100+ Free EGE Math Profile Practice Questions

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Solve the inequality 2^x > 8 (Task 15, exponential inequality).

x > 4

x < 3

x > 1/3

x > 3

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Key Facts: EGE Math Profile Exam

The EGE Mathematics Profile is a 235-minute, 19-task advanced maths exam (12 short-answer + 7 extended-answer tasks, max 32 primary points) required for STEM university admission in Russia.

Sample EGE Math Profile Practice Questions

Try these sample questions to test your EGE Math Profile exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1On squared paper with cells 1 cm by 1 cm, a right triangle has legs of length 6 cm and 8 cm lying along the grid lines. What is the area of the triangle (Task 1, planimetry)?

A.24 cm^2

B.48 cm^2

C.14 cm^2

D.28 cm^2

Explanation: The area of a right triangle equals half the product of its legs: A = (1/2)*6*8 = 24 cm^2. Task 1 of the Profile EGE tests basic planimetry on a coordinate/grid figure.

2Vectors a = (3, 4) and b = (1, 0) are given in the coordinate plane. What is the dot (scalar) product a · b (Task 2, vectors, added in 2026)?

A.4

B.3

C.7

D.0

Explanation: The scalar product of (x1, y1) and (x2, y2) is x1*x2 + y1*y2 = 3*1 + 4*0 = 3. Task 2 on vectors was newly added to the Profile EGE in 2026.

3A rectangular box (cuboid) has edges 2, 3 and 4. What is its volume (Task 3, stereometry)?

A.9

B.12

C.24

D.26

Explanation: The volume of a cuboid is the product of its three edge lengths: V = 2*3*4 = 24. This is a standard Task 3 stereometry computation.

4A fair six-sided die is rolled once. What is the probability that the number rolled is greater than 4 (Task 4, classical probability)?

A.1/6

B.1/2

C.2/3

D.1/3

Explanation: Outcomes greater than 4 are {5, 6}, i.e. 2 favorable outcomes out of 6 equally likely. The probability is 2/6 = 1/3, using the classical definition P = favorable / total.

5Events A and B are independent with P(A) = 0.6 and P(B) = 0.5. What is P(A and B) (Task 5, probability theorems)?

A.1.1

B.0.3

C.0.1

D.0.8

Explanation: For independent events the probability of both occurring is the product: P(A and B) = P(A)*P(B) = 0.6*0.5 = 0.3. The multiplication rule for independent events is a core Task 5 skill.

6Solve the equation 3^(x-2) = 27 (Task 6, simplest equations).

A.x = 3

B.x = 9

C.x = 5

D.x = 1

Explanation: Write 27 as 3^3, so 3^(x-2) = 3^3 gives x - 2 = 3, hence x = 5. Equating exponents of equal bases is the standard method for the simplest exponential equations.

7Find the value of log_2(8) + log_3(9) (Task 7, value of an expression).

A.17

B.6

C.4

D.5

Explanation: log_2(8) = 3 because 2^3 = 8, and log_3(9) = 2 because 3^2 = 9. Their sum is 3 + 2 = 5. Task 7 tests evaluation of logarithmic and trigonometric expressions.

8A point moves along a line so that its coordinate is x(t) = t^3 - 3t^2 + 5 (metres). Find its instantaneous velocity at t = 2 s (Task 8, derivative as rate).

A.0 m/s

B.6 m/s

C.1 m/s

D.12 m/s

Explanation: Velocity is the derivative of position: x'(t) = 3t^2 - 6t. At t = 2, x'(2) = 3*4 - 6*2 = 12 - 12 = 0 m/s. The physical meaning of the derivative is a key Task 8 idea.

9The line y = 4x - 7 is tangent to the graph of y = x^2 + bx + 3. Using the geometric meaning of the derivative, what is the slope of the tangent at the point of tangency (Task 8/9, tangent line)?

A.7

B.3

C.4

D.1

Explanation: The slope of a tangent line equals the value of the derivative at the point of tangency, which is the coefficient of x in the tangent's equation. Here y = 4x - 7 has slope 4, so the derivative at the tangency point is 4.

10A body is thrown upward and its height (metres) is h(t) = 20t - 5t^2. Using calculus, at what time does it reach maximum height (Task 9, applied/derivative problem)?

A.t = 1 s

B.t = 4 s

C.t = 5 s

D.t = 2 s

Explanation: Maximum height occurs where the velocity h'(t) = 20 - 10t equals zero, giving t = 2 s. At that point the second derivative h'' = -10 < 0 confirms a maximum.

