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100+ Free IME Practice Questions

Prepare for the Concurso de Admissão ao Curso de Formação e Graduação (CFG/IME) exam with instant access — no signup required.

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2026 Statistics

Key Facts: IME Exam

40

1st Phase MCQs

IME Edital

6 / 6 / 4

Min Correct Math/Phys/Chem

DECEx / IME

2 Phases

Exam Format

IME Rio de Janeiro

R$ 140

Registration Fee

Exército Brasileiro

The IME entrance exam opens with a 5-hour, 40-question 1st Phase split 15 Mathematics, 15 Physics and 10 Chemistry, requiring 20 correct overall plus 6/6/4 per subject, before four days of discursive papers; this bank provides 100 English-language practice questions across all blueprint engineering science topics.

Sample IME Practice Questions

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

1Let $z \in \mathbb{C}$ be a root of $z^5 + z^4 + z^3 + z^2 + z + 1 = 0$ with $\operatorname{Im}(z) > 0$. What is the value of $\operatorname{Re}(z) + \operatorname{Re}(z^2) + \operatorname{Re}(z^3) + \operatorname{Re}(z^4) + \operatorname{Re}(z^5)$?
A.-1
B.0
C.1/2
D.-1/2
Explanation: The left-hand side is the geometric sum $(z^6 - 1)/(z - 1)$, so the roots are the sixth roots of unity other than 1. The equation itself states that $z + z^2 + z^3 + z^4 + z^5 = -1$, and since the real part of a sum is the sum of the real parts, the required value is $\operatorname{Re}(-1) = -1$.
2Consider the polynomial $P(x) = x^4 - 6x^3 + 13x^2 - 12x + 4$, which factors into two distinct real roots each of multiplicity 2. What is the sum of the distinct roots of $P(x)$?
A.3
B.4
C.5
D.6
Explanation: The polynomial is a perfect square: $P(x) = (x^2 - 3x + 2)^2 = (x-1)^2(x-2)^2$. Its distinct roots are 1 and 2, so their sum is 3. Note that Girard's relation gives 6 for the sum counted with multiplicity, which is a different quantity.
3What is the determinant of the Vandermonde matrix $V = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 4 & 9 & 16 \\ 1 & 8 & 27 & 64 \end{pmatrix}$?
A.12
B.24
C.6
D.48
Explanation: A Vandermonde determinant with nodes $x_1, \dots, x_n$ equals $\prod_{i<j}(x_j - x_i)$. With nodes 1, 2, 3 and 4 this is $(2-1)(3-1)(4-1)(3-2)(4-2)(4-3) = 1 \cdot 2 \cdot 3 \cdot 1 \cdot 2 \cdot 1 = 12$.
4Let $\alpha, \beta, \gamma$ be the roots of $x^3 - 3x^2 + 4x - 5 = 0$. What is the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2}$?
A.-14/25
B.14/25
C.-6/25
D.6/25
Explanation: Girard's relations give $\sum\alpha = 3$, $\sum\alpha\beta = 4$ and $\alpha\beta\gamma = 5$. Then $\sum 1/\alpha = 4/5$ and $\sum 1/(\alpha\beta) = 3/5$, so $\sum 1/\alpha^2 = (4/5)^2 - 2(3/5) = 16/25 - 30/25 = -14/25$. A negative value is possible because two of the roots are complex conjugates.
5What is the eccentricity of the ellipse $9x^2 + 25y^2 - 18x + 50y - 191 = 0$?
A.4/5
B.3/5
C.2/5
D.$\sqrt{5}/3$
Explanation: Completing the squares gives $9(x-1)^2 + 25(y+1)^2 = 225$, that is $\frac{(x-1)^2}{25} + \frac{(y+1)^2}{9} = 1$. So $a = 5$, $b = 3$ and $c = 4$, giving $e = c/a = 4/5$.
6What is the distance from the point $P(1, 2, -1)$ to the plane $\pi: 2x - 2y + z - 5 = 0$ in $\mathbb{R}^3$?
A.8/3
B.2
C.7/3
D.3
Explanation: The distance formula gives $d = \frac{|2(1) - 2(2) + (-1) - 5|}{\sqrt{4 + 4 + 1}} = \frac{|-8|}{3} = \frac{8}{3}$.
7What is the value of the trigonometric product $P = \cos\left(\frac{\pi}{7}\right)\cos\left(\frac{2\pi}{7}\right)\cos\left(\frac{3\pi}{7}\right)$?
A.1/8
B.1/4
C.1/16
D.$\sqrt{7}/8$
Explanation: Multiplying and dividing by $8\sin(\pi/7)$ and applying $\sin(2\theta) = 2\sin\theta\cos\theta$ three times telescopes the numerator to $\sin(8\pi/7) = -\sin(\pi/7)$. Careful sign tracking through the collapse leaves $P = 1/8$.
8How many derangements (chaotic permutations) does a set of 5 distinct elements have?
A.44
B.120
C.53
D.24
Explanation: By inclusion-exclusion, $D_n = n!\sum_{k=0}^{n}\frac{(-1)^k}{k!}$, so $D_5 = 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120) = 44$. The recurrence $D_n = n\,D_{n-1} + (-1)^n$ gives the same value from $D_4 = 9$.
9The sequence $(a_n)$ is defined by $a_1 = 1$, $a_2 = 3$ and $a_{n+2} = 3a_{n+1} - 2a_n$ for every $n \ge 1$. What is the general term $a_n$?
A.$2^n - 1$
B.$2^{n-1} + 1$
C.$3^{n-1}$
D.$2^{n+1} - 3$
Explanation: The characteristic equation $r^2 - 3r + 2 = 0$ has roots 1 and 2, so $a_n = A + B\cdot 2^n$. The initial conditions give $A + 2B = 1$ and $A + 4B = 3$, hence $B = 1$ and $A = -1$, so $a_n = 2^n - 1$.
10A convex polyhedron has only triangular and hexagonal faces, has 32 vertices, and exactly 3 edges meet at each vertex. How many hexagonal faces does it have?
A.14
B.20
C.18
D.16
Explanation: With $t$ triangles and $h$ hexagons, counting edge-ends gives $E = (3t + 6h)/2$ and counting vertex-ends with degree 3 gives $V = (3t + 6h)/3$. Euler's relation $V - E + F = 2$ then forces $t = 4$ for any such solid. From $V = t + 2h = 32$ we get $h = 14$, and the checks confirm $E = 48$ and $F = 18$, since $32 - 48 + 18 = 2$.

