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2026 Statistics

Key Facts: ESA Exam

50

Total Questions

ESA Edital

50%

Min Section Score

DECEx / Exército

4 hours

Exam Duration

ESA Três Corações

R$ 95

Registration Fee

Exército Brasileiro

The ESA sergeant admission exam consists of 50 questions across Mathematics (14), Portuguese (14), History (6), Geography (6), and English (10) plus an essay; this bank provides 100 English-language practice questions across all blueprint subjects.

Sample ESA Practice Questions

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

1In an arithmetic progression (PA), the 3rd term is a3 = 11 and the 7th term is a7 = 27. What is the 20th term (a20) of this progression?
A.75
B.79
C.83
D.87
Explanation: In an arithmetic progression, the difference between any two terms is given by an - am = (n - m) * r. Here, a7 - a3 = 4r => 27 - 11 = 16 => r = 4. The first term is a1 = a3 - 2r = 11 - 8 = 3. Therefore, the 20th term is a20 = a1 + 19r = 3 + 19(4) = 3 + 76 = 79.
2The sum of the infinite terms of a decreasing geometric progression (PG) is equal to 12. If the first term is a1 = 8, what is the common ratio (q) of this progression?
A.1/4
B.2/3
C.1/3
D.1/2
Explanation: The sum of the terms of an infinite geometric progression with |q| < 1 is given by S = a1 / (1 - q). Substituting the given values: 12 = 8 / (1 - q) => 1 - q = 8/12 = 2/3 => q = 1 - 2/3 = 1/3.
3A projectile is fired such that its height h (in meters) after t seconds is modeled by the quadratic function h(t) = -5t^2 + 40t + 10. What is the maximum height reached by the projectile?
A.90 meters
B.80 meters
C.100 meters
D.70 meters
Explanation: The maximum height of a vertical parabolic trajectory occurs at the vertex of the quadratic function. The time to reach maximum height is tv = -b / (2a) = -40 / (2 * -5) = 4 seconds. Substituting tv into h(t): h(4) = -5(4)^2 + 40(4) + 10 = -5(16) + 160 + 10 = -80 + 160 + 10 = 90 meters.
4What is the real solution set for the logarithmic equation log2(x + 3) + log2(x - 1) = 5?
A.{-7, 5}
B.{7}
C.{3}
D.{5}
Explanation: First, consider the existence conditions: x + 3 > 0 => x > -3 and x - 1 > 0 => x > 1, so the domain is x > 1. Applying logarithmic properties: log2((x + 3)(x - 1)) = 5 => (x + 3)(x - 1) = 2^5 = 32 => x^2 + 2x - 3 = 32 => x^2 + 2x - 35 = 0. Factoring gives (x + 7)(x - 5) = 0, yielding x = -7 or x = 5. Since x must be greater than 1, x = -7 is extraneous, leaving S = {5}.
5Solve the exponential inequality (1/3)^(x^2 - 2x) >= 1/27 in the real numbers R. What is the solution set?
A.[-1, 3]
B.(-inf, -1] U [3, +inf)
C.[-3, 1]
D.[1, 3]
Explanation: Rewriting the right side with base 1/3 gives 1/27 = (1/3)^3. Since the base 0 < 1/3 < 1, the inequality direction reverses when comparing exponents: x^2 - 2x <= 3 => x^2 - 2x - 3 <= 0. The roots of the quadratic equation x^2 - 2x - 3 = 0 are x = -1 and x = 3. Because the parabola opens upwards (a = 1 > 0), the expression is less than or equal to zero between the roots, yielding the closed interval S = [-1, 3].
6What is the sum of all real solutions to the modular equation |2x - 6| = x + 3?
A.10
B.8
C.12
D.6
Explanation: For the equation to have real solutions, the right-hand side must be non-negative: x + 3 >= 0 => x >= -3. Case 1: 2x - 6 = x + 3 => x = 9 (valid since 9 >= -3). Case 2: 2x - 6 = -(x + 3) = -x - 3 => 3x = 3 => x = 1 (valid since 1 >= -3). Both roots are valid. The sum of the solutions is 9 + 1 = 10.
7Given the real functions f(x) = 3x - 5 and g(x) = x^2 + 2, what is the value of the composite function (g o f)(2)?
A.11
B.3
C.7
D.5
Explanation: By definition of composite functions, (g o f)(2) = g(f(2)). First, evaluate f(2): f(2) = 3(2) - 5 = 6 - 5 = 1. Next, substitute this value into g(x): g(1) = (1)^2 + 2 = 1 + 2 = 3.
8Given the 2x2 square matrix A = [[2, k], [3, 4]], for what value of k is the determinant of matrix A equal to -4?
A.2
B.-2
C.6
D.4
Explanation: The determinant of a 2x2 matrix A = [[a, b], [c, d]] is computed as det(A) = ad - bc. For matrix A = [[2, k], [3, 4]], det(A) = (2)(4) - (3)(k) = 8 - 3k. Setting det(A) = -4 gives 8 - 3k = -4 => 3k = 12 => k = 4.
9Consider the linear system in variables x and y: { 2x + my = 4 ; 4x + 6y = 8 }. For which value of m does this system have infinitely many solutions (sistema possivel e indeterminado - SPI)?
A.m = 3
B.m = -3
C.m = 6
D.m ≠ 3
Explanation: A 2x2 linear system a1*x + b1*y = c1 and a2*x + b2*y = c2 is indeterminate (SPI) with infinitely many solutions when the coefficient ratios are all equal: a1/a2 = b1/b2 = c1/c2. Here, 2/4 = m/6 = 4/8 => 1/2 = m/6 = 1/2 => m = 3. When m = 3, the two equations represent the exact same line in R^2.
10A military squad consists of 8 soldiers. How many distinct 3-soldier patrol teams can be formed from this squad?
A.336
B.56
C.24
D.112
Explanation: Since the order of soldiers in a patrol team does not matter, this is a simple combination problem of 8 elements chosen 3 at a time: C(8, 3) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 336 / 6 = 56 patrol teams.

