles
- B. 1.6 × 1010 miles
- C. 3.2 × 1010 miles
- D. 4.6 × 1013 miles
Question 3 (Hard)
In triangle RST, the measure of angle R is 10 degrees and the measure of angle T is 50 degrees. Point L lies on RS, point K lies on ST, and LK is parallel to RT. What is the measure, in degrees, of angle SKL? (Disregard the degree symbol when entering your answer.)
- A. 35
- B. 40
- C. 45
- D. 50
Question 4 (Hard)
The graph of the quadratic function y = f(x) in the xy-plane intersects the x-axis when x = 39 and when x = p, where p is a constant. The maximum value of y = f(x) occurs at the point (14, m), where m is a constant.
What is the value of p?
- A. -15
- B. -11
- C. -5
- D. 0
Question 5 (Hard)
f(x)= |59-2x|
The function f is defined by the given equation. For which of the following values of k does f(k)=3k?
The function f is defined by the given equation. For which of the following values of k does f(k)=3k?
- A. 59/9
- B. 59/5
- C. 177/5
- D. 59
Answer key and solutions
1. A (None) Subtract 2: |k − 3| = −1. An absolute value can never be negative, so there are no solutions. Tip: Isolate the absolute value first. If it equals a negative number, the answer is zero solutions. Absolute Value Equations notes
2. B (1.6 × 1010 miles) One day has 24 × 60 × 60 = 86,400 seconds. 186,000 × 86,400 ≈ 1.6 × 1010 miles. Tip: Convert the time to the same unit as the rate before you multiply. Rates and Unit Conversions notes
3. D (50) LK is parallel to RT, so angle SKL and angle T are corresponding angles. They are equal, so angle SKL = 50°. The 10° angle is extra information. Tip: Parallel lines cut by a side of a triangle copy the triangle’s angles. Angles notes
4. B (-11) A parabola is symmetric about its vertex. The vertex is at x = 14, so the two x-intercepts are the same distance from 14. 39 is 25 to the right, so p = 14 − 25 = −11. Tip: The vertex x-value is the average of the two x-intercepts: (39 + p) / 2 = 14. Quadratic Functions notes
5. B (59/5) Since 3k equals an absolute value, k cannot be negative. Case 1: 59 − 2k = 3k, so k = 59/5. Case 2: 59 − 2k = −3k gives k = −59, which is negative, so it fails. Only 59/5 works. Tip: Solve both cases, then check each answer in the original equation. Absolute Value Equations notes
Want more questions at this level?
Targeted Practice builds sets from the skills you miss.
How to get better at hard SAT math
- Look for the shortcut first. Most of these have a short path, like plugging in a root, squaring both sides, or using symmetry. Hard questions reward the idea, not long algebra.
- Learn each skill from the notes linked in the solutions, then drill it in Targeted Practice.
- Watch it explained. The free Advanced Math video series with Mr. O walks through the Advanced Math domain.
- Test yourself under time with a full-length test in the question bank.
Want 5 more with step-by-step solutions? See hard SAT math questions.
Frequently asked questions
What makes an SAT math question hard?
Usually one extra step or one trap, not harder math. Most hard questions use Algebra and Advanced Math skills you already know, with a twist like an unknown constant or a hidden identity.
How many math questions are on the SAT?
44, in two modules of 22 questions and 35 minutes each. How you do on Module 1 decides whether Module 2 is harder or easier.
Can I use a calculator?
Yes, on every Math question. Desmos is built into Bluebook.
Sources
- College Board, How the SAT Is Structured.
- College Board, The Math Section: Overview.
Frequently Asked Questions
What makes an SAT math question hard?
Usually one extra step or one trap, not harder math.
How many math questions are on the SAT?
44, in two modules of 22 questions and 35 minutes each.
Can I use a calculator?
Yes, on every Math question. Desmos is built into Bluebook.