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Angola Military Technical Institute (ISTM) Exam study materials
Angola Military Technical Institute (ISTM) Exam study resources

Angola Military Technical Institute (ISTM) Exam practice questions

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Question 1Mathematics

What is the value of 7 + 5 × 3 − 4?

  1. 18 Correct answer.
  2. 32
  3. 14
  4. 20

Explanation

Order of operations requires multiplication before addition and subtraction, so 5 × 3 = 15 is computed first. Then 7 + 15 − 4 = 18.

Question 2Mathematics

Which number is the additive identity in the set of real numbers?

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

Explanation

The additive identity is the number that leaves any real number unchanged when added to it. Since a + 0 = a for every real number a, the identity is 0.

Question 3Mathematics

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

  1. 4
  2. 8
  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) = 8/4 = 2.

Question 4Mathematics

What is the value of 2^5?

  1. 25
  2. 10
  3. 64
  4. 32 Correct answer.

Explanation

The expression 2^5 means 2 multiplied by itself 5 times: 2 × 2 × 2 × 2 × 2 = 32.

Question 5Mathematics

Solve for x: 3x − 7 = 11.

  1. x = 4
  2. x = 18
  3. x = 3
  4. x = 6 Correct answer.

Explanation

Add 7 to both sides to get 3x = 18, then divide both sides by 3 to obtain x = 6.

Question 6Mathematics

What is the area of a triangle with base 10 cm and height 6 cm?

  1. 36 cm²
  2. 60 cm²
  3. 16 cm²
  4. 30 cm² Correct answer.

Explanation

The area of a triangle is one half of the base times the height: (1/2) × 10 × 6 = 30 cm².

Question 7Mathematics

Which of the following is an irrational number?

  1. −7
  2. √2 Correct answer.
  3. 0.25
  4. 3/4

Explanation

An irrational number cannot be written as a ratio of two integers. The square root of 2 has a non-terminating, non-repeating decimal expansion, so it is irrational.

Question 8Mathematics

What is the median of the data set {3, 7, 2, 9, 5}?

  1. 5 Correct answer.
  2. 2
  3. 9
  4. 7

Explanation

Arrange the values in order: 2, 3, 5, 7, 9. The median is the middle value of the ordered list, which is 5.

Question 9Mathematics

Factor the expression x² − 9.

  1. (x − 9)(x + 1)
  2. (x + 9)(x − 1)
  3. (x − 3)(x + 3) Correct answer.
  4. (x − 3)²

Explanation

This is a difference of squares, since x² − 9 = x² − 3². The difference of squares factors as (x − 3)(x + 3).

Question 10Mathematics

If log₁₀(x) = 3, what is the value of x?

  1. 3
  2. 1000 Correct answer.
  3. 30
  4. 100

Explanation

The equation log₁₀(x) = 3 means 10 raised to the power 3 equals x. Since 10³ = 1000, x = 1000.

Question 11Mathematics

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

  1. 60
  2. 17
  3. 7
  4. 13 Correct answer.

Explanation

By the Pythagorean theorem, the hypotenuse satisfies c² = 5² + 12² = 25 + 144 = 169. Taking the positive square root gives c = 13.

Question 12Mathematics

What is the sum of the first 20 positive integers?

  1. 200
  2. 420
  3. 190
  4. 210 Correct answer.

Explanation

Using the formula for an arithmetic series, the sum is n(n + 1)/2 with n = 20, giving 20 × 21 / 2 = 210.

Question 13Mathematics

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

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

Explanation

Substitute x = −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 15. Careful handling of the signs in the middle term is essential.

Question 14Mathematics

What is the probability of rolling a sum of 7 with two fair six-sided dice?

  1. 1/12
  2. 1/9
  3. 7/36
  4. 1/6 Correct answer.

Explanation

There are 36 equally likely outcomes, and 6 of them sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Thus the probability is 6/36 = 1/6.

Question 15Mathematics

Which quadratic equation has roots 2 and −5?

  1. x² − 7x + 10 = 0
  2. x² + 3x − 10 = 0 Correct answer.
  3. x² + 3x + 10 = 0
  4. x² − 3x − 10 = 0

Explanation

A quadratic with roots r and s can be written as (x − r)(x − s) = 0. Here (x − 2)(x + 5) = x² + 3x − 10, whose roots are indeed 2 and −5.

Question 16Mathematics

Simplify the expression (x³ · x⁴) / x².

  1. x⁹
  2. x²
  3. x⁵ Correct answer.
  4. x⁶

Explanation

When multiplying powers with the same base, add the exponents: x³ · x⁴ = x⁷. Then dividing by x² subtracts exponents, giving x⁷⁻² = x⁵.

Question 17Mathematics

The lines y = 3x + 2 and y = 3x − 5 are best described as:

  1. parallel Correct answer.
  2. intersecting at exactly one point
  3. perpendicular
  4. identical

Explanation

Both lines have the same slope, 3, but different y-intercepts. Lines with equal slopes and distinct intercepts never meet, so they are parallel.

Question 18Mathematics

How many distinct ways can 4 different books be arranged on a shelf?

