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Military Academy Belgrade Entrance Exam study materials
Military Academy Belgrade Entrance Exam study resources

Military Academy Belgrade Entrance Exam practice questions

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Question 1Mathematics

What is the value of 3² + 2³?

  1. 13
  2. 36
  3. 17 Correct answer.
  4. 25

Explanation

Evaluate each power first: 3² = 9 and 2³ = 8. Adding gives 9 + 8 = 17, since exponentiation is applied before addition.

Question 2Mathematics

Solve the equation 5x − 7 = 3x + 9.

  1. x = 2
  2. x = 16
  3. x = 1
  4. x = 8 Correct answer.

Explanation

Subtract 3x from both sides to get 2x − 7 = 9, then add 7 to obtain 2x = 16. Dividing both sides by 2 gives x = 8.

Question 3Mathematics

A triangle has sides of lengths 6 cm and 8 cm with a right angle between them. What is the length of the third side?

  1. 10 cm Correct answer.
  2. 12 cm
  3. 2 cm
  4. 14 cm

Explanation

The two given sides are the legs of a right triangle, so the third side is the hypotenuse. By the Pythagorean theorem, c² = 6² + 8² = 36 + 64 = 100, hence c = 10 cm.

Question 4Mathematics

What is the value of log₂ 32?

  1. 4
  2. 6
  3. 5 Correct answer.
  4. 16

Explanation

A logarithm answers the question of what exponent of the base gives the argument. Since 2⁵ = 32, log₂ 32 = 5.

Question 5Mathematics

Which of the following numbers is irrational?

  1. √2 Correct answer.
  2. 0.75
  3. −3
  4. 22/7

Explanation

An irrational number cannot be written as a ratio of two integers. The square root of 2 is not a perfect square, so √2 cannot be expressed as a fraction and is irrational.

Question 6Mathematics

The sum of the interior angles of a convex polygon is 900°. How many sides does the polygon have?

  1. 7 Correct answer.
  2. 6
  3. 9
  4. 8

Explanation

The interior angle sum of an n-gon is (n − 2)·180°. Setting (n − 2)·180 = 900 gives n − 2 = 5, so n = 7.

Question 7Mathematics

If f(x) = 2x² − 3x + 1, what is f(−2)?

  1. 3
  2. −1
  3. 15 Correct answer.
  4. 11

Explanation

Substitute x = −2 carefully: 2(−2)² = 2·4 = 8, then −3(−2) = +6, then +1. Summing gives 8 + 6 + 1 = 15.

Question 8Mathematics

What is the slope of the line passing through the points (2, 3) and (6, 11)?

  1. 8
  2. 2 Correct answer.
  3. 4
  4. 1/2

Explanation

Slope equals the change in y divided by the change in x. Here (11 − 3)/(6 − 2) = 8/4 = 2.

Question 9Mathematics

What is the value of sin 30°?

  1. 1/2 Correct answer.
  2. 1
  3. √2/2
  4. √3/2

Explanation

For the standard acute angles, sin 30° = 1/2, coming from the half-equilateral triangle where the side opposite 30° is half the hypotenuse.

Question 10Mathematics

What is the discriminant of the quadratic equation 2x² − 4x + 5 = 0, and what does it tell us about the roots?

  1. −24, so the roots are complex conjugates Correct answer.
  2. 24, so the roots are two distinct real numbers
  3. −24, so the roots are two distinct real numbers
  4. 16, so there is a double real root

Explanation

The discriminant is b² − 4ac = (−4)² − 4·2·5 = 16 − 40 = −24. A negative discriminant means the quadratic has no real roots, so its two roots form a complex conjugate pair.

Question 11Mathematics

In an arithmetic sequence the first term is 7 and the common difference is 4. What is the 15th term?

  1. 60
  2. 63 Correct answer.
  3. 67
  4. 59

Explanation

The nth term of an arithmetic sequence is a₁ + (n − 1)d. Here 7 + 14·4 = 7 + 56 = 63.

Question 12Mathematics

How many different ways can 3 students be chosen from a group of 8 to form a committee?

  1. 336
  2. 56 Correct answer.
  3. 24
  4. 120

Explanation

Order does not matter in a committee, so we use combinations: C(8,3) = 8·7·6 / (3·2·1) = 56.

Question 13Mathematics

A fair six-sided die is rolled once. What is the probability of rolling a number greater than 4?