About the IME Exam

The IME Concurso de Admissão is the elite admission competition for the Instituto Militar de Engenharia in Rio de Janeiro/RJ, Brazil's premier military engineering institution. The examination admits candidates to the Curso de Formação e Graduação (CFG) for both active-duty military engineering careers (Ativa) and reserve civilian tracks (Reserva). The selection features a demanding two-phase structure: the 1st Phase comprises 40 objective STEM questions (15 Mathematics, 15 Physics, 10 Chemistry) in a 5-hour test, while the 2nd Phase spreads discursive Mathematics, Physics and Chemistry papers plus a combined Portuguese/English day over four consecutive 4-hour sittings. Note: the official exam is set and answered in Portuguese and its decisive second phase is discursive; this bank is an English-language MCQ study adaptation of the published syllabus, not an official translation or a simulation of the discursive papers.

Questions

40 scored questions

Time Limit

5 hours (1st Phase objective exam); 4 hours per 2nd Phase day

Passing Score

1st Phase >= 5.00/10.00 with 6/6/4 minimums in Math/Physics/Chemistry

Exam Fee

R$ 140 (Exército Brasileiro / Departamento de Educação e Cultura do Exército (DECEx))

IME Exam Content Outline

35%

Matemática

Polynomials, complex numbers, 3D analytic geometry, trigonometry, and combinatorics

35%

Física

Rotational mechanics, fluids, thermodynamics, wave optics, and electromagnetism

20%

Química

Thermodynamics, chemical equilibrium, electrochemistry, and organic reactions

10%

Língua Portuguesa e Inglesa

Advanced syntax, textual analysis, and technical English comprehension

How to Pass the IME Exam

What You Need to Know

  • Passing score: 1st Phase >= 5.00/10.00 with 6/6/4 minimums in Math/Physics/Chemistry
  • Exam length: 40 questions
  • Time limit: 5 hours (1st Phase objective exam); 4 hours per 2nd Phase day
  • Exam fee: R$ 140

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

IME Study Tips from Top Performers

1Practice deep analytical problem-solving involving complex numbers, polynomial roots, and 3D spatial geometry
2Derive physics formulas from first principles, focusing on rigid body dynamics, thermodynamics cycles, and Maxwell's equations
3Master quantitative chemical equilibrium calculations, Nernst equation problems, and organic mechanisms
4Solve past official IME 1st Phase and 2nd Phase papers from 2010 to 2025 to develop advanced stamina

Frequently Asked Questions

What is the structure of the IME admission examination?

The IME selection process is divided into two academic phases. Article 38 of the CFG/Ativa edital sets the 1st Phase as a 5-hour eliminatory objective paper of exactly 40 questions — 15 Mathematics, 15 Physics and 10 Chemistry — all worth the same number of points. The 2nd Phase runs over four consecutive days of 4 hours each: discursive Mathematics, then Physics, then Chemistry, then a combined Portuguese (objective questions plus Redação) and English day.

What engineering specializations are offered at IME?

IME offers undergraduate engineering degrees in Fortification and Construction (Civil), Electrical, Electronics, Communications, Mechanical, Armaments, Chemical, Materials, Cartographic, and Computer Engineering.

What are the passing criteria for IME?

Article 40 of the edital eliminates any candidate whose 1st Phase score falls below 5.00 out of 10.00 — fewer than 20 correct answers out of 40 — or who gets fewer than 6 right in Mathematics, fewer than 6 in Physics, or fewer than 4 in Chemistry. In the 2nd Phase a mark below 4.00 in any of the five papers, or an INAPTO essay, also eliminates the candidate; the 2nd Phase weights are Mathematics 3.0, Physics 2.5, Chemistry 2.5, Portuguese 1.0 and English 1.0.

What is the registration fee for IME?

The registration fee is R$ 140, with fee waivers available for eligible candidates registered in CadÚnico or recognized as bone marrow donors.