About the EGE Math Profile Exam

The EGE Mathematics Profile Level (Математика профильного уровня) is the advanced mathematics paper of Russia's Unified State Exam, required for admission to technical and STEM university programmes. The 2026 paper has 19 tasks in two parts: Part 1 (tasks 1-12) requires short numeric answers, and Part 2 (tasks 13-19) requires full written solutions graded by expert examiners. Content spans algebra, equations and inequalities (logarithmic, exponential, trigonometric), functions and the beginnings of analysis (derivatives, antiderivatives, extrema, tangents), planimetry, vectors and stereometry, probability and statistics, applied word problems, financial mathematics, parameters and number theory. The maximum primary score in 2026 is 32, converted to a 100-point test scale, with a minimum threshold of 27 test points for university entry. The exam lasts 3 hours 55 minutes and is administered free of charge by Rosobrnadzor.

Questions

100 scored questions

Time Limit

3 hours 55 minutes (235 minutes)

Passing Score

Minimum 27 test points (out of 100) for university admission; grades 27-49 = '3', 50-63 = '4', 64+ = '5'

Exam Fee

Free for Russian school graduates (state-funded final certification, no fee) (Rosobrnadzor; content materials (KIMs) developed by FIPI)

EGE Math Profile Exam Content Outline

25%

Algebra: equations & inequalities

Exponential, logarithmic, trigonometric, rational and irrational equations and inequalities, and systems (Tasks 6, 13, 15)

18%

Calculus & analysis

Derivatives, antiderivatives/integrals, tangents, rates of change, extrema and applied optimization (Tasks 8-9)

12%

Functions & function investigation

Graphs, properties, monotonicity, max/min on intervals (Tasks 11-12)

22%

Planimetry, vectors & stereometry

Areas, volumes, angles, triangles, circles, vectors and 3-D solids (Tasks 1-3, 14, 17)

10%

Probability & statistics

Classical probability, addition and multiplication rules, conditional probability (Tasks 4-5)

Word & financial problems

Motion, work-rate, percentage and loan/deposit financial-math problems (Tasks 10, 16)

Parameters & number theory

Equations with parameters and properties of integers (Tasks 18-19)

How to Pass the EGE Math Profile Exam

What You Need to Know

Passing score: Minimum 27 test points (out of 100) for university admission; grades 27-49 = '3', 50-63 = '4', 64+ = '5'
Exam length: 100 questions
Time limit: 3 hours 55 minutes (235 minutes)
Exam fee: Free for Russian school graduates (state-funded final certification, no fee)

Keys to Passing

Complete 500+ practice questions
Score 80%+ consistently before scheduling
Focus on highest-weighted sections
Use our AI tutor for tough concepts

EGE Math Profile Study Tips from Top Performers

1Automate Part 1 (tasks 1-12): aim to solve each short-answer task quickly and accurately, since each is worth 1 primary point.

2Master derivatives and antiderivatives early — they appear in tasks 8, 9 and the function-investigation tasks 11-12.

3Practice writing full justified solutions for Part 2 (tasks 13-19); experts grade reasoning, not just the final answer.

4Memorize key trigonometric values, logarithm rules, and area/volume formulas for fast, error-free computation.

5Drill financial-math (task 16) and parameter (task 18) problems, the highest-value extended tasks worth 3-4 points each.

6Use FIPI demo versions and open-bank tasks to match the authentic 2026 format and difficulty.

Frequently Asked Questions

How many tasks are on the EGE Mathematics Profile exam in 2026?

There are 19 tasks: Part 1 has 12 short-answer tasks (1-12) and Part 2 has 7 extended-answer tasks (13-19) requiring full written solutions. A new vector task (task 2) was added in 2026.

What is the passing score for the EGE Mathematics Profile?

The minimum threshold for university admission is 27 test points out of 100 (about 6 primary points). Competitive STEM programmes typically require far higher scores, often 70+.

How long is the EGE Mathematics Profile exam?

The exam lasts 3 hours 55 minutes (235 minutes), enough time for the 12 short-answer and 7 extended-answer tasks.

Is the EGE Mathematics Profile free?

Yes. The EGE is a state final certification exam taken free of charge by Russian secondary-school graduates; the government funds its administration.

Who administers the EGE?

The EGE is overseen by Rosobrnadzor (the Federal Service for Supervision in Education and Science), with exam materials (KIMs) developed by FIPI, the Federal Institute of Pedagogical Measurements.

What topics does the Profile exam cover?

Algebra and equations, functions and beginnings of analysis (derivatives and integrals), planimetry, vectors and stereometry, probability and statistics, word and financial problems, parameters and number theory.

100+ Free EGE Math Profile Practice Questions

Russia EGE - Mathematics (Profile Level)