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

Estonian Upper-Secondary Narrow Mathematics Exam practice questions

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Question 1Geometry

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

  1. 54 Correct answer.
  2. 9
  3. 36
  4. 27

Explanation

A cube has 6 faces, each with area s²=9. Total surface area = 6×9 = 54 cm².

Question 2Geometry

Find the total surface area of a rectangular box with dimensions 3 × 4 × 5.

  1. 94 Correct answer.
  2. 47
  3. 60
  4. 69

Explanation

Surface area = 2(lw + lh + wh) = 2(12 + 15 + 20) = 2(47) = 94.

Question 3Geometry

A prism has a triangular base that is a right triangle with legs 6 and 8, and the prism's length (height) is 10. Find the prism's volume.

  1. 240 Correct answer.
  2. 480
  3. 24
  4. 120

Explanation

Base area = (1/2)(6)(8) = 24. Volume = base area × length = 24 × 10 = 240.

Question 4Geometry

Find the total surface area of a cylinder (including both circular ends) with radius 5 and height 12, in terms of π.

  1. 170π Correct answer.
  2. 120π
  3. 50π
  4. 85π

Explanation

Total surface area = 2πr² + 2πrh = 2π(25) + 2π(5)(12) = 50π + 120π = 170π.

Question 5Geometry

Find the length of the space diagonal of a rectangular box with dimensions 3, 4, and 12.

  1. 13 Correct answer.
  2. 19
  3. 169
  4. 5

Explanation

The space diagonal is d = √(3² + 4² + 12²) = √(9+16+144) = √169 = 13.

Question 6Geometry

A cylindrical water tank has radius 2 m and height 5 m, and is filled to 60% of its capacity. How many cubic meters of water are in the tank, in terms of π?

  1. 12π Correct answer.
  2. 20π
  3. 6π
  4. 8π

Explanation

Full volume = πr²h = π(4)(5) = 20π. At 60% capacity: 0.6 × 20π = 12π cubic meters.

Question 7Geometry

A regular square pyramid has a base edge of 10 cm and a slant height (from the apex to the midpoint of a base edge) of 13 cm. Find the height of the pyramid.

  1. 12 Correct answer.
  2. 13
  3. 144
  4. 8

Explanation

The height, half the base edge (5), and the slant height (13) form a right triangle: h² + 5² = 13², so h² = 169 - 25 = 144, giving h = 12 cm.

Question 8Geometry

A cone has height 9 cm and volume 48π cm³. Find its radius.

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

Explanation

Using V = (1/3)πr²h: 48π = (1/3)πr²(9) = 3πr², so r² = 16, giving r = 4 cm.

Question 9Geometry

A right circular cone has a base radius of 6 cm and a slant height of 10 cm. Find the height of the cone.

  1. 8 Correct answer.
  2. 64
  3. 12
  4. 4

Explanation

The radius, height, and slant height form a right triangle: h² + 6² = 10², so h² = 100-36=64, giving h = 8 cm.

Question 10Algebra Functions

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

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

Explanation

Subtract 2x from both sides to get x - 7 = 5, then add 7 to both sides to find x = 12. Always move variable terms to one side and constant terms to the other before combining.

Question 11Algebra Functions

Evaluate: |-3| + |2 - 5|.

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

Explanation

The absolute value of -3 is 3, and 2 - 5 = -3, whose absolute value is also 3. Adding these gives 3 + 3 = 6.

Question 12Algebra Functions

Simplify: x^3 * x^(-1), for x ≠ 0.

  1. x^2 Correct answer.
  2. x^3
  3. x^(-3)
  4. x^4

Explanation

When multiplying powers with the same base, add the exponents: 3 + (-1) = 2, giving x^2.

Question 13Algebra Functions

Solve for x: 2^x = 8.

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

Explanation

Since 8 = 2^3, the equation 2^x = 2^3 gives x = 3 by matching exponents on equal bases.

Question 14Algebra Functions

Solve for x: x^2 - 5x + 6 = 0.

  1. x = 2 or x = 3 Correct answer.
  2. x = -2 or x = -3
  3. x = 1 or x = 6
  4. x = 2 or x = -3

Explanation

Factor the quadratic as (x-2)(x-3)=0, so the solutions are x=2 and x=3, since these are the two numbers that multiply to 6 and add to 5.

