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Estonian Upper-Secondary Broad Mathematics Exam study materials
Estonian Upper-Secondary Broad Mathematics Exam study resources

Estonian Upper-Secondary Broad Mathematics Exam practice questions

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Question 1Calculus

Find the derivative of f(x) = x³.

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

Explanation

By the power rule, the derivative of xⁿ is n·x^(n−1), so the derivative of x³ is 3x².

Question 2Calculus

Evaluate the limit: lim(x→2) (x + 3).

  1. 5 Correct answer.
  2. 2
  3. 3
  4. 0

Explanation

Since x + 3 is continuous everywhere, the limit as x approaches 2 is simply found by direct substitution: 2 + 3 = 5.

Question 3Calculus

Find the derivative of the constant function f(x) = 5.

  1. 0 Correct answer.
  2. 5
  3. 1
  4. x

Explanation

A constant function never changes as x varies, so its rate of change - and hence its derivative - is 0 everywhere.

Question 4Calculus

Find f'(x) for f(x) = 4x³ − 2x + 7.

  1. 12x² − 2 Correct answer.
  2. 12x² + 7
  3. 4x² − 2
  4. 12x²

Explanation

Differentiating term by term: the derivative of 4x³ is 12x², the derivative of −2x is −2, and the derivative of the constant 7 is 0. Combined, f'(x) = 12x² − 2.

Question 5Calculus

Evaluate the limit: lim(x→3) (x² − 9)/(x − 3).

  1. 6 Correct answer.
  2. 0
  3. 3
  4. 9

Explanation

Factoring gives (x − 3)(x + 3)/(x − 3), which simplifies to x + 3 for x ≠ 3. Taking the limit as x → 3 gives 3 + 3 = 6.

Question 6Calculus

Find the equation of the tangent line to f(x) = x² at x = 3.

  1. y = 6x − 9 Correct answer.
  2. y = 6x + 9
  3. y = 9x − 6
  4. y = 6x − 18

Explanation

Since f(3) = 9 and f'(x) = 2x gives f'(3) = 6, the tangent line is y − 9 = 6(x − 3), which simplifies to y = 6x − 9.

Question 7Calculus

Use the product rule to find f'(x) for f(x) = x² sin(x).

  1. 2x sin(x) + x² cos(x) Correct answer.
  2. 2x cos(x)
  3. x² cos(x)
  4. 2x sin(x) − x² cos(x)

Explanation

The product rule states (uv)' = u'v + uv'. With u = x² and v = sin(x), this gives 2x·sin(x) + x²·cos(x).

Question 8Calculus

Use the chain rule to find the derivative of f(x) = (3x + 1)⁴.

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

Explanation

By the chain rule, the derivative is 4(3x + 1)³ times the derivative of the inner function (3x + 1), which is 3. This gives 4 × 3 × (3x + 1)³ = 12(3x + 1)³.

Question 9Calculus

Evaluate the limit: lim(x→0) sin(x)/x.

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

Explanation

This is a well-known standard limit in calculus: as x approaches 0, sin(x)/x approaches exactly 1, even though direct substitution gives the indeterminate form 0/0.

Question 10Calculus

Find the derivative of f(x) = e^(2x).

  1. 2e^(2x) Correct answer.
  2. e^(2x)
  3. e^(2x)/2
  4. 2xe^(2x−1)

Explanation

By the chain rule, the derivative of e^(u) is e^(u)·u'. Here u = 2x, so u' = 2, giving f'(x) = 2e^(2x).

Question 11Calculus

Find the second derivative of f(x) = x⁴ − 3x² evaluated at x = 1.

  1. 6 Correct answer.
  2. 12
  3. −6
  4. 18

Explanation

The first derivative is f'(x) = 4x³ − 6x, and the second derivative is f''(x) = 12x² − 6. At x = 1, this gives 12(1) − 6 = 6.

Question 12Calculus

Evaluate the limit: lim(x→∞) (3x² + 2x)/(x² − 5).

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

Explanation

Dividing numerator and denominator by x² gives (3 + 2/x)/(1 − 5/x²). As x → ∞, the terms with x in the denominator vanish, leaving 3/1 = 3.

