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Angola Air Force Academy Entrance Exam study materials
Angola Air Force Academy Entrance Exam study resources

Angola Air Force Academy Entrance Exam practice questions

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Question 1Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Solve for x: 3x - 7 = 2x + 5.

  1. 12 Correct answer.
  2. 6
  3. -12
  4. 2

Explanation

Collect the x-terms on one side and the constants on the other: 3x - 2x = 5 + 7, which gives x = 12. Substituting back, the left side is 3(12) - 7 = 29 and the right side is 2(12) + 5 = 29, so the solution checks out.

Question 2Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What are the solutions of x² - 5x + 6 = 0?

  1. -2 and -3
  2. 1 and 6
  3. 5 and 6
  4. 2 and 3 Correct answer.

Explanation

Factor the quadratic by finding two numbers whose product is the constant term 6 and whose sum is 5, namely 2 and 3. This gives (x - 2)(x - 3) = 0, so x = 2 or x = 3. Both values satisfy the original equation.

Question 3Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the exact value of sin 30°?

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

Explanation

In a 30°-60°-90° triangle the side opposite the 30° angle is half the hypotenuse, so sin 30° = 1/2. This value is one of the standard exact trigonometric values worth memorising.

Question 4Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

The angles of a triangle are in the ratio 1 : 2 : 3. What is the size of the largest angle?

  1. 45°
  2. 90° Correct answer.
  3. 30°
  4. 60°

Explanation

Let the angles be x, 2x and 3x. Since the angles of a triangle sum to 180°, 6x = 180°, so x = 30° and the largest angle is 3x = 90°.

Question 5Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Find the mean of the data set 4, 7, 9, 10, 15.

  1. 10
  2. 8
  3. 11
  4. 9 Correct answer.

Explanation

The mean equals the sum of the values divided by how many values there are. Here the sum is 4 + 7 + 9 + 10 + 15 = 45 and there are 5 values, so the mean is 45 ÷ 5 = 9.

Question 6Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the distance between the points (1, 2) and (4, 6) in the coordinate plane?

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

Explanation

Apply the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²). The differences are 3 and 4, so d = √(9 + 16) = √25 = 5.

Question 7Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A fair six-sided die is rolled once. What is the probability of obtaining an even number?

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

Explanation

There are 6 equally likely outcomes, and the even results are 2, 4 and 6, giving 3 favourable outcomes. The probability is therefore 3/6 = 1/2.

Question 8Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Given f(x) = 2x² - 3, what is f(-2)?

  1. 5 Correct answer.
  2. 11
  3. 1
  4. -11

Explanation

Substitute x = -2, remembering to square before multiplying: (-2)² = 4, then 2(4) - 3 = 5.

Question 9Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

For which values of x is the function f(x) = 1/√(5 - x) defined as a real number?

  1. x ≤ 5
  2. x < 5 Correct answer.
  3. x > 5
  4. x ≠ 5

Explanation

The expression under the square root must be non-negative, so 5 - x ≥ 0, and because that expression sits in a denominator it cannot equal zero, so 5 - x > 0. Together these give x < 5.

Question 10Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Evaluate log₂48 - log₂3.

  1. 45
  2. 16
  3. 2
  4. 4 Correct answer.

Explanation

Use the quotient rule for logarithms: log₂48 - log₂3 = log₂(48/3) = log₂16. Since 16 = 2⁴, the value is 4.

Question 11Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

An arithmetic progression has first term 7 and common difference 4. What is its 15th term?

  1. 59
  2. 63 Correct answer.
  3. 56
  4. 67

Explanation

The nth term of an arithmetic progression is a + (n - 1)d. With a = 7, d = 4 and n = 15, the term is 7 + 14 × 4 = 7 + 56 = 63.

Question 12Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

If x + y = 10 and x - y = 4, what is the value of xy?

  1. 25
  2. 16
  3. 24
  4. 21 Correct answer.

Explanation

Adding the two equations eliminates y: 2x = 14, so x = 7, and then y = 3. Their product is 7 × 3 = 21.

Question 13Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Solve for x: 2^(x + 1) = 32.

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

Explanation

Write 32 as a power of 2: 32 = 2⁵. Since the bases are equal, the exponents must be equal, so x + 1 = 5 and x = 4.

Question 14Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

For which positive value of k does the equation x² + kx + 9 = 0 have two equal real roots?

  1. 9
  2. 36
  3. 6 Correct answer.
  4. 3

Explanation

A quadratic has equal roots when its discriminant b² - 4ac is zero. Here k² - 4(1)(9) = 0, so k² = 36 and k = ±6; the positive value is 6.