Question 15Algebra Functions

Solve the inequality: x^2 - 4x + 3 > 0.

  1. x < 1 or x > 3 Correct answer.
  2. 1 < x < 3
  3. x ≤ 1 or x ≥ 3
  4. x < -1 or x > -3

Explanation

Factor as (x-1)(x-3)>0. Since the parabola opens upward, the expression is positive outside the roots, so the solution is x<1 or x>3.

Question 16Algebra Functions

Solve the system: 2x + y = 7 and x - y = 2.

  1. x = 3, y = 1 Correct answer.
  2. x = 1, y = 3
  3. x = 3, y = -1
  4. x = 2, y = 3

Explanation

Adding the two equations eliminates y: 3x = 9, so x = 3. Substituting back gives y = 7 - 2(3) = 1.

Question 17Algebra Functions

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

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

Explanation

Multiply both sides by (x-1): x + 3 = 2(x-1) = 2x - 2, so 5 = x. Checking, x=5 does not make the denominator zero, so it is valid.

Question 18Algebra Functions

Solve for x: √(2x + 3) = 5.

  1. x = 11 Correct answer.
  2. x = 14
  3. x = 1
  4. x = 22

Explanation

Square both sides: 2x + 3 = 25. Subtract 3: 2x = 22, so x = 11.

Question 19Algebra Functions

Solve for x: |2x - 5| = 7.

  1. x = 6 or x = -1 Correct answer.
  2. x = 6 or x = 1
  3. x = -6 or x = 1
  4. x = -6 or x = -1

Explanation

Split into two cases: 2x-5=7 gives x=6, and 2x-5=-7 gives x=-1. Both values are valid since absolute value equations produce two possible solutions.

Question 20Algebra Functions

Solve the inequality: 5x^2 - 2x + 9 > (x + 3)^2.

  1. x < 0 or x > 2 Correct answer.
  2. 0 < x < 2
  3. x ≤ 0 or x ≥ 2
  4. x < -2 or x > 0

Explanation

Expand (x+3)^2 = x^2+6x+9, giving 5x^2-2x+9 > x^2+6x+9. Simplify to 4x^2-8x>0, or 4x(x-2)>0. Since the parabola opens upward, the inequality holds outside the roots: x<0 or x>2.

Question 21Algebra Functions

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

  1. x = 2/5 Correct answer.
  2. x = -2/5
  3. x = 6/5
  4. x = 2

Explanation

Multiply both sides by (x-2)(x+2): x(x+2) + 3(x-2) = (x-2)(x+2). This simplifies to 5x - 6 = -4, so 5x = 2 and x = 2/5. This value does not make either denominator zero, so it is valid.

Question 22Algebra Functions

Solve for x: 2(x - 3) - (x + 1) = 4.

  1. x = 11 Correct answer.
  2. x = 3
  3. x = -11
  4. x = 7

Explanation

Distribute to get 2x - 6 - x - 1 = 4, which simplifies to x - 7 = 4, so x = 11.

Question 23Algebra Functions

For the function f(x) = 1/(x - 4), what value of x must be excluded from the domain?

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

Explanation

The denominator x-4 cannot equal zero, so x=4 must be excluded from the domain to avoid division by zero.

Question 24Algebra Functions

A shirt costs 50 euros and is discounted by 20%. What is the sale price?

  1. 40 euros Correct answer.
  2. 30 euros
  3. 10 euros
  4. 45 euros

Explanation

A 20% discount means the price is reduced to 80% of the original: 50 × 0.8 = 40 euros.

Question 25Algebra Functions

A quantity increases from 80 to 100. What is the percent increase?

  1. 25% Correct answer.
  2. 20%
  3. 125%
  4. 5%

Explanation

Percent increase = (change/original) × 100% = (100-80)/80 × 100% = 20/80 × 100% = 25%.

Question 26Algebra Functions

A car worth 20,000 euros loses 10% of its value each year. What is its value after 2 years?

  1. 16,200 euros Correct answer.
  2. 16,000 euros
  3. 18,000 euros
  4. 14,400 euros

Explanation

Each year the value is multiplied by 0.9: 20,000 × (0.9)² = 20,000 × 0.81 = 16,200 euros.