Question 13Calculus

Find the derivative of f(x) = ln(x² + 1), then evaluate f'(1).

  1. 1 Correct answer.
  2. 2
  3. 0.5
  4. 4

Explanation

By the chain rule, f'(x) = 2x/(x² + 1). At x = 1, this gives 2(1)/(1 + 1) = 2/2 = 1.

Question 14Calculus

Use the quotient rule to find f'(x) for f(x) = x/(x + 1), then evaluate f'(0).

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

Explanation

The quotient rule gives f'(x) = [(1)(x + 1) − x(1)] / (x + 1)² = 1/(x + 1)². At x = 0, this equals 1/1² = 1.

Question 15Algebra Functions

Simplify the expression (x² − 9) / (x − 3) for x ≠ 3.

  1. x + 3 Correct answer.
  2. x − 3
  3. x² + 3
  4. 9 − x

Explanation

The numerator is a difference of squares: x² − 9 = (x − 3)(x + 3). Dividing by (x − 3) cancels the common factor, leaving x + 3.

Question 16Algebra Functions

Solve for x: 3x − 7 = 2x + 5.

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

Explanation

Subtract 2x from both sides to get x − 7 = 5, then add 7 to both sides to get x = 12.

Question 17Algebra Functions

The quadratic equation x² − 5x + 6 = 0 has two real roots. What is the larger root?

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

Explanation

Factoring gives (x − 2)(x − 3) = 0, so the roots are x = 2 and x = 3. The larger of the two roots is 3.

Question 18Algebra Functions

Write √50 in simplest radical form.

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

Explanation

50 = 25 × 2, and √25 = 5, so √50 = √25 · √2 = 5√2.

Question 19Algebra Functions

Solve for x: 2/(x − 1) + 3 = 5/(x − 1).

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

Explanation

Subtracting 2/(x − 1) from both sides gives 3 = 3/(x − 1), so x − 1 = 1, meaning x = 2. Checking x = 2 in the original equation confirms it does not make any denominator zero.

Question 20Algebra Functions

Solve the system 2x + y = 11 and x − y = 1. What is the value of x?

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

Explanation

From the second equation, x = y + 1. Substituting into the first gives 2(y + 1) + y = 11, so 3y = 9 and y = 3, meaning x = 4.

Question 21Algebra Functions

Simplify (a³ − a) / (a² − 1) for a ≠ ±1.

  1. a Correct answer.
  2. a²
  3. a − 1
  4. 1/a

Explanation

Factor the numerator as a(a² − 1). Dividing by (a² − 1) cancels the common factor, leaving just a.

Question 22Algebra Functions

A rectangle's length is 3 cm more than twice its width. If the perimeter is 36 cm, find the width.

  1. 5 cm Correct answer.
  2. 4.5 cm
  3. 6 cm
  4. 15 cm

Explanation

Let w be the width, so the length is 2w + 3. Half the perimeter equals w + (2w + 3) = 3w + 3 = 18, giving 3w = 15, so w = 5 cm.

Question 23Algebra Functions

Let y = x². Solving x⁴ − 5x² + 4 = 0 gives several real roots for x. What is the largest one?

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

Explanation

Substituting y = x² gives y² − 5y + 4 = 0, which factors as (y − 1)(y − 4) = 0, so y = 1 or y = 4. This means x = ±1 or x = ±2, and the largest of these values is 2.

Question 24Algebra Functions

Simplify (2³ · 2⁻¹) / 2² as a power of 2, then evaluate it.

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

Explanation

Using exponent rules, 2³ · 2⁻¹ = 2², and dividing by 2² gives 2⁰, which equals 1.

Question 25Vectors

Given points A(1, 2) and B(4, 6), find the vector AB as a pair of components.

  1. (3, 4) Correct answer.
  2. (5, 8)
  3. (−3, −4)
  4. (3, −4)

Explanation

Vector AB is found by subtracting the coordinates of A from B: (4 − 1, 6 − 2) = (3, 4).

Question 26Vectors

Find the midpoint of the segment with endpoints (2, 3) and (8, 7).

