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Unofficial practice questions, not affiliated with the Massachusetts Department of Elementary and Secondary Education or Pearson. Bank checked October 9, 2026.
A circuit calculation divides the complex number 7 − 4i by 2 + i. What is the quotient, written in the form a + bi?Answer: 2 − 3i. Source: MTEL Mathematics (63) test objectives, subarea I Number Sense and Operations
A lab note records a ratio as 125^(2/3) ÷ 32^(3/5). What is its value?Answer: 25/8. Source: MTEL Mathematics (63) test objectives, subarea I Number Sense and Operations
A tiler wants the largest square tile that covers a floor of 1,071 cm by 462 cm exactly, with no tile cut. By the Euclidean algorithm, what is the side of that tile?Answer: 21 cm. Source: MTEL Mathematics (63) test objectives, subarea I Number Sense and Operations
Matrix M has rows (4, 7) and (2, 5). The linear transformation it represents maps the unit square onto a parallelogram. What is the area of that parallelogram?Answer: 6. Source: MTEL Mathematics (63) test objectives, subarea I Number Sense and Operations
Which of these sets of numbers is closed under division?Answer: The nonzero rational numbers. Source: MTEL Mathematics (63) test objectives, subarea I Number Sense and Operations
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Subarea II: Relations, functions and algebra 15
A culture of cells grows exponentially from 500 cells to 4,000 cells in 6 hours. What is its doubling time?Answer: 2 hours. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
A polynomial of degree 3 has real coefficients, leading coefficient 1 and zeros 2 and 1 + 3i. What is its constant term?Answer: −20. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
A thermostat keeps a room within 1.5 degrees of 68°F. Which inequality describes every temperature t that the room may reach?Answer: |t − 68| ≤ 1.5. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
The arc of a fountain jet follows h(x) = −2x² + 12x − 7, with x and h in feet. What is the vertex of the parabola?Answer: (3, 11). Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
For 0 ≤ x < 2π, which list gives every solution of tan²x = 3?Answer: π/3, 2π/3, 4π/3 and 5π/3. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
For f(x) = (2x + 1)/(x − 3), with x ≠ 3, which expression is the inverse function f⁻¹(x)?Answer: (3x + 1)/(x − 2). Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
For f(x) = log₁₀ (x + 4) − 1, which pair gives the vertical asymptote and the x-intercept of the graph?Answer: x = −4 and (6, 0). Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
For which value of k does the system 3x − 2y = 4 and 6x + ky = 9 have no solution?Answer: −4. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
For y = 3 sin(2x − π/2) + 1, what are the period and the phase shift?Answer: Period π, shifted π/4 to the right. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
The third term of an arithmetic sequence is 11 and the eighth term is 31. What is the twentieth term?Answer: 79. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
The time needed to fill a tank varies inversely with the pump's flow rate. At 12 gallons per minute the tank fills in 40 minutes. How long does it take at 15 gallons per minute?Answer: 32 minutes. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
Using the angle-sum identity with 60° and 45°, what is the exact value of cos 105°?Answer: (√2 − √6)/4. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
Which features does the graph of f(x) = (x² − 9)/(2x² + 2x − 12) have?Answer: A hole at x = −3; vertical asymptote x = 2; horizontal asymptote y = 1/2. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
Which set holds every real solution of x² − x − 6 > 0?Answer: x < −2 or x > 3. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
Which value or values of x satisfy log₂ x + log₂ (x − 2) = 3?Answer: x = 4 only. Source: MTEL Mathematics (63) test objectives, subarea II Relations, Functions, and Algebra
Subarea III: Geometry and measurement 10
A cube has bottom face ABCD and top face EFGH, with E above A, F above B, G above C and H above D. Which edge is skew to edge AB?Answer: CG. Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
A plane slices a sphere of radius 5 cm at a distance of 3 cm from the center. What is the area of the circular cross section?Answer: 16π cm². Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
A triangle with an area of 5 square units is dilated by a scale factor of 3 about the origin and then translated 4 units to the left. What is the area of the image?Answer: 45 square units. Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
A water tank holds 40 liters. A second tank has the same shape, with every linear dimension 2.5 times as large. How much does the second tank hold?Answer: 625 liters. Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
Chord AB of a circle subtends a central angle of 110°. Point C lies on the major arc AB. What is the measure of inscribed angle ACB?Answer: 55°. Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
On a sphere, three arcs of great circles form a triangle. Which statement about the sum of its three angles is true?Answer: It is always greater than 180°. Source: Wikipedia, "Spherical geometry" (read 2026-10-09): the sum of the angles of a triangle is greater than 180°. MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
pH is the negative base-10 logarithm of the hydrogen ion concentration. Solution A has a pH of 3 and solution B a pH of 6. How does the hydrogen ion concentration of A compare with that of B?Answer: It is 1,000 times as great. Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
Two sides of a triangular garden bed measure 7 m and 8 m, and the angle between them is 60°. To the nearest tenth of a meter, how long is the third side?Answer: 7.5 m. Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
Which conic section is the graph of 4x² + 9y² − 16x + 18y − 11 = 0, and where is its center?Answer: An ellipse centered at (2, −1). Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
