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100+ Free EsPCEx Practice Questions

Prepare for the Concurso de Admissão à Escola Preparatória de Cadetes do Exército (EsPCEx) exam with instant access — no signup required.

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2026 Statistics

Key Facts: EsPCEx Exam

100

Total Questions

EsPCEx Edital

50 / 100

Redação APTO Cut

EsPCEx Edital

2 Days (9h)

Total Duration

EsPCEx Campinas

R$ 100

Registration Fee

Exército Brasileiro

The EsPCEx entrance exam consists of 100 objective questions over two days covering Portuguese, Physics, Chemistry, Math, Geography, History, and English plus an essay; this bank provides 100 English-language practice questions across all blueprint subjects.

Sample EsPCEx Practice Questions

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

1Let f: ℝ − {2} → ℝ − {3} be the bijective function defined by f(x) = (3x + 1)/(x − 2). What are the rule of its inverse function f⁻¹(x) and the value of f⁻¹(5), respectively?
A.f⁻¹(x) = (3x − 1)/(x + 2) and f⁻¹(5) = 2
B.f⁻¹(x) = (2x + 1)/(x − 3) and f⁻¹(5) = 11/2
C.f⁻¹(x) = (x − 1)/(x − 3) and f⁻¹(5) = 2
D.f⁻¹(x) = (x + 2)/(3x − 1) and f⁻¹(5) = 1/2
Explanation: Set y = (3x + 1)/(x − 2) and isolate x: y(x − 2) = 3x + 1, so yx − 3x = 2y + 1 and x(y − 3) = 2y + 1, giving x = (2y + 1)/(y − 3). Renaming the variables gives f⁻¹(x) = (2x + 1)/(x − 3), and f⁻¹(5) = 11/2. The excluded value 3 in the codomain of f reappears as the excluded value in the domain of f⁻¹.
2In the Cartesian plane, what is the distance between the point P(3, 4) and the line r with general equation 3x + 4y − 10 = 0?
A.5
B.1
C.3
D.15/7
Explanation: The distance from P(x₀, y₀) to Ax + By + C = 0 is d = |Ax₀ + By₀ + C|/√(A² + B²). Substituting gives d = |9 + 16 − 10|/√25 = 15/5 = 3.
3Consider the complex number z = cos(π/3) + i·sin(π/3). What is the value of the expression z⁶ + z³ + 1?
A.0
B.−1
C.3
D.1
Explanation: By De Moivre's theorem, zⁿ = cos(nπ/3) + i·sin(nπ/3). For n = 6 the argument is 2π, so z⁶ = 1; for n = 3 the argument is π, so z³ = −1. Therefore z⁶ + z³ + 1 = 1 − 1 + 1 = 1.
4The roots of the polynomial P(x) = x³ − 15x² + 71x − 105 form an arithmetic progression. What is the sum of the squares of those roots?
A.83
B.75
C.89
D.225
Explanation: Girard's relations give a root sum of 15 and a pairwise-product sum of 71. Writing the roots as (a − r), a, (a + r) gives 3a = 15, so a = 5. The identity (Σr)² − 2Σrᵢrⱼ then gives the sum of squares directly: 15² − 2(71) = 225 − 142 = 83. The roots themselves are 3, 5 and 7.
5How many distinct anagrams of the letters in the Portuguese word CADETE keep all the vowels together?
A.144
B.72
C.36
D.120
Explanation: CADETE has 6 letters: the consonants C, D, T and the vowels A, E, E. Treating the vowels as one block, the block can be internally arranged in 3!/2! = 3 ways because the E repeats. The block plus the three consonants make 4 distinct items, permutable in 4! = 24 ways. The total is 24 × 3 = 72.
6Let A be a 4 × 4 square matrix with det(A) = 8. What is the determinant of the matrix 2A⁻¹?
A.1/2
B.4
C.2
D.8
Explanation: For an n × n matrix, det(k·M) = kⁿ·det(M), and here n = 4, so det(2A⁻¹) = 2⁴·det(A⁻¹). Since det(A⁻¹) = 1/det(A) = 1/8, the result is 16 × (1/8) = 2.
7For every real x where the functions are defined and cos(x) ≠ 0, the trigonometric expression sin(2x)/(1 + cos(2x)) is identical to which function?
A.cot(x)
B.sin(x)
C.cos(x)
D.tan(x)
Explanation: Use the double-angle identities sin(2x) = 2·sin(x)·cos(x) and cos(2x) = 2·cos²(x) − 1. The denominator becomes 1 + 2cos²(x) − 1 = 2cos²(x), so the fraction reduces to [2·sin(x)·cos(x)]/[2·cos²(x)] = sin(x)/cos(x) = tan(x).
8Over the real numbers, what is the solution set of the logarithmic equation log₂(x − 1) + log₂(x + 3) = 5?
A.{5}
B.{−7, 5}
C.{7}
D.{3}
Explanation: The existence conditions require x − 1 > 0 and x + 3 > 0, so x > 1. The product rule gives (x − 1)(x + 3) = 2⁵ = 32, that is x² + 2x − 35 = 0, which factors as (x + 7)(x − 5) = 0. Only x = 5 satisfies x > 1, so −7 is an extraneous root.
9A right circular cone has a base radius of 6 cm and a height of 8 cm. A sphere is inscribed in that cone. What is the volume of the sphere, in cm³?
A.18π
B.36π
C.48π
D.72π
Explanation: The slant height is √(36 + 64) = 10 cm, so the meridian cross-section is an isosceles triangle of base 12 cm, equal sides 10 cm and height 8 cm. Its area is 48 cm² and its semiperimeter is 16 cm, so the inradius is r = 48/16 = 3 cm. The sphere's volume is (4/3)π(27) = 36π cm³.
10What is the sum of the first 20 terms of an arithmetic progression whose first term is a₁ = 3 and whose common difference is r = 4?
A.800
B.840
C.820
D.790
Explanation: The twentieth term is a₂₀ = 3 + 19(4) = 79. The sum of the first n terms is S = (a₁ + aₙ)·n/2 = (3 + 79) × 20/2 = 82 × 10 = 820.