Question 27Algebra Functions

Solve for x: 3^x = 81.

  1. 4 Correct answer.
  2. 3
  3. 27
  4. 81

Explanation

Since 3⁴ = 81, the equation 3^x = 3⁴ gives x = 4 by matching exponents.

Question 28Algebra Functions

Solve for x: log₂(x) + log₂(x - 2) = 3.

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

Explanation

Combine using the product rule: log₂[x(x-2)] = 3, so x(x-2) = 2³ = 8. This gives x²-2x-8=0, factoring to (x-4)(x+2)=0, so x=4 or x=-2. Since log₂(x-2) requires x>2, the solution x=-2 is rejected, leaving x=4.

Question 29Algebra Functions

A bacteria population doubles every 3 hours, starting at 500. Using P(t) = 500·2^(t/3), how many hours until the population reaches 4000?

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

Explanation

Set 4000 = 500·2^(t/3), so 2^(t/3) = 8 = 2³. This means t/3 = 3, so t = 9 hours.

Question 30Algebra Functions

The graph of f(x) = x² is shifted 3 units right and 2 units down. What is the new function g(x)?

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

Explanation

A shift right by 3 replaces x with (x-3), and a shift down by 2 subtracts 2 from the result, giving g(x) = (x-3)² - 2.

Question 31Geometry

A right triangle has legs of length 6 and 8. Find the length of the hypotenuse.

  1. 10 Correct answer.
  2. 14
  3. 24
  4. 100

Explanation

By the Pythagorean theorem: c = √(6² + 8²) = √(36+64) = √100 = 10.

Question 32Geometry

Triangle ABC is similar to triangle DEF with a scale factor of 3 (DEF is 3 times larger in linear dimensions). If the area of ABC is 5 cm², what is the area of DEF?

  1. 45 cm² Correct answer.
  2. 15 cm²
  3. 5 cm²
  4. 135 cm²

Explanation

When linear dimensions scale by a factor k, area scales by k². Here k=3, so area scales by 3²=9: 5×9=45 cm².

Question 33Geometry

A rectangular garden measures 12 m by 8 m and has a 2-meter-wide path running around the inside of its border. Find the area of the path.

  1. 64 m² Correct answer.
  2. 32 m²
  3. 36 m²
  4. 88 m²

Explanation

Outer area = 12×8 = 96 m². The inner area (inside the path) has dimensions reduced by 2m on each side: (12-4)×(8-4) = 8×4 = 32 m². Path area = 96 - 32 = 64 m².

Question 34Geometry

A 13-meter ladder leans against a wall, with its base 5 meters from the wall. How high up the wall does the ladder reach?

  1. 12 m Correct answer.
  2. 144 m
  3. 14 m
  4. 8 m

Explanation

By the Pythagorean theorem: height = √(13² - 5²) = √(169-25) = √144 = 12 meters.

Question 35Calculus

Evaluate the definite integral: ∫ from 1 to 3 of 2x dx.

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

Explanation

The antiderivative of 2x is x². Evaluating from 1 to 3: 3² - 1² = 9 - 1 = 8.

Question 36Calculus

Find the area under the curve f(x) = x² between x = 0 and x = 2.

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

Explanation

The antiderivative of x² is x³/3. Evaluating from 0 to 2: (2³/3) - (0³/3) = 8/3 - 0 = 8/3 ≈ 2.67.

Question 37Vectors

Find the equation of the line with slope 2 that passes through the point (0, 5).

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

Explanation

Using slope-intercept form y = mx + b, with slope m=2 and y-intercept b=5 (since the point (0,5) is on the y-axis), the equation is y = 2x + 5.

Question 38Vectors

For what value of k are vectors a = (2, 4) and b = (k, 8) collinear?

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

Explanation

Vectors are collinear when their components are proportional: 2/k = 4/8, so 4k = 16, giving k = 4.

Question 39Vectors

Find the equation of a circle with center (2, -3) and radius 5.

