• 49 questions

CSET Math Subtest 1 Practice Test

Unofficial practice for CSET: Mathematics Subtest I, test code 211: number and quantity, and algebra.

Evaluate 27^(2/3) × 9^(−1/2) ÷ 3^(−2) by the laws of exponents.

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

Print covers the next 35 questions. Share this set copies a link to the same questions in the same order.

Exam mode: 35 questions, no time limit.

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Unofficial practice questions, not affiliated with the California Commission on Teacher Credentialing or Pearson. Bank checked October 9, 2026.

Constructed-response prompts to print 6 lines, made to print

Subtest I extended constructed-response practice

Name ____ Date ____

Write a full answer to each prompt on the lines, showing every step and the reason for it. Unofficial practice prompts; none is from the real test.

Number and quantity

  • Prove that √3 is irrational. State what you assume, and name the property of the integers that each step uses.
  • Use the principle of mathematical induction to prove that 1 + 2 + 2² + … + 2ⁿ = 2ⁿ⁺¹ − 1 for every natural number n. Mark the base case and the inductive step.

Algebra

  • Find every complex root of x⁴ − x³ − 5x² − x − 6. Use the Rational Root Theorem to choose candidates and the Factor Theorem to confirm them, and show the division.
  • Derive the quadratic formula by completing the square on ax² + bx + c = 0, where a is not 0. Justify each step.
  • Prove that two nonzero vectors in the plane are perpendicular if and only if their dot product is 0.
  • Show that the 2 × 2 real matrices form a ring that is not a field: name one field property that fails and give matrices that show it.

Source: Own prompts. The real subtest has 3 extended constructed-response questions, 1 on Number and Quantity and 2 on Algebra (https://www.ctcexams.nesinc.com/Content/StudyGuide/CA_SG_obj_211.htm); the topics follow the content domains of Subtest I (https://www.ctcexams.nesinc.com/content/docs/CSET_Prep/CS_211subtestdescription.pdf), both read 2026-10-09. Checked October 9, 2026.

How Subtest I is scored 8 lines, made to print

How the written answers count

What the program publishes about scoring, in our words. Unofficial; not a scoring guide of the real test.

The score

  • Two sections, one score The 35 multiple-choice questions carry 70% of the Subtest I score and the 3 extended written answers carry 30%.
  • The scale Raw scores are converted to a 100 to 300 scale, and 220 is the minimum passing score.
  • One sitting A passing score has to be earned at a single administration; sections cannot be combined across sittings.

What scorers look at in a written answer

  • Purpose How far the answer does what the assignment asks.
  • Subject matter knowledge Whether the mathematics is accurate and is applied to the problem.
  • Support Whether the supporting evidence, such as the steps and reasons of a proof, is fitting and of good quality.
  • Depth and breadth of understanding A fourth characteristic, named for some assignments only.
  • Each written answer is scored by two or more qualified California educators.

Source: Test Results Report Information (https://www.ctcexams.nesinc.com/content/docs/CS_ScoreReport_Insert.pdf) and the test structure of CSET: Mathematics (https://www.ctcexams.nesinc.com/Content/StudyGuide/CA_SG_obj_211.htm), both read 2026-10-09. Checked October 9, 2026.

Next step Another teacher math test: Praxis 5165

All 49 questions

Number and quantity: real and complex numbers 8

  • A pump moves 4.5 liters of water per minute. How many cubic meters does it move in 24 hours? (1 m³ = 1,000 L) Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).
  • The complex number z = 3 + 4i is multiplied by i. Where does the product lie in the complex plane? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).
  • The decimal 0.272727… repeats without end. Written as a fraction in lowest terms, what is it? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).
  • Evaluate 27^(2/3) × 9^(−1/2) ÷ 3^(−2) by the laws of exponents. Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).
  • Exactly one of these four expressions simplifies to a rational number. Which one? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).
Show all 49 questions (44 more)

Number and quantity: real and complex numbers, continued

  • In the complex plane, how far is the point for 5 − 12i from the origin? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).
  • Multiply the complex numbers 2 + 5i and 3 − i. Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).
  • What is (4 + 2i) ÷ (1 − i) in the form a + bi? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0001 The Real and Complex Number Systems (SMR 1.1).