About the EsPCEx Exam

The EsPCEx Concurso de Admissão is the annual national entrance examination for the Escola Preparatória de Cadetes do Exército in Campinas/SP. It serves as the initial year and exclusive pathway for cadets progressing to the Academia Militar das Agulhas Negras (AMAN) in Resende/RJ to become combat officer commanders (Oficiais Combatentes) in the Brazilian Army. The exam spans two consecutive 4.5-hour days with 100 multiple-choice questions of five alternatives each (Day 1: 20 Portuguese, 12 Physics, 12 Chemistry and the Redação; Day 2: 20 Mathematics, 12 Geography, 12 History, 12 English). Subjects are weighted rather than equally counted: Portuguese and Mathematics carry peso 2, Physics and English peso 1.5, and Chemistry, Geography and History peso 1. The Redação is eliminatory only. 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

100 scored questions

Time Limit

2 days: 4h30m per day (real exam)

Passing Score

Competitive ranking; Redação eliminatory only at 50/100

Exam Fee

R$ 100 (Exército Brasileiro / Departamento de Educação e Cultura do Exército (DECEx))

EsPCEx Exam Content Outline

20%

Matemática

Functions, analytic geometry, trigonometry, complex numbers, and combinatorics

20%

Língua Portuguesa

Textual analysis, advanced syntax, normative grammar, and Brazilian literature

12%

Física

Classical mechanics, thermodynamics, wave optics, and electromagnetism

12%

Química

Stoichiometry, chemical equilibrium, thermochemistry, and organic chemistry

12%

História

Brazilian and General World history from antiquity to the contemporary era

12%

Geografia

Brazilian regional geography, geopolitics, global economy, and environmental systems

12%

Língua Inglesa

Reading comprehension, syntactic structures, verb forms, and contextual vocabulary

How to Pass the EsPCEx Exam

What You Need to Know

  • Passing score: Competitive ranking; Redação eliminatory only at 50/100
  • Exam length: 100 questions
  • Time limit: 2 days: 4h30m per day (real exam)
  • Exam fee: R$ 100

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

EsPCEx Study Tips from Top Performers

1Prioritize Mathematics and Portuguese as they carry the highest individual question counts (20 questions each)
2Practice multi-step calculation problems in stoichiometry, chemical equilibrium, and kinematic mechanics
3Review Brazilian and world history chronologies with attention to geopolitical conflicts and constitutional evolution
4Simulate the complete two-day testing experience using past official EsPCEx exams from 2017 to 2025

Frequently Asked Questions

What is the format and schedule of the EsPCEx exam?

The exam is held over two consecutive days (4 hours and 30 minutes each). Day 1 includes 20 Portuguese, 12 Physics, 12 Chemistry, and an essay. Day 2 includes 20 Mathematics, 12 Geography, 12 History, and 12 English questions.

What is the relationship between EsPCEx and AMAN?

EsPCEx is the 1st year of the 5-year Brazilian Army officer formation curriculum. After completing the year in Campinas/SP, successful cadets automatically advance to the Academia Militar das Agulhas Negras (AMAN) in Resende/RJ for the remaining 4 years.

What are the passing score requirements for EsPCEx?

There is no fixed pass mark per subject. Classification comes from the weighted objective score — Portuguese and Mathematics carry peso 2, Physics and English peso 1.5, and Chemistry, Geography and History peso 1 — and candidates are ranked against the 440 available places. The Redação is purely eliminatory: it is worth 0 to 100 points, requires at least 50 to be judged APTO, must run 25 to 30 lines, and contributes nothing to the ranking. Only candidates above the median objective score have their essays marked at all.

What is the registration fee for EsPCEx?

The registration fee is R$ 100, with fee waivers available for eligible candidates enrolled in CadÚnico or recognized as bone marrow donors.