Question 15Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

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

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

Explanation

Use the identity sin²θ + cos²θ = 1. Then cos²θ = 1 - 9/25 = 16/25, and since θ is acute, cos θ is positive, so cos θ = 4/5. This corresponds to the 3-4-5 right triangle.

Question 16Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A circle has circumference 12π cm. What is its area in square centimetres?

  1. 144π
  2. 6π
  3. 36π Correct answer.
  4. 12π

Explanation

From C = 2πr, the radius is 12π ÷ (2π) = 6 cm. The area is then πr² = π × 36 = 36π cm².

Question 17Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A right-angled triangle has legs of length 9 cm and 12 cm. How long is the hypotenuse?

  1. 13 cm
  2. 225 cm
  3. 15 cm Correct answer.
  4. 21 cm

Explanation

By Pythagoras' theorem the hypotenuse satisfies h² = 9² + 12² = 81 + 144 = 225. Taking the positive square root gives h = 15 cm.

Question 18Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the median of the data set 3, 5, 7, 7, 9, 12, 15?

  1. 9
  2. 7 Correct answer.
  3. 8
  4. 7.5

Explanation

The data are already in ascending order and there are 7 values, an odd count, so the median is the 4th value. Counting to the middle position gives 7.

Question 19Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the remainder when x³ - 2x + 4 is divided by x - 1?

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

Explanation

By the remainder theorem, the remainder is the value of the polynomial at x = 1. Substituting gives 1³ - 2(1) + 4 = 1 - 2 + 4 = 3.

Question 20Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Solve the inequality 2x - 5 < 3x + 1.

  1. x < 4
  2. x < -6
  3. x > -6 Correct answer.
  4. x > 6

Explanation

Subtract 2x from both sides to get -5 < x + 1, then subtract 1 to obtain -6 < x, which is the same as x > -6. No sign flip is needed because the inequality was not multiplied or divided by a negative number.

Question 21Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Given f(x) = 2x + 1 and g(x) = x² - 3, which expression equals g(f(x))?

  1. 4x² - 2
  2. 4x² + 4x - 2 Correct answer.
  3. 2x² - 5
  4. 4x² + 4x + 4

Explanation

The composite g(f(x)) means substituting f(x) into g, so replace every x in x² - 3 with (2x + 1). Expanding gives (2x + 1)² - 3 = 4x² + 4x + 1 - 3 = 4x² + 4x - 2. Note that the order matters, since f(g(x)) would give a different result.

Question 22Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

How many solutions does 2sin²x - sin x - 1 = 0 have in the interval 0° ≤ x < 360°?

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

Explanation

Treat the equation as a quadratic in sin x and factor: (2sin x + 1)(sin x - 1) = 0, giving sin x = -1/2 or sin x = 1. In the given interval sin x = 1 occurs only at 90°, while sin x = -1/2 occurs at 210° and 330°. That makes three solutions in total.

Question 23Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Solve log₃x + log₃(x - 2) = 1 for x.

  1. 1
  2. 9
  3. -1
  4. 3 Correct answer.

Explanation

Combine the logarithms using the product rule: log₃[x(x - 2)] = 1, so x(x - 2) = 3¹ = 3. This gives x² - 2x - 3 = 0, that is (x - 3)(x + 1) = 0, yielding x = 3 or x = -1; the negative root is rejected because logarithms require positive arguments.

Question 24Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

In a geometric sequence the 3rd term is 12 and the 6th term is 96. What is the sum of the first five terms?

  1. 45
  2. 189
  3. 93 Correct answer.
  4. 96

Explanation

Dividing the 6th term by the 3rd gives r³ = 96/12 = 8, so r = 2, and then the first term satisfies a·2² = 12, giving a = 3. The sum of the first five terms is a(r⁵ - 1)/(r - 1) = 3(32 - 1)/1 = 93.

Question 25Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A sphere of radius 6 cm fits exactly inside a cylinder of radius 6 cm and height 12 cm, touching both bases. What is the ratio of the cylinder's volume to the sphere's volume?

  1. 3 : 2 Correct answer.
  2. 9 : 4
  3. 4 : 3
  4. 2 : 1

Explanation

The cylinder volume is πr²h = π(6²)(12) = 432π cm³ and the sphere volume is (4/3)πr³ = (4/3)π(216) = 288π cm³. Dividing gives 432π : 288π = 3 : 2, a ratio that holds for any sphere inscribed in a cylinder of the same radius and height equal to the diameter.

Question 26Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Simplify the product (3x²y)(4xy³).