Number and quantity: number theory 6

  • A proof by induction shows 1 + 3 + 5 + … + (2n − 1) = n². Assuming it for n = k, which expression must the inductive step show equal to (k + 1)²? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0002 Number Theory (SMR 1.2).
  • The Euclidean algorithm is run on 1071 and 462. What greatest common divisor does it give? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0002 Number Theory (SMR 1.2).
  • For which digit d is the four-digit number 5d38 divisible by 9? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0002 Number Theory (SMR 1.2).
  • How many positive integers divide 360 evenly? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0002 Number Theory (SMR 1.2).
  • What is the least common multiple of 84 and 90? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0002 Number Theory (SMR 1.2).
  • What is the remainder when 7^100 is divided by 5? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Number and Quantity 0002 Number Theory (SMR 1.2).

Algebra: algebraic structures 8

  • A chemist wants 30 liters of acid at 40% strength and has stock at 25% and at 50%. How many liters of the 25% stock go into the mix? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).
  • If a < b and c < 0, which statement must be true? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).
  • In an ordered field every nonzero square is positive. Which fact shows that no such ordering of the complex numbers exists? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).
  • Squaring both sides of √(x + 7) = x − 5 leads to x = 2 or x = 9. Which of these values solve the original equation? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).
  • Which expression is x² − 6x + 2 written in the form (x − h)² + k? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).
  • Which is the complete factorization of 2x³ − 18x over the integers? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).
  • Which of these elements has a multiplicative inverse inside its own set? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).
  • Which values of x satisfy |2x − 3| ≤ 7? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0003 Algebraic Structures (SMR 2.1).

Algebra: polynomial equations and inequalities 9

  • A cubic polynomial with real coefficients has roots 2 and 3 + i. What is its third root? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • By the Rational Root Theorem, a rational root of 3x³ + 4x² − 7x + 2 is a factor of 2 over a factor of 3. Which of these is a root? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • Counted with multiplicity, how many complex roots does x⁵ − 3x³ + x − 7 have? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • Dividing p(x) = x⁴ − 3x² + 5 by x + 2 leaves which remainder, by the Remainder Theorem? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • For which value of k is x − 2 a factor of x³ + kx² − 4x + 8? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • On which set of real numbers is the product (x + 3)(x − 2)(x − 5) positive? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • The roots of 2x² − 7x + 4 = 0 are r and s. What is r² + s²? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • What is the coefficient of x³ in the expansion of (2x − 1)⁵? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).
  • Which pair of complex numbers are the roots of x² − 6x + 25? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0004 Polynomial Equations and Inequalities (SMR 2.2).

Algebra: functions 10

  • $2,000 is invested at 6% annual interest compounded monthly. Which expression gives its value after 5 years? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • A flare is fired upward, and t seconds later its height in meters is h(t) = −5t² + 30t + 2. How high does the flare rise? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • A line meets 3x − 4y = 8 at a right angle and goes through (6, −1). What is its equation? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • Both y > 2x − 1 and x + y ≤ 4 must hold. Which of these points satisfies the pair? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • Compose g after f, where f(x) = 2x − 3 and g(x) = x² + 1. Which polynomial results? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • Find the inverse function of f(x) = (3x − 2)/(x + 4). Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • For which real numbers x is f(x) = √(6 − 2x) ÷ (x + 1) defined? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • What is the solution of log₂(x) + log₂(x − 6) = 4? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • Which describes the graph of f(x) = (2x² − 8)/(x² − x − 6)? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).
  • Which function is one-to-one on the real numbers? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0005 Functions (SMR 2.3).