  1. 1/3 Correct answer.
  2. 1/2
  3. 2/3
  4. 1/6

Explanation

The outcomes greater than 4 are 5 and 6, giving 2 favorable outcomes out of 6 equally likely ones. The probability is 2/6 = 1/3.

Question 14Mathematics

Simplify the expression (x³·x⁵)/x² for x ≠ 0.

  1. x¹⁰
  2. x⁸
  3. x⁴
  4. x⁶ Correct answer.

Explanation

When multiplying powers of the same base, add the exponents: x³·x⁵ = x⁸. When dividing, subtract the exponents: x⁸/x² = x⁶.

Question 15Mathematics

The line 2x + 3y = 12 intersects the x-axis at which point?

  1. (0, 4)
  2. (0, 6)
  3. (4, 0)
  4. (6, 0) Correct answer.

Explanation

A point on the x-axis has y = 0. Substituting gives 2x = 12, so x = 6, and the intercept is (6, 0).

Question 16Mathematics

The roots of a quadratic equation are 3 and −5. Which equation has exactly these roots?

  1. x² − 2x − 15 = 0
  2. x² + 2x − 15 = 0 Correct answer.
  3. x² + 2x + 15 = 0
  4. x² − 8x + 15 = 0

Explanation

If the roots are r and s, then the sum is −b/a and the product is c/a. Here the sum is −2, so the x-coefficient is +2, and the product is −15, so the constant is −15, giving x² + 2x − 15 = 0.

Question 17Mathematics

What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + …?

  1. 2/3
  2. 3
  3. 3/2 Correct answer.
  4. 1

Explanation

For an infinite geometric series with first term a and ratio r where |r| < 1, the sum is a/(1 − r). Here a = 1 and r = 1/3, giving 1/(1 − 1/3) = 1/(2/3) = 3/2.

Question 18Mathematics

If sin θ = 3/5 and θ is an acute angle, what is cos θ?

  1. 5/3
  2. 3/4
  3. 5/4
  4. 4/5 Correct answer.

Explanation

Using the identity sin²θ + cos²θ = 1, we get cos²θ = 1 − 9/25 = 16/25. Since θ is acute, cos θ is positive, so cos θ = 4/5.

Question 19Mathematics

What is the domain of the function f(x) = √(x − 3)?

  1. x ≤ 3
  2. all real numbers
  3. x ≥ 3 Correct answer.
  4. x > 3

Explanation

The expression under an even root must be non-negative, so x − 3 ≥ 0. That gives x ≥ 3, including x = 3 where the root is zero.

Question 20Mathematics

A circle has radius 5. What is the length of an arc subtending a central angle of 72°?

  1. π/2
  2. 10π
  3. 2π Correct answer.
  4. 5π

Explanation

Arc length equals rθ with θ in radians. Converting 72° gives 72·π/180 = 2π/5 radians, so the arc is 5·(2π/5) = 2π.

Question 21Mathematics

Two similar triangles have corresponding sides in the ratio 3 : 5. What is the ratio of their areas?

  1. 6 : 10
  2. 27 : 125
  3. 3 : 5
  4. 9 : 25 Correct answer.

Explanation

Areas of similar figures scale as the square of the linear ratio. Squaring 3 : 5 gives 9 : 25.

Question 22Mathematics

For what values of k does the equation x² + kx + 9 = 0 have exactly one real solution?

  1. k = 6 only
  2. k = 6 or k = −6 Correct answer.
  3. k = 3 or k = −3
  4. k = 9 or k = −9

Explanation

A quadratic has a double root exactly when its discriminant is zero. Here b² − 4ac = k² − 36 = 0, so k² = 36 and k = ±6.

Question 23Mathematics

Simplify the expression (1 + tan²θ)·cos²θ for angles where it is defined.

  1. sin²θ
  2. cos²θ
  3. 1 Correct answer.
  4. sec²θ

Explanation

The identity 1 + tan²θ = sec²θ holds for all defined θ. Multiplying by cos²θ gives sec²θ·cos²θ = 1, since sec θ = 1/cos θ.

Question 24Mathematics

A geometric sequence has second term 6 and fifth term 48. What is the common ratio?

  1. 8
  2. 2 Correct answer.
  3. 3
  4. 4

Explanation

In a geometric sequence aₙ = a₁rⁿ⁻¹, so a₅/a₂ = r³. Here 48/6 = 8 = r³, giving r = 2.

Question 25Mathematics

How many distinct arrangements of the letters in the word BANANA are there?