  1. (x-2)² + (y+3)² = 25 Correct answer.
  2. (x+2)² + (y-3)² = 25
  3. (x-2)² + (y+3)² = 5
  4. (x-2)² + (y-3)² = 25

Explanation

The standard circle equation is (x-h)² + (y-k)² = r², where (h,k) is the center. With center (2,-3) and r=5, this gives (x-2)² + (y+3)² = 25.

Question 40Vectors

Find the point of intersection of the lines y = 2x - 1 and y = -x + 5.

  1. (2, 3) Correct answer.
  2. (3, 2)
  3. (2, -3)
  4. (-2, 3)

Explanation

Setting the equations equal: 2x - 1 = -x + 5, so 3x = 6, giving x = 2. Substituting back: y = 2(2) - 1 = 3.

Question 41Vectors

A line has slope 3/4. What is the slope of a line perpendicular to it?

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

Explanation

Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of 3/4 is -4/3.

Question 42Vectors

A circle has center A(1, 2) and passes through point B(3, 3). Find the equation of the line tangent to the circle at point B.

  1. y = -2x + 9 Correct answer.
  2. y = 2x - 3
  3. y = 0.5x + 2.5
  4. y = -2x + 3

Explanation

The radius AB has slope (3-2)/(3-1) = 1/2. A tangent line is perpendicular to the radius at the point of tangency, so its slope is the negative reciprocal, -2. Using point B(3,3): y - 3 = -2(x-3), which simplifies to y = -2x + 9.

Question 43Vectors

Line 1 passes through (0,0) and (5,5). Line 2 passes through (0,6) and (6,0). Find their point of intersection.

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

Explanation

Line 1 has equation y = x. Line 2 has slope -1 and equation y = -x + 6. Setting them equal: x = -x + 6, so 2x = 6, giving x = 3 and y = 3.

Question 44Sequences Series

An arithmetic sequence has first term a₁ = 5 and common difference d = 3. Find the 4th term.

  1. 14 Correct answer.
  2. 17
  3. 11
  4. 8

Explanation

The nth term formula is aₙ = a₁ + (n-1)d. For n=4: a₄ = 5 + 3(3) = 5 + 9 = 14.

Question 45Sequences Series

A geometric sequence has first term 2 and common ratio 3. Find the 3rd term.

  1. 18 Correct answer.
  2. 6
  3. 54
  4. 12

Explanation

The nth term formula is aₙ = a₁·r^(n-1). For n=3: a₃ = 2·3² = 2×9 = 18.

Question 46Sequences Series

An arithmetic sequence has a₁ = 7 and d = -2. Find a₁₀.

  1. -11 Correct answer.
  2. -13
  3. 25
  4. -4

Explanation

Using aₙ = a₁ + (n-1)d: a₁₀ = 7 + 9(-2) = 7 - 18 = -11.

Question 47Sequences Series

Find the sum of the first 20 terms of an arithmetic sequence with a₁ = 3 and d = 4.

  1. 820 Correct answer.
  2. 860
  3. 82
  4. 800

Explanation

Using Sₙ = (n/2)[2a₁ + (n-1)d]: S₂₀ = (20/2)[2(3) + 19(4)] = 10[6 + 76] = 10(82) = 820.

Question 48Sequences Series

Find the sum of the first 4 terms of a geometric sequence with a₁ = 3 and r = 2.

  1. 45 Correct answer.
  2. 21
  3. 48
  4. 90

Explanation

The terms are 3, 6, 12, 24. Using Sₙ = a₁(rⁿ-1)/(r-1): S₄ = 3(2⁴-1)/(2-1) = 3(15)/1 = 45.

Question 49Sequences Series

An arithmetic sequence has a₁ = 100 and d = -5. Find a₁₅.

  1. 30 Correct answer.
  2. 170
  3. 25
  4. 35

Explanation

Using aₙ = a₁ + (n-1)d: a₁₅ = 100 + 14(-5) = 100 - 70 = 30.

Question 50Calculus

Find f'(x) for f(x) = x³ - 4x² + 2.

  1. 3x² - 8x Correct answer.
  2. 3x² - 4x
  3. x² - 8x
  4. 3x² - 8x + 2

Explanation

Using the power rule term by term: the derivative of x³ is 3x², the derivative of -4x² is -8x, and the derivative of the constant 2 is 0. So f'(x) = 3x² - 8x.