About the ESA Exam

The ESA Concurso de Admissão is the annual national entrance examination for the Escola de Sargentos das Armas (ESA) in Três Corações/MG, training 3rd Sergeants (3º Sargento) of the Brazilian Army across Combat Arms (Infantry, Cavalry, Artillery, Engineering, Communications), Logistics (Material Bélico, Intendência, Topografia, Manutenção), and Health/Music tracks. For the Área Geral/Aviação the written test features 50 multiple-choice questions with five alternatives each (14 Math, 14 Portuguese, 6 Brazil History, 6 Brazil Geography, 10 English) plus an essay in a single 4-hour session; candidates in the Saúde and Música areas sit the same 50-question paper rebalanced to 10 Math, 10 Portuguese, 8 combined History and Geography, 10 English and 12 area-specific questions. This bank targets the Área Geral blueprint. Note: the official exam is set and answered in Portuguese; this bank is an English-language MCQ study adaptation of the published syllabus, not an official translation or a simulation of the essay.

Questions

50 scored questions

Time Limit

4 hours (real exam)

Passing Score

50% per subject block; Redação APTO at 5.0/10.0 (eliminatory only)

Exam Fee

R$ 95 (Exército Brasileiro / DECEx / ESA)

ESA Exam Content Outline

28%

Matemática

Progressions, functions, geometry, trigonometry, logarithms, and probability

28%

Língua Portuguesa

Text comprehension, period syntax, morphology, agreement, and literature

12%

História do Brasil

Colonial periods, Empire, Republic, Vargas Era, and military history

12%

Geografia do Brasil

Territory, relief, biomes, climate, demography, and economic geography

20%

Língua Inglesa

Grammar rules, reading comprehension, verb tenses, and contextual usage

How to Pass the ESA Exam

What You Need to Know

  • Passing score: 50% per subject block; Redação APTO at 5.0/10.0 (eliminatory only)
  • Exam length: 50 questions
  • Time limit: 4 hours (real exam)
  • Exam fee: R$ 95

Keys to Passing

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

ESA Study Tips from Top Performers

1Master algebra, trigonometry, and analytic geometry, as Mathematics represents 28% of the objective test
2Thoroughly review Brazilian history timelines from the Colony to the 1988 Constitution and FEB in World War II
3Study Brazilian physical and economic geography, including regional biomes, hydrographic basins, and agricultural frontiers
4Practice normative Portuguese grammar with special focus on verbal concord and pronoun placement (colocação pronominal)

Frequently Asked Questions

How is the ESA CFGS written examination structured?

The ESA exam is a 4-hour single-day test with 50 objective questions: 14 in Mathematics, 14 in Portuguese, 6 in History of Brazil, 6 in Geography of Brazil, and 10 in English, plus a 20-to-30 line dissertative essay.

What is the passing score for ESA?

A candidate must reach at least half the available marks in each subject block, with History and Geography counted together as one 12-question block, and must be judged APTO on the Redação with at least 5.0 out of 10.0. Since the 2023 concurso the essay has been eliminatory only — its mark no longer feeds the classification average, so a 5.0 counts exactly the same as a 9.0.

What rank do graduates achieve after ESA?

Graduates of the two-year CFGS course are appointed as Third Sergeants (3º Sargento) of the Brazilian Army with a recognized higher technologist degree.

What is the registration fee for the ESA concurso?

The registration fee is R$ 95, with exemptions granted to candidates registered in CadÚnico or registered bone marrow donors.