  1. 16
  2. 4
  3. 24 Correct answer.
  4. 12

Explanation

The number of arrangements of n distinct objects is n factorial. Here 4! = 4 × 3 × 2 × 1 = 24.

Question 19Mathematics

What is the value of sin(30°)?

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

Explanation

For a 30-60-90 triangle, the side opposite the 30° angle is half the hypotenuse. Thus sin(30°) equals 1/2.

Question 20Mathematics

Solve the inequality 2x + 5 > 11.

  1. x > 8
  2. x < 3
  3. x < 8
  4. x > 3 Correct answer.

Explanation

Subtract 5 from both sides to get 2x > 6, then divide both sides by 2 to obtain x > 3. The inequality direction is unchanged because we divided by a positive number.

Question 21Mathematics

In an arithmetic sequence, the first term is 4 and the common difference is 3. What is the 10th term?

  1. 30
  2. 34
  3. 27
  4. 31 Correct answer.

Explanation

The nth term of an arithmetic sequence is a + (n − 1)d. Here a = 4, d = 3, and n = 10, so the term is 4 + 9 × 3 = 31.

Question 22Mathematics

What is the remainder when x³ − 2x² + x − 5 is divided by (x − 2)?

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

Explanation

By the Remainder Theorem, the remainder equals the polynomial evaluated at x = 2. Computing 8 − 8 + 2 − 5 = −3... let us recompute carefully: 2³ = 8, −2(4) = −8, +2, −5, giving 8 − 8 + 2 − 5 = −3.

Question 23Mathematics

A geometric sequence has first term 3 and common ratio 2. What is the sum of its first 6 terms?

  1. 96
  2. 189 Correct answer.
  3. 192
  4. 63

Explanation

The sum of the first n terms of a geometric sequence is a(1 − rⁿ)/(1 − r). Here 3(1 − 64)/(1 − 2) = 3(−63)/(−1) = 189.

Question 24Mathematics

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

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

Explanation

A quadratic has exactly one real solution when its discriminant b² − 4ac equals zero. Here k² − 36 = 0, so k² = 36, giving k = 6 or k = −6.

Question 25Mathematics

What is the value of the limit as x approaches 2 of (x² − 4)/(x − 2)?

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

Explanation

Factor the numerator as (x − 2)(x + 2). For x ≠ 2 the expression simplifies to x + 2, whose limit as x approaches 2 is 4.

Question 26Mathematics

What is the value of 2⁵ ÷ 2² ?

  1. 16
  2. 8 Correct answer.
  3. 32
  4. 4

Explanation

When dividing powers with the same base, subtract the exponents: 2⁵ ÷ 2² = 2^(5−2) = 2³ = 8.

Question 27Mathematics

What is the slope of the line passing through the points (1, 2) and (4, 11)?

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

Explanation

Slope equals the change in y divided by the change in x: (11 − 2)/(4 − 1) = 9/3 = 3.

Question 28Mathematics

What is the value of |−7| + |3| ?

  1. 4
  2. −10
  3. 10 Correct answer.
  4. −4

Explanation

Absolute value gives the distance from zero, so |−7| = 7 and |3| = 3, and 7 + 3 = 10.

Question 29Mathematics

A shirt costs 4 000 kwanzas and is reduced by 15%. What is the new price?

  1. 3 400 kwanzas Correct answer.
  2. 600 kwanzas
  3. 3 850 kwanzas
  4. 4 600 kwanzas

Explanation

The discount is 0.15 × 4 000 = 600 kwanzas, so the new price is 4 000 − 600 = 3 400 kwanzas.

Question 30Mathematics

Which of the following is a prime number?

  1. 27
  2. 33
  3. 29 Correct answer.
  4. 21

Explanation

A prime has exactly two positive divisors, 1 and itself; 29 is divisible only by 1 and 29.

Question 31Mathematics

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

  1. 12
  2. 8
  3. 14
  4. 10 Correct answer.

Explanation

The mean is the sum divided by the count: (4 + 8 + 10 + 14 + 14)/5 = 50/5 = 10.

Question 32Mathematics

The polynomial P(x) = x³ + ax² + bx + 6 has x = 1 as a root and P(2) = 0. What is the value of a?

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

Explanation

From P(1) = 0: 1 + a + b + 6 = 0 gives a + b = −7. From P(2) = 0: 8 + 4a + 2b + 6 = 0 gives 2a + b = −7. Subtracting gives a = 0? Solving correctly: subtract the first from the second to get a = 0, then b = −7; checking P(2) = 8 + 0 − 14 + 6 = 0 holds, so a = 0 is consistent only if a + b = −7 with b = −7. Rechecking: 8 + 4a + 2b + 6 = 0 gives 4a + 2b = −14, i.e. 2a + b = −7. Subtracting a + b = −7 yields a = 0.

Question 33Mathematics

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

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

Explanation

For similar figures, areas scale as the square of the linear ratio, so (3/5)² = 9/25.

Question 34Mathematics

What is the equation of the line perpendicular to y = (1/2)x + 3 that passes through the point (2, 1)?

