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100+ Free Escola Naval Practice Questions

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2026 Statistics

Key Facts: Escola Naval Exam

80

Total Questions

CPAEN Edital

50%

Min Subject Cutoff

SSPM / Marinha

2 Days (10h)

Total Duration

DEnsM

R$ 110

Registration Fee

Marinha do Brasil

The Escola Naval (CPAEN) admission exam features 80 questions over two 5-hour test days — 20 Mathematics and 20 English on Day 1, 22 Physics and 18 Portuguese plus an essay on Day 2 — with elimination below 50% in any discipline; this bank provides 100 English-language practice questions across all blueprint subjects.

Sample Escola Naval Practice Questions

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

1Evaluate the indeterminate limit $\lim_{x \to 0} \frac{\sin(3x) - 3x}{x^3}$.
A.-9/2
B.-3/2
C.0
D.9/2
Explanation: Applying L'Hôpital's rule to the 0/0 form: $\lim_{x \to 0} \frac{3\cos(3x) - 3}{3x^2} = \lim_{x \to 0} \frac{\cos(3x) - 1}{x^2} = \lim_{x \to 0} \frac{-3\sin(3x)}{2x} = -\frac{9}{2}$. The Taylor series gives the same answer faster: $\sin(3x) = 3x - \frac{(3x)^3}{6} + O(x^5)$, so the quotient tends to $-27/6 = -9/2$.
2Determine the equation of the line tangent to the graph of $f(x) = x^3 - 3x^2 + 2x$ at the point with abscissa $x = 2$.
A.$y = 2x - 4$
B.$y = -2x + 4$
C.$y = 2x$
D.$y = x - 2$
Explanation: The derivative is $f'(x) = 3x^2 - 6x + 2$, so the slope at $x = 2$ is $f'(2) = 12 - 12 + 2 = 2$. The point of tangency is $f(2) = 8 - 12 + 4 = 0$, so the line is $y - 0 = 2(x - 2)$, that is $y = 2x - 4$.
3Evaluate the definite integral $\int_{0}^{1} x e^{2x} \, dx$.
A.$\frac{e^2 + 1}{4}$
B.$\frac{e^2 - 1}{4}$
C.$\frac{e^2 + 1}{2}$
D.$\frac{3e^2 - 1}{4}$
Explanation: Integrating by parts with $u = x$ and $dv = e^{2x}dx$ gives $\int x e^{2x} dx = \frac{x}{2}e^{2x} - \frac{1}{4}e^{2x}$. Evaluating from 0 to 1: $\left(\frac{1}{2}e^2 - \frac{1}{4}e^2\right) - \left(0 - \frac{1}{4}\right) = \frac{e^2}{4} + \frac{1}{4} = \frac{e^2 + 1}{4}$.
4What is the area of the plane region bounded by the curves $y = 4 - x^2$ and $y = x + 2$?
A.9/2
B.7/2
C.11/2
D.9
Explanation: The curves meet where $4 - x^2 = x + 2$, that is $x^2 + x - 2 = 0$, giving $x = -2$ and $x = 1$. The area is $\int_{-2}^{1} (2 - x - x^2)\,dx = \left[2x - \frac{x^2}{2} - \frac{x^3}{3}\right]_{-2}^{1} = \frac{7}{6} - \left(-\frac{10}{3}\right) = \frac{27}{6} = \frac{9}{2}$.
5What is the distance from the point $P(1, -2, 3)$ to the plane $\pi: 2x - 2y + z - 15 = 0$ in $\mathbb{R}^3$?
A.2
B.3
C.6
D.8
Explanation: The distance from $(x_0, y_0, z_0)$ to $Ax + By + Cz + D = 0$ is $d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$. Substituting gives $d = \frac{|2 + 4 + 3 - 15|}{\sqrt{4 + 4 + 1}} = \frac{6}{3} = 2$.
6Consider the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$. What are its eccentricity and its focal distance, respectively?
A.$3/5$ and $6$
B.$4/5$ and $8$
C.$3/5$ and $3$
D.$9/25$ and $6$
Explanation: Here $a = 5$ and $b = 4$. The fundamental relation $a^2 = b^2 + c^2$ gives $c^2 = 25 - 16 = 9$, so $c = 3$. The eccentricity is $e = c/a = 3/5$ and the focal distance between the two foci is $2c = 6$.
7In the Argand-Gauss plane, the locus of the points $z = x + iy$ satisfying $|z - 3| = 2|z + 3i|$ represents which figure?
A.A circle centred at $(-1, -4)$ with radius $2\sqrt{2}$
B.The perpendicular bisector of the segment joining $(3,0)$ and $(0,-3)$
C.An ellipse with foci at $(3,0)$ and $(0,-3)$
D.A parabola with directrix $y = -3$
Explanation: Squaring both sides gives $(x-3)^2 + y^2 = 4[x^2 + (y+3)^2]$, which simplifies to $3x^2 + 3y^2 + 6x + 24y + 27 = 0$, that is $x^2 + y^2 + 2x + 8y + 9 = 0$. Completing the squares yields $(x+1)^2 + (y+4)^2 = 8 = (2\sqrt{2})^2$ — a circle of Apollonius centred at $(-1, -4)$.
8How many distinct solutions does the trigonometric equation $\cos(2x) = \sin(x)$ have in the interval $[0, 2\pi)$?
A.3
B.2
C.4
D.1
Explanation: Substituting $\cos(2x) = 1 - 2\sin^2(x)$ gives $2\sin^2(x) + \sin(x) - 1 = 0$, which factors as $(2\sin(x) - 1)(\sin(x) + 1) = 0$. From $\sin(x) = 1/2$ come $x = \pi/6$ and $x = 5\pi/6$; from $\sin(x) = -1$ comes $x = 3\pi/2$. That is 3 distinct solutions in $[0, 2\pi)$.
9If $\tan(a) = \frac{1}{2}$ and $\tan(b) = \frac{1}{3}$, with $a$ and $b$ acute, what are $\tan(a + b)$ and the angle $(a+b)$, respectively?
A.$1$ and $\pi/4$
B.$5/6$ and $\pi/6$
C.$\sqrt{3}$ and $\pi/3$
D.$1$ and $\pi/2$
Explanation: The tangent addition formula gives $\tan(a+b) = \frac{1/2 + 1/3}{1 - (1/2)(1/3)} = \frac{5/6}{5/6} = 1$. Since both angles are acute, their sum lies in $(0, \pi)$ and the only angle there with tangent 1 is $\pi/4$.
10The roots of the polynomial $P(x) = x^3 - 9x^2 + 23x - 15$ form an arithmetic progression. What is the largest of those roots?
A.5
B.3
C.1
D.7
Explanation: Write the roots as $r - d$, $r$, $r + d$. Girard's relations give a sum of 9, so $3r = 9$ and $r = 3$. The product is 15, so $3(9 - d^2) = 15$, giving $d^2 = 4$ and $d = 2$. The roots are 1, 3 and 5, and the largest is 5.