  1. 120
  2. 60 Correct answer.
  3. 720
  4. 20

Explanation

There are 6 letters with repeated letters: A appears 3 times and N appears 2 times. Dividing the total permutations by the factorials of the repeats gives 6!/(3!·2!) = 720/12 = 60.

Question 26Mathematics

What is the value of cos 60°?

  1. √2/2
  2. 1/2 Correct answer.
  3. 1
  4. √3/2

Explanation

For the standard angle 60°, the cosine equals 1/2, as seen in the 30-60-90 triangle where the side adjacent to 60° is half the hypotenuse.

Question 27Mathematics

If A = {1, 2, 3} and B = {3, 4}, what is A ∩ B?

  1. {1, 2, 3, 4}
  2. {1, 2}
  3. {3} Correct answer.
  4. {4}

Explanation

The intersection A ∩ B contains exactly the elements that belong to both sets. The only element present in both A and B is 3.

Question 28Mathematics

What is the value of the determinant of the matrix [[2, 5], [1, 3]]?

  1. 1 Correct answer.
  2. −1
  3. 11
  4. 6

Explanation

For a 2×2 matrix [[a, b], [c, d]] the determinant equals ad − bc. Here that is 2·3 − 5·1 = 6 − 5 = 1.

Question 29Mathematics

What is the modulus of the complex number z = 3 − 4i?

  1. 1
  2. 7
  3. 5 Correct answer.
  4. 25

Explanation

The modulus of a + bi is √(a² + b²). Here √(3² + (−4)²) = √(9 + 16) = √25 = 5.

Question 30Mathematics

Which of the following is the derivative of f(x) = x³ − 4x with respect to x?

  1. 3x² − 4 Correct answer.
  2. x² − 4
  3. 3x³ − 4
  4. 3x² − 4x

Explanation

Differentiate term by term using the power rule: the derivative of x³ is 3x², and the derivative of −4x is −4. The constant term would vanish, but here none is present.

Question 31Mathematics

What is the value of the integral ∫₀¹ 2x dx?

  1. 1 Correct answer.
  2. 0
  3. 2
  4. 1/2

Explanation

An antiderivative of 2x is x². Evaluating from 0 to 1 gives 1² − 0² = 1.

Question 32Mathematics

If log₃ x = 2, what is x?

  1. 9 Correct answer.
  2. 6
  3. 8
  4. 5

Explanation

By definition of the logarithm, log₃ x = 2 means 3² = x. Therefore x = 9.

Question 33Mathematics

For the function f(x) = x² − 6x + 5, what is the x-coordinate of the vertex of its graph?

  1. 3 Correct answer.
  2. 6
  3. −6
  4. −3

Explanation

For a parabola y = ax² + bx + c, the vertex x-coordinate is −b/(2a). Here −(−6)/(2·1) = 3.

Question 34Mathematics

What is the value of sin 2θ if sin θ = 4/5 and θ is acute?

  1. 7/25
  2. 12/25
  3. 8/5
  4. 24/25 Correct answer.

Explanation

Since θ is acute with sin θ = 4/5, we get cos θ = 3/5 from the Pythagorean identity. Then sin 2θ = 2 sin θ cos θ = 2·(4/5)·(3/5) = 24/25.

Question 35Mathematics

How many 4-digit numbers can be formed from the digits 1, 2, 3, 4, 5 if repetition of digits is not allowed?

  1. 120 Correct answer.
  2. 24
  3. 625
  4. 20

Explanation

We are arranging 4 distinct digits chosen from 5, which gives 5·4·3·2 = 120 ordered arrangements.

Question 36Mathematics

The vectors a = (2, −1) and b = (k, 3) are perpendicular. What is k?

  1. 3/2 Correct answer.
  2. −6
  3. 6
  4. −3/2

Explanation

Two vectors are perpendicular exactly when their dot product is zero. Setting 2k + (−1)(3) = 0 gives 2k = 3, so k = 3/2.

Question 37Mathematics

What is the sum of the first 20 positive odd integers 1 + 3 + 5 + … + 39?

  1. 210
  2. 400 Correct answer.
  3. 420
  4. 380

Explanation

The sum of the first n odd integers equals n². With n = 20, the sum is 20² = 400.

Question 38Mathematics

The polynomial P(x) = x³ − 2x² − 5x + 6 has x = 1 as a root. What are the other two roots?

  1. −2 and −3
  2. 2 and −3 Correct answer.
  3. 2 and 3
  4. −2 and 3

Explanation

Dividing P(x) by (x − 1) gives x² − x − 6, which factors as (x − 3)(x + 2). The remaining roots are therefore 3 and −2.