  1. (5, 5) Correct answer.
  2. (10, 10)
  3. (6, 4)
  4. (3, 2)

Explanation

The midpoint formula averages each coordinate: ((2 + 8)/2, (3 + 7)/2) = (5, 5).

Question 27Vectors

Find the dot product of vectors u = (2, −3) and v = (4, 5).

  1. −7 Correct answer.
  2. 23
  3. −2
  4. 2

Explanation

The dot product is u·v = (2)(4) + (−3)(5) = 8 − 15 = −7.

Question 28Vectors

Find the equation of the line through the point (2, 3) with slope 4.

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

Explanation

Using point-slope form, y − 3 = 4(x − 2), which simplifies to y = 4x − 8 + 3 = 4x − 5.

Question 29Vectors

Find the equation of the circle with center (1, −2) and radius 3.

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

Explanation

The standard circle equation (x − h)² + (y − k)² = r² with center (h, k) = (1, −2) and radius r = 3 gives (x − 1)² + (y + 2)² = 9.

Question 30Vectors

Vectors a = (1, 2) and b = (k, 4) are parallel. Find the value of k.

  1. 2 Correct answer.
  2. −1
  3. 8
  4. 0.5

Explanation

Parallel vectors have proportional components: 1/k = 2/4, which cross-multiplies to 4 = 2k, giving k = 2.

Question 31Vectors

Find the distance between the points (2, 3) and (7, 15).

  1. 13 Correct answer.
  2. 17
  3. 169
  4. 12

Explanation

The differences are (7 − 2, 15 − 3) = (5, 12), so the distance is √(5² + 12²) = √(25 + 144) = √169 = 13.

Question 32Vectors

Vectors a and b have magnitudes 3 and 5, and the angle between them is 60°. Find the dot product a·b.

  1. 7.5 Correct answer.
  2. 15
  3. 3.75
  4. 13.0

Explanation

The dot product formula a·b = |a||b|cos(θ) gives 3 × 5 × cos(60°) = 15 × 0.5 = 7.5.

Question 33Probability Statistics

How many ways can 5 distinct books be arranged in a row on a shelf?

  1. 120 Correct answer.
  2. 20
  3. 25
  4. 60

Explanation

The number of arrangements of 5 distinct objects is 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 34Probability Statistics

A fair six-sided die is rolled once. What is the probability of rolling a number less than 5?

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

Explanation

The numbers less than 5 are 1, 2, 3 and 4 - four favorable outcomes out of 6 total, so the probability is 4/6 = 2/3.

Question 35Probability Statistics

A bag contains 3 red balls and 5 blue balls. What is the probability of drawing a blue ball at random?

  1. 5/8 Correct answer.
  2. 3/8
  3. 5/3
  4. 1/2

Explanation

There are 5 blue balls out of 8 total balls, so the probability of drawing a blue ball is 5/8.

Question 36Probability Statistics

How many distinct committees of 3 people can be chosen from a group of 8 people?

  1. 56 Correct answer.
  2. 336
  3. 24
  4. 168

Explanation

Since order does not matter for a committee, this is a combination: C(8,3) = 8!/(3!5!) = 56.

Question 37Probability Statistics

Two fair dice are rolled. What is the probability that their sum is 7?

  1. 1/6 Correct answer.
  2. 1/12
  3. 1/36
  4. 1/8

Explanation

There are 6 outcomes that sum to 7 out of 36 possible dice combinations: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). This gives 6/36 = 1/6.

Question 38Probability Statistics

A committee of 4 is formed from 5 men and 6 women. In how many ways can it include exactly 2 men and 2 women?

  1. 150 Correct answer.
  2. 330
  3. 25
  4. 50

Explanation

Choose 2 men from 5 in C(5,2) = 10 ways, and 2 women from 6 in C(6,2) = 15 ways. Since these choices are made together, multiply: 10 × 15 = 150.

Question 39Probability Statistics

Events A and B are independent, with P(A) = 0.3 and P(B) = 0.4. Find P(A and B).

  1. 0.12 Correct answer.
  2. 0.7
  3. 0.24
  4. 0.1

Explanation

For independent events, P(A and B) = P(A) × P(B) = 0.3 × 0.4 = 0.12.