Which single condition is enough to prove that a quadrilateral is a parallelogram?Answer: Its diagonals bisect each other. Source: MTEL Mathematics (63) test objectives, subarea III Geometry and Measurement
Subarea IV: Probability, statistics, calculus and discrete mathematics 10
A carnival game costs $4 to play. A fair six-sided die is rolled: a 6 pays $12, a 4 or a 5 pays $3, and any other roll pays nothing. What is a player's expected net gain per play?Answer: −$1.00. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
A cart's velocity is v(t) = 6 − 2t meters per second for 0 ≤ t ≤ 5. What total distance does the cart travel in those 5 seconds?Answer: 13 m. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
A committee of 4 is chosen from 4 teachers and 5 parents. How many possible committees have exactly 2 teachers and 2 parents?Answer: 60. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
A crate holds 6 ripe and 4 unripe mangoes. A buyer takes 2 at random, without replacement. What is the probability that at least one of the two is unripe?Answer: 2/3. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
A data set has a median of 40 and an interquartile range of 12. Each value x is converted by the rule y = 0.5x − 3. What are the median and the interquartile range of the converted data?Answer: Median 17, interquartile range 6. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
A particle moves along a line with position s(t) = t³ − 6t² + 9t meters at t seconds. At which time is the particle at rest with positive acceleration?Answer: t = 3. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
A proof by induction shows that 1 + 3 + 5 + … + (2n − 1) = n². The inductive step assumes that the first k odd numbers total k². Which identity completes the step?Answer: k² + (2k + 1) = (k + 1)². Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
In a random sample of 400 voters, 60% favor a ballot question. Using z = 1.96, about what is the margin of error of a 95% confidence interval for the proportion?Answer: About 4.8 percentage points. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
What is the area of the region enclosed by the curve y = 4x − x² and the x-axis?Answer: 32/3. Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
What is the derivative of f(x) = x²e^(3x)?Answer: xe^(3x)(2 + 3x). Source: MTEL Mathematics (63) test objectives, subarea IV Probability, Statistics, Calculus, and Discrete Mathematics
How this list was compiled: Our own questions, written to the test objectives of MTEL Mathematics, field 63 (the tests list names it "Mathematics (Secondary) (63)"), as the test information guide prints them on the Massachusetts Tests for Educator Licensure website (https://www.mtel.nesinc.com/Content/StudyGuide/MA_SG_OBJ_63.htm and https://www.mtel.nesinc.com/TestView.aspx?f=HTML_FRAG/MA063_TestPage.html, read 2026-10-09): 85 multiple-choice questions and 2 open-response items in 4 hours; multiple-choice subareas I Number Sense and Operations (10 to 12 questions, 10% of the score, objectives 0001 and 0002), II Relations, Functions, and Algebra (31 to 33, 30%, objectives 0003 to 0008), III Geometry and Measurement (20 to 22, 20%, objectives 0009 to 0012) and IV Probability, Statistics, Calculus, and Discrete Mathematics (20 to 22, 20%, objectives 0013 to 0017), and the open-response subarea, 20%. This bank holds 5, 15, 10 and 10. Every answer is arithmetic or reasoning from the numbers in the question, and each was worked out a second time before the bank was written; the one stated theorem, on the angles of a spherical triangle, was read in the Wikipedia article named under its question. Licence: Questions, choices and explanations are original work of this site.
How it works
This MTEL math practice test follows Mathematics (63), the secondary test: 40 questions of our own in its four multiple-choice subareas. It is unofficial and not affiliated with, endorsed by or sponsored by the Massachusetts Department of Elementary and Secondary Education or Pearson.
Read the full guide
Which MTEL math test this is
Of the mathematics tests on the MTEL tests list, this page follows Mathematics (Secondary) (63). Mathematics (Middle School) (65) and Mathematics (Elementary) (68) have their own practice pages, linked below.
The four subareas
By the test objectives: number sense and operations, 10 to 12 questions; relations, functions and algebra, 31 to 33; geometry and measurement, 20 to 22; probability, statistics, calculus and discrete mathematics, 20 to 22. This bank holds 5, 15, 10 and 10.
Worked answers
The working stands under each answer, with the error that leads to a tempting wrong choice. Every answer was worked a second time before it was printed.
The real test supplies a scientific calculator and a formula page on screen. The MTEL site lets you read the formulas before test day.
The two open-response assignments are written, and the MTEL site says a response may be handwritten and scanned. They are not practiced here.
Questions and answers
How many questions are on the MTEL math test?
Mathematics (63) has 85 multiple-choice questions and 2 open-response assignments in 4 hours of testing time. The MTEL site adds that a test may include questions that do not count toward the score.
What is a passing score on the MTEL math test?
The MTEL site gives a qualifying score of 240. Multiple-choice questions are 80% of the score and the open-response assignments 20%, so the number right on this page predicts nothing.
What math is on the MTEL Mathematics test?
Number sense, complex numbers and matrices; algebra and functions through trigonometry; Euclidean, coordinate and transformational geometry; probability, statistics, differential and integral calculus, and discrete mathematics. The test overview sets the level at that of a college major.
Is there a free MTEL math practice test?
This page is free, with no sign-up. The MTEL site lists its own materials for field 63: a test information guide with the objectives and sample questions, the formula page and a practice test. The questions here are ours.