Question 39Mathematics

What is the equation of the circle with center (2, −3) and radius 4?

  1. (x − 2)² + (y − 3)² = 16
  2. (x − 2)² + (y + 3)² = 16 Correct answer.
  3. (x − 2)² + (y + 3)² = 4
  4. (x + 2)² + (y − 3)² = 16

Explanation

A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r². Substituting h = 2, k = −3, r = 4 gives (x − 2)² + (y + 3)² = 16.

Question 40Mathematics

A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

  1. 4/25
  2. 1/5
  3. 2/15 Correct answer.
  4. 8/45

Explanation

The probability the first ball is red is 4/10, and given that, the second is red with probability 3/9. Multiplying gives (4/10)(3/9) = 12/90 = 2/15.

Question 41Mathematics

What is the value of the limit lim(x→2) (x² − 4)/(x − 2)?

  1. 4 Correct answer.
  2. 2
  3. 0
  4. does not exist

Explanation

Factor the numerator as (x − 2)(x + 2) and cancel the common factor (x − 2), valid for x ≠ 2. The expression becomes x + 2, whose limit as x → 2 is 4.

Question 42Mathematics

Which of the following is the inverse of the function f(x) = 2x + 3?

  1. f⁻¹(x) = 2x − 3
  2. f⁻¹(x) = (x + 3)/2
  3. f⁻¹(x) = (x − 3)/2 Correct answer.
  4. f⁻¹(x) = x/2 − 3

Explanation

Set y = 2x + 3 and solve for x: x = (y − 3)/2. Interchanging x and y gives f⁻¹(x) = (x − 3)/2.

Question 43Mathematics

What is the value of the expression C(6, 2) + C(6, 3)?

  1. 35 Correct answer.
  2. 15
  3. 20
  4. 30

Explanation

C(6, 2) = 15 and C(6, 3) = 20, so the sum is 35. This also equals C(7, 3) by Pascal's identity.

Question 44Mathematics

What is the sum of the roots of the equation 3x² − 12x + 7 = 0?

  1. 7/3
  2. 12
  3. 4 Correct answer.
  4. −4

Explanation

By Vieta's formulas, the sum of the roots of ax² + bx + c = 0 is −b/a. Here −(−12)/3 = 4.

Question 45Mathematics

If z = 1 + i, what is z²?

  1. −2i
  2. 1 + 2i
  3. 2i Correct answer.
  4. 2

Explanation

Expand (1 + i)² = 1 + 2i + i². Since i² = −1, this simplifies to 1 + 2i − 1 = 2i.

Question 46Mathematics

A function f satisfies f'(x) = 3x² and f(0) = 2. What is f(1)?

  1. 2
  2. 1
  3. 5
  4. 3 Correct answer.

Explanation

Integrating f'(x) = 3x² gives f(x) = x³ + C. The condition f(0) = 2 forces C = 2, so f(x) = x³ + 2 and f(1) = 3.

Question 47Mathematics

What is the coefficient of x³ in the expansion of (2 + x)⁵?

  1. 32
  2. 80
  3. 10
  4. 40 Correct answer.

Explanation

The term containing x³ is C(5, 3)·2²·x³ = 10·4·x³ = 40x³, so the coefficient is 40.

Question 48Mathematics

What is the value of the integral ∫₁ᵉ (1/x) dx?

  1. 1/e
  2. e
  3. 1 Correct answer.
  4. e − 1

Explanation

An antiderivative of 1/x is ln x. Evaluating from 1 to e gives ln e − ln 1 = 1 − 0 = 1.

Question 49Mathematics

A sequence satisfies a₁ = 1 and aₙ₊₁ = 2aₙ + 1. What is a₄?

  1. 13
  2. 31
  3. 7
  4. 15 Correct answer.

Explanation

Apply the recurrence step by step: a₂ = 2·1 + 1 = 3, a₃ = 2·3 + 1 = 7, a₄ = 2·7 + 1 = 15.

Question 50Mathematics

For which value of m does the system 2x + 3y = 5, 4x + my = 10 have infinitely many solutions?

  1. 5
  2. 6 Correct answer.
  3. no value of m
  4. 3

Explanation

Infinitely many solutions occur when the two equations are proportional. The second equation is exactly twice the first when m = 2·3 = 6 and the constant 10 = 2·5, giving a dependent system.