  1. 12x⁵y³
  2. 12x³y⁴ Correct answer.
  3. 7x³y⁴
  4. 12x²y⁴

Explanation

Multiply the numerical coefficients: 3 × 4 = 12. When multiplying powers of the same base, add the exponents, so x^(2+1) = x³ and y^(1+3) = y⁴. The result is therefore 12x³y⁴.

Question 27Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Which factorisation of x² − 7x + 12 is correct?

  1. (x − 2)(x − 6)
  2. (x − 3)(x − 4) Correct answer.
  3. (x + 3)(x + 4)
  4. (x − 1)(x − 12)

Explanation

Two numbers must multiply to +12 and add to −7. The pair −3 and −4 satisfies both conditions, since (−3)(−4) = 12 and −3 + (−4) = −7. Hence x² − 7x + 12 = (x − 3)(x − 4).

Question 28Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the value of sin 30° + cos 60°?

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

Explanation

The exact values are sin 30° = 1/2 and cos 60° = 1/2. Adding them gives 1/2 + 1/2 = 1.

Question 29Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

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

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

Explanation

The slope is the change in y divided by the change in x: (11 − 3)/(6 − 2). This equals 8/4 = 2.

Question 30Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the arithmetic mean of the numbers 4, 8, 10 and 14?

  1. 10
  2. 9 Correct answer.
  3. 8
  4. 12

Explanation

The mean is the sum of the values divided by how many there are. The sum is 4 + 8 + 10 + 14 = 36, and 36 ÷ 4 = 9.

Question 31Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Written in simplest radical form, √50 is equal to which expression?

  1. 25√2
  2. 2√5
  3. 10√5
  4. 5√2 Correct answer.

Explanation

Write 50 as 25 × 2, so √50 = √25 × √2. Since √25 = 5, the result is 5√2.

Question 32Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What are the coordinates of the vertex of the parabola y = (x − 2)² + 5?

  1. (5, 2)
  2. (−2, 5)
  3. (2, 5) Correct answer.
  4. (2, −5)

Explanation

The form y = (x − h)² + k has vertex (h, k). Reading the equation gives h = 2 and k = 5, so the vertex is (2, 5).

Question 33Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the sum of the interior angles of a convex octagon?

  1. 720°
  2. 1260°
  3. 1080° Correct answer.
  4. 900°

Explanation

For a polygon with n sides the interior angles add to (n − 2) × 180°. For an octagon n = 8, so the sum is 6 × 180° = 1080°.

Question 34Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Solving 2x² − 6x + 1 = 0 with the quadratic formula gives which pair of roots?

  1. x = 3 ± √7
  2. x = (−3 ± √7)/2
  3. x = (3 ± √7)/4
  4. x = (3 ± √7)/2 Correct answer.

Explanation

Here a = 2, b = −6, c = 1, so the discriminant is b² − 4ac = 36 − 8 = 28. The formula gives x = (6 ± √28)/4 = (6 ± 2√7)/4, which simplifies to (3 ± √7)/2.

Question 35Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

The function f is defined by f(x) = 3x − 4. What is the value of f⁻¹(8)?

  1. 4 Correct answer.
  2. 2
  3. 12
  4. 20

Explanation

The inverse satisfies f(x) = 8, so solve 3x − 4 = 8. Adding 4 gives 3x = 12, hence x = 4.

Question 36Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Given that cos θ = 12/13 and θ lies in the fourth quadrant, what is the value of tan θ?

  1. 12/5
  2. 5/12
  3. −5/12 Correct answer.
  4. −12/5

Explanation

In the fourth quadrant sine is negative and cosine is positive. Using sin²θ = 1 − cos²θ = 1 − 144/169 = 25/169 gives sin θ = −5/13, so tan θ = sin θ / cos θ = −5/12.

Question 37Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Two sides of a triangle measure 7 cm and 9 cm, and the angle between them is 60°. What is the length of the third side?

  1. √67 cm Correct answer.
  2. √193 cm
  3. √130 cm
  4. 11 cm

Explanation

By the cosine rule, c² = a² + b² − 2ab cos C = 49 + 81 − 2(7)(9)(1/2) = 130 − 63 = 67. The third side is therefore √67 cm, about 8.19 cm.

Question 38Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

For the data set 2, 4, 4, 4, 5, 5, 7, 9, treating it as a population, what is the variance?

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

Explanation

The mean is 40 ÷ 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, whose sum is 32. Dividing by 8 gives a variance of 4.

Question 39Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Solve for x: 9^x = 27^(x − 1).