Question 40Probability Statistics

Find the median of the data set: 4, 8, 15, 16, 23, 42.

  1. 15.5 Correct answer.
  2. 15
  3. 16
  4. 18

Explanation

With 6 values (an even count), the median is the average of the two middle values (15 and 16), which is (15 + 16)/2 = 15.5.

Question 41Probability Statistics

A binomial experiment has n = 10 trials with success probability p = 0.2. What is the probability of exactly 2 successes (to the nearest hundredth)?

  1. ≈0.30 Correct answer.
  2. ≈0.20
  3. ≈0.12
  4. ≈0.40

Explanation

The binomial formula gives P(X=2) = C(10,2)(0.2)²(0.8)⁸ = 45 × 0.04 × 0.1678 ≈ 0.30.

Question 42Probability Statistics

In a normal distribution with mean 100 and standard deviation 15, what is the z-score for a value of 130?

  1. 2.0 Correct answer.
  2. 1.5
  3. 30
  4. 0.5

Explanation

The z-score formula is z = (x − μ)/σ = (130 − 100)/15 = 30/15 = 2.0.

Question 43Geometry

Find the surface area of a cube with side length 3 cm.

  1. 54 cm² Correct answer.
  2. 27 cm²
  3. 36 cm²
  4. 18 cm²

Explanation

A cube has 6 identical square faces, so its surface area is 6 × 3² = 6 × 9 = 54 cm².

Question 44Geometry

Find the volume of a rectangular box with dimensions 2 m × 3 m × 5 m.

  1. 30 m³ Correct answer.
  2. 10 m³
  3. 15 m³
  4. 21 m³

Explanation

The volume of a rectangular box is length × width × height, so 2 × 3 × 5 = 30 m³.

Question 45Geometry

Find the volume of a cylinder with radius 3 cm and height 10 cm (use π ≈ 3.14159).

  1. ≈282.7 cm³ Correct answer.
  2. ≈94.2 cm³
  3. ≈188.5 cm³
  4. ≈565.5 cm³

Explanation

The volume formula V = πr²h gives π × 3² × 10 = π × 90 ≈ 282.7 cm³.

Question 46Geometry

Find the volume of a sphere with radius 3 cm, rounded to the nearest whole number (use π ≈ 3.14159).

  1. 113 cm³ Correct answer.
  2. 85 cm³
  3. 339 cm³
  4. 28 cm³

Explanation

The sphere volume formula V = (4/3)πr³ gives (4/3) × 3.14159 × 27 ≈ 113 cm³.

Question 47Geometry

A right circular cone has radius 4 cm and height 9 cm. Find its volume, rounded to the nearest whole number (use π ≈ 3.14159).

  1. 151 cm³ Correct answer.
  2. 452 cm³
  3. 75 cm³
  4. 302 cm³

Explanation

The cone volume formula V = (1/3)πr²h gives (1/3) × 3.14159 × 16 × 9 ≈ 151 cm³.

Question 48Geometry

Find the total surface area of a right circular cylinder with radius 2 cm and height 5 cm, rounded to the nearest tenth (use π ≈ 3.14159).

  1. ≈88.0 cm² Correct answer.
  2. ≈62.8 cm²
  3. ≈125.7 cm²
  4. ≈50.3 cm²

Explanation

Total surface area is the two circular bases plus the lateral surface: 2πr² + 2πrh = 2π(4) + 2π(2)(5) = 8π + 20π = 28π ≈ 88.0 cm².

Question 49Geometry

A regular square pyramid has a base edge of 6 cm and a height of 4 cm. Find its volume.

  1. 48 cm³ Correct answer.
  2. 144 cm³
  3. 24 cm³
  4. 96 cm³

Explanation

The pyramid volume formula V = (1/3) × base area × height gives (1/3) × 36 × 4 = 48 cm³.

Question 50Geometry

A cylinder is inscribed in a sphere of radius 5 cm, and the cylinder's height is 8 cm. Find the radius of the cylinder.

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

Explanation

The sphere's radius relates to the cylinder by R² = r² + (h/2)², so 25 = r² + 16, giving r² = 9 and r = 3 cm.