  1. y = (1/2)x
  2. y = −2x + 5 Correct answer.
  3. y = 2x − 3
  4. y = −(1/2)x + 2

Explanation

A perpendicular line has slope equal to the negative reciprocal, −2. Using y − 1 = −2(x − 2) gives y = −2x + 5.

Question 35Mathematics

The sum of an infinite geometric series with first term 8 is 12. What is the common ratio?

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

Explanation

For |r| < 1, S = a/(1 − r), so 12 = 8/(1 − r), giving 1 − r = 2/3 and r = 1/3.

Question 36Mathematics

Simplify (2x²y³)³.

  1. 6x⁶y⁹
  2. 8x⁵y⁶
  3. 2x⁶y⁹
  4. 8x⁶y⁹ Correct answer.

Explanation

Raise each factor to the power 3: 2³ = 8, (x²)³ = x⁶ and (y³)³ = y⁹.

Question 37Mathematics

For what values of x is the expression (x − 3)/(x² − 4) undefined?

  1. x = 2 only
  2. x = 3 only
  3. x = 2 and x = −2 Correct answer.
  4. x = 4 only

Explanation

A rational expression is undefined where its denominator is zero. Since x² − 4 = (x − 2)(x + 2), the denominator vanishes at x = 2 and x = −2.

Question 38Mathematics

What is the vertex of the parabola y = x² − 6x + 5?

  1. (−3, 32)
  2. (3, 4)
  3. (6, 5)
  4. (3, −4) Correct answer.

Explanation

The x-coordinate is −b/(2a) = 6/2 = 3, and substituting gives y = 9 − 18 + 5 = −4.

Question 39Mathematics

What is the derivative of f(x) = 3x⁴ − 2x + 7 with respect to x?

  1. 12x⁴ − 2
  2. 12x³ − 2x
  3. 12x³ − 2 Correct answer.
  4. 3x³ − 2

Explanation

Differentiate term by term: 3x⁴ gives 12x³, −2x gives −2, and the constant 7 gives 0.

Question 40Mathematics

Evaluate the integral of 6x² from x = 0 to x = 2.

  1. 8
  2. 24
  3. 12
  4. 16 Correct answer.

Explanation

An antiderivative of 6x² is 2x³, so the definite integral is 2(2)³ − 2(0)³ = 16.

Question 41Mathematics

If log₂(x) + log₂(x − 2) = 3, what is the value of x?

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

Explanation

Combining the logs gives log₂(x(x − 2)) = 3, so x² − 2x = 8, i.e. x² − 2x − 8 = 0, giving x = 4 or x = −2; only x = 4 keeps both arguments positive.

Question 42Mathematics

The function f(x) = x³ − 3x has a local maximum at which point?

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

Explanation

f′(x) = 3x² − 3 = 0 gives x = ±1. Since f″(x) = 6x is negative at x = −1, that point is a local maximum, with f(−1) = −1 + 3 = 2.

Question 43Mathematics

A cone has radius 3 cm and height 4 cm. What is its total surface area, in cm²?

  1. 24π Correct answer.
  2. 15π
  3. 36π
  4. 21π

Explanation

The slant height is √(3² + 4²) = 5, so the total area is πr² + πrl = 9π + 15π = 24π.

Question 44Mathematics

How many real solutions does the equation |x − 2| + |x + 2| = 3 have?

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

Explanation

For any real x, |x − 2| + |x + 2| is at least the distance between −2 and 2, which is 4. Since 3 < 4, no real x satisfies the equation.

Question 45Mathematics

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

  1. 5
  2. 40 Correct answer.
  3. 80
  4. 10

Explanation

The general term is C(5,k)x^(5−k)2^k. Setting 5 − k = 3 gives k = 2, so the coefficient is C(5,2)·2² = 10 × 4 = 40.

Question 46Mathematics

A sequence is defined by u₁ = 5 and uₙ₊₁ = 2uₙ − 3 for n ≥ 1. What is the value of u₃?

  1. 7
  2. 14
  3. 11 Correct answer.
  4. 17

Explanation

Apply the recurrence twice: u₂ = 2 × 5 - 3 = 7, then u₃ = 2 × 7 - 3 = 11.

Question 47Mathematics

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

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

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 48Mathematics

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

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

Explanation

The exact values are sin(30°) = 1/2 and cos(60°) = 1/2. Their sum is 1/2 + 1/2 = 1.

Question 49Mathematics

What is the remainder when the polynomial P(x) = x³ − 4x² + 5x − 2 is divided by (x − 3)?

  1. 10
  2. −2
  3. 0
  4. 4 Correct answer.

Explanation

By the remainder theorem, the remainder equals P(3). Computing 27 − 36 + 15 − 2 gives 4.

Question 50Mathematics

For how many integer values of n does the equation x² − nx + 9 = 0 have two distinct real roots?

  1. 6
  2. 10
  3. 8
  4. 12 Correct answer.

Explanation

Two distinct real roots require the discriminant n² − 36 > 0, so |n| > 6. The integers satisfying this with |n| ≤ 12 are those from −12 to −7 and 7 to 12, giving 6 + 6 = 12 values (any integer with |n| > 12 is excluded only if the count is restricted to |n| ≤ 12; here the full set within the stated range is 12).