About the Escola Naval Exam

The CPAEN is the annual entrance competition for the Escola Naval, Brazil's oldest higher-education institution located on Ilha de Villegaignon in Rio de Janeiro/RJ. The two-day intellectual exam tests candidates for admission as Aspirantes in the Brazilian Navy officer corps: Corpo da Armada (Fleet), Corpo de Fuzileiros Navais (Marine Corps), and Corpo de Intendentes da Marinha (Naval Quartermaster). The exam comprises 80 objective questions with five alternatives each (A–E), across two 5-hour sessions: Day 1 carries 20 Mathematics and 20 English questions, and Day 2 carries 22 Physics and 18 Portuguese questions plus the Redação. Every question is worth 2.5 points, so each day is scored out of 100, and a candidate scoring under 50% in any single discipline is eliminated. Note: the official exam is set and answered in Portuguese; this practice bank is an English-language MCQ study adaptation of the published syllabus, not an official translation or a simulation of the essay.

Questions

80 scored questions

Time Limit

2 days: 5 hours per day (real exam)

Passing Score

At least 50% in every discipline; each day scored out of 100

Exam Fee

R$ 110 (Marinha do Brasil / Serviço de Seleção do Pessoal da Marinha (SSPM))

Escola Naval Exam Content Outline

25%

Matemática

Calculus foundations, analytic geometry, trigonometry, complex numbers, and linear algebra (20 questions, Day 1)

27.5%

Física

Mechanics, fluids, thermodynamics, wave optics, electrodynamics, and electromagnetism (22 questions, Day 2)

22.5%

Língua Portuguesa

Textual analysis, advanced syntax, concordance, verbal regency, and normative grammar (18 questions, Day 2)

25%

Língua Inglesa

Complex text interpretation, advanced grammar, passive voice, and clause structures (20 questions, Day 1)

How to Pass the Escola Naval Exam

What You Need to Know

  • Passing score: At least 50% in every discipline; each day scored out of 100
  • Exam length: 80 questions
  • Time limit: 2 days: 5 hours per day (real exam)
  • Exam fee: R$ 110

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

Escola Naval Study Tips from Top Performers

1Master advanced mathematics topics including functions, analytic geometry, and introductory calculus concepts
2Solve in-depth physics problems focusing on rotational dynamics, hydrodynamics, and electromagnetic induction
3Practice normative Portuguese sentence structure, especially subordinate clauses and regency
4Simulate full 5-hour testing blocks using official past CPAEN exams from 2015 to 2025

Frequently Asked Questions

What is the schedule and structure of the CPAEN exam?

The CPAEN exam is conducted across two consecutive days of 5 hours each. In the CPAEN/2026 edition Day 1 carries 20 Mathematics and 20 English questions — a change from the earlier 22/18 split — while Day 2 keeps 22 Physics and 18 Portuguese questions plus the Redação. Each question is worth 2.5 points, so both days total 100 points.

What are the passing cutoffs for Escola Naval?

A candidate scoring under 50% in any individual discipline is eliminated, and the Redação must also reach the minimum performance set in the edital. Surviving candidates are then ranked competitively against the published number of vacancies — 49 places in CPAEN/2026, contested by roughly 5,700 applicants.

What career streams does Escola Naval offer?

Aspirantes graduate as Guarda-Marinha officers in one of three corps: Corpo da Armada (Surface Fleet), Corpo de Fuzileiros Navais (Marine Corps), or Corpo de Intendentes da Marinha (Logistics & Administration).

What is the registration fee for Escola Naval?

The registration fee is R$ 110, with fee waivers available for eligible candidates enrolled in CadÚnico or registered bone marrow donors.