Algebra: linear algebra 8

  • A bake sale sold 120 items for $330 in all. Pies cost $5 each and boxes of cookies $2 each. How many pies were sold? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).
  • Evaluate the 3 × 3 determinant whose rows are (2, −1, 0), (1, 3, 2) and (0, 4, 1). Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).
  • For which value of k are the vectors ⟨k, 3⟩ and ⟨4, −6⟩ perpendicular? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).
  • For which values of k does the system kx + 2y = 4, 3x + (k − 1)y = 6 fail to have exactly one solution? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).
  • If u = ⟨2, −1⟩ and v = ⟨−3, 4⟩, what is 2u − 3v? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).
  • If v = ⟨−2, 1, 2⟩, what is the magnitude of −3v? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).
  • Matrices multiply row by column. With A = rows (1, 2), (0, −1) and B = rows (3, 0), (1, 4), find the product AB. Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).
  • What is the inverse of the matrix with rows (4, 7) and (2, 4)? Worked twice. Outline: CSET: Mathematics Subtest I content specifications, Algebra 0006 Linear Algebra (SMR 2.4).

How this list was compiled: Our own questions, written to the published content domains of CSET: Mathematics Subtest I, test code 211, read 2026-10-09 on the program site of the California Educator Credentialing Assessments (test page https://www.ctcexams.nesinc.com/TestView.aspx?f=HTML_FRAG/CA_CSET211_TestPage.html; test structure https://www.ctcexams.nesinc.com/Content/StudyGuide/CA_SG_obj_211.htm; subtest description https://www.ctcexams.nesinc.com/content/docs/CSET_Prep/CS_211subtestdescription.pdf): 35 multiple-choice questions, 10 on Number and Quantity and 25 on Algebra, and 3 extended constructed-response questions. The bank holds 14 questions on number and quantity (8 on the real and complex number systems, 6 on number theory) and 35 on algebra (8 on algebraic structures, 9 on polynomial equations and inequalities, 10 on functions, 8 on linear algebra). Every answer was worked twice. Licence: Questions, choices and explanations are original work of this site.

How it works

This free CSET Math Subtest 1 practice test asks 49 questions of our own on number and quantity and on algebra. It is unofficial and not affiliated with the California Commission on Teacher Credentialing or Pearson.

Read the full guide

What Subtest I covers

CSET: Mathematics Subtest I is test code 211. Its 35 multiple-choice questions split 10 on Number and Quantity and 25 on Algebra (test structure read October 9, 2026). The bank keeps that ratio, 14 to 35, and exam mode draws 35.

Inside the two domains

Number and quantity: the real and complex numbers, then number theory with the Euclidean algorithm and induction. Algebra: algebraic structures, polynomial equations and inequalities, functions, and linear algebra with vectors, matrices and determinants.

The written part

The real subtest also sets three extended constructed-response questions, one on number and quantity and two on algebra, which this page does not mark. Six prompts of our own print from under the questions.

Questions and answers

Is this CSET Math Subtest 1 practice test the real exam?

No. The 49 questions were written here from the published content domains of Subtest I. None is taken or reworded from the real subtest, its sample questions or the official practice test, and a result here does not forecast your score.

How many questions are on CSET Math Subtest 1?

35 multiple-choice and 3 extended constructed-response questions, by the published test structure. One session of 2 hours and 30 minutes covers both parts.

What is the passing score for CSET Math Subtest 1?

220, on a scale that runs from 100 to 300. The multiple-choice section carries 70% of the score and the written answers 30%, so a count of right answers here is not a pass or a fail.

Can you use a calculator on CSET Math Subtest 1?

No. The test page says only Subtest II allows a graphing calculator. Subtest I does include assignments that let you handwrite a response and scan it.

What is on CSET Math Subtest 2 and Subtest 3?

Subtest II (212) is geometry, and probability and statistics. Subtest III (213) is calculus. The Single Subject Teaching Credential in Mathematics takes a pass on all three, and each is registered for as a session of its own.

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