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

Explanation

Write both sides with base 3: 9^x = 3^(2x) and 27^(x − 1) = 3^(3x − 3). Equating exponents gives 2x = 3x − 3, so x = 3. Checking: 9³ = 729 = 27².

Question 40Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

What is the value of log₄ 8?

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

Explanation

Let log₄ 8 = y, so 4^y = 8. Writing both sides in base 2 gives 2^(2y) = 2³, hence 2y = 3 and y = 3/2.

Question 41Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A sequence is defined by a₁ = 2 and aₙ₊₁ = 3aₙ − 1 for n ≥ 1. What is the value of a₃?

  1. 11
  2. 17
  3. 14 Correct answer.
  4. 15

Explanation

Apply the rule twice. First a₂ = 3(2) − 1 = 5, then a₃ = 3(5) − 1 = 14.

Question 42Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

An arc of a circle of radius 9 cm subtends an angle of 120° at the centre. What is the length of the arc?

  1. 6π cm Correct answer.
  2. 12π cm
  3. 3π cm
  4. 9π cm

Explanation

Arc length equals (θ/360) × 2πr. Here (120/360) × 2π × 9 = (1/3) × 18π = 6π cm.

Question 43Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A right circular cone has base radius 3 cm and height 8 cm. What is its volume?

  1. 36π cm³
  2. 24π cm³ Correct answer.
  3. 8π cm³
  4. 72π cm³

Explanation

The volume of a cone is (1/3)πr²h. Substituting r = 3 and h = 8 gives (1/3)π(9)(8) = 24π cm³.

Question 44Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Which interval gives the complete solution set of the inequality |2x − 3| ≤ 7?

  1. x ≤ 5
  2. −2 ≤ x ≤ 5 Correct answer.
  3. −5 ≤ x ≤ 2
  4. x ≥ −2

Explanation

The inequality means −7 ≤ 2x − 3 ≤ 7. Adding 3 to all parts gives −4 ≤ 2x ≤ 10, and dividing by 2 gives −2 ≤ x ≤ 5.

Question 45Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

In how many different ways can a committee of 3 students be chosen from a group of 8 students?

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

Explanation

Because the order of selection does not matter, the count is the combination C(8,3) = 8!/(3!5!) = (8 × 7 × 6)/(3 × 2 × 1) = 56.

Question 46Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

Angles α and β are acute, with sin α = 5/13 and cos β = 3/5. What is the value of sin(α + β)?

  1. 16/65
  2. 63/65 Correct answer.
  3. 56/65
  4. 33/65

Explanation

Since both angles are acute, cos α = 12/13 and sin β = 4/5. Then sin(α + β) = sin α cos β + cos α sin β = (5/13)(3/5) + (12/13)(4/5) = 15/65 + 48/65 = 63/65.

Question 47Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

The polynomial x³ + kx² − 4x + 4 has x − 2 as a factor. What is the value of k?

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

Explanation

By the factor theorem, substituting x = 2 must give zero. This gives 8 + 4k − 8 + 4 = 4k + 4, so 4k + 4 = 0 and k = −1.

Question 48Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A bacterial culture starts with 100 cells and doubles every 3 hours. After how many hours will the population first reach 800 cells?

  1. 12 hours
  2. 9 hours Correct answer.
  3. 6 hours
  4. 24 hours

Explanation

The population after t hours is 100 × 2^(t/3). Setting this equal to 800 gives 2^(t/3) = 8 = 2³, so t/3 = 3 and t = 9 hours.

Question 49Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

In a right-angled triangle the two legs measure 6 cm and 8 cm. What is the length of the altitude drawn from the right angle to the hypotenuse?

  1. 2.4 cm
  2. 4.8 cm Correct answer.
  3. 6 cm
  4. 5 cm

Explanation

The hypotenuse is √(36 + 64) = 10 cm. The altitude h to the hypotenuse satisfies area = (1/2)(6)(8) = (1/2)(10)h, so h = 48/10 = 4.8 cm.

Question 50Mathematics (algebra, functions, trigonometry, geometry, basic statistics)

A data set has mean 20 and standard deviation 4. Every value is multiplied by 3 and then 5 is added to each result. What are the mean and standard deviation of the new data set?

  1. Mean 25, standard deviation 7
  2. Mean 65, standard deviation 12 Correct answer.
  3. Mean 65, standard deviation 17
  4. Mean 60, standard deviation 12

Explanation

Multiplying by 3 multiplies the mean by 3, and adding 5 then adds 5, giving 3(20) + 5 = 65. Spread is affected only by the multiplication, so the standard deviation becomes |3| × 4 = 12.