I85PREP
⌘ K
Tottori Prefectural Public High School Entrance Examination - Mathematics study materials
Tottori Prefectural Public High School Entrance Examination - Mathematics study resources

Tottori Prefectural Public High School Entrance Examination - Mathematics practice questions

One question at a time. Build your Tottori Prefectural Public High School Entrance Examination - Mathematics understanding.

First 50 of 200 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Numbers and Algebraic Expressions

What is the prime factorisation of 540?

  1. 2³ × 3³ × 5
  2. 2² × 3² × 5²
  3. 2² × 3³ × 5 Correct answer.
  4. 2² × 3³ × 5²

Explanation

540 = 4 × 135 = 2² × 27 × 5 = 2² × 3³ × 5. Dividing successively by primes confirms 540 = 4 × 135 = 4 × 27 × 5.

Question 2Numbers and Algebraic Expressions

Find the greatest common divisor (GCD) of 72 and 108.

  1. 12
  2. 24
  3. 36 Correct answer.
  4. 18

Explanation

72 = 2³ × 3² and 108 = 2² × 3³. The GCD takes the minimum power of each common prime: 2² × 3² = 4 × 9 = 36.

Question 3Numbers and Algebraic Expressions

Find the least common multiple (LCM) of 18 and 30.

  1. 30
  2. 90 Correct answer.
  3. 180
  4. 60

Explanation

18 = 2 × 3² and 30 = 2 × 3 × 5. The LCM takes the highest power of each prime: 2 × 3² × 5 = 90.

Question 4Numbers and Algebraic Expressions

Evaluate (-2)³.

  1. 6
  2. -8 Correct answer.
  3. 8
  4. -6

Explanation

(-2)³ = (-2) × (-2) × (-2) = 4 × (-2) = -8. An odd power of a negative number is negative.

Question 5Numbers and Algebraic Expressions

Evaluate 7 - (-3) + (-4).

  1. 6 Correct answer.
  2. 10
  3. -6
  4. -10

Explanation

Subtracting a negative is addition: 7 - (-3) = 7 + 3 = 10, then 10 + (-4) = 6.

Question 6Numbers and Algebraic Expressions

Simplify √12 + √27.

  1. √39
  2. 5√9
  3. 6√3
  4. 5√3 Correct answer.

Explanation

√12 = 2√3 and √27 = 3√3, so √12 + √27 = 2√3 + 3√3 = 5√3. Only like surds can be combined.

Question 7Numbers and Algebraic Expressions

Which of these numbers is irrational (無理数)?

  1. √16
  2. 0.25
  3. 22/7
  4. √5 Correct answer.

Explanation

An irrational number cannot be written as a fraction of two integers, and its decimal neither terminates nor repeats. 5 is not a perfect square, so √5 = 2.2360... never terminates or repeats. A square root is rational only when the number under it is a perfect square.

Question 8Numbers and Algebraic Expressions

Simplify (-2a²b)³.

  1. -8a⁵b²
  2. -8a⁶b³ Correct answer.
  3. 8a⁶b³
  4. -6a⁵b³

Explanation

Each factor is cubed: (-2)³ = -8, (a²)³ = a⁶, and (b)³ = b³, giving -8a⁶b³. Powers are multiplied when raised to a power.

Question 9Numbers and Algebraic Expressions

Simplify 3x² × 4x³.

  1. 12x⁶
  2. 7x⁶
  3. 12x⁵ Correct answer.
  4. 7x⁵

Explanation

Multiply the coefficients 3 × 4 = 12 and add the exponents x² · x³ = x⁵, giving 12x⁵.

Question 10Numbers and Algebraic Expressions

Expand (x + 3)(x + 5).

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

Explanation

Using FOIL: x·x = x², x·5 + 3·x = 5x + 3x = 8x, and 3·5 = 15, giving x² + 8x + 15.

Question 11Numbers and Algebraic Expressions

Expand (x + 4)².

  1. x² + 8x + 16 Correct answer.
  2. x² + 16
  3. x² + 8x + 8
  4. x² + 4x + 16

Explanation

(x + 4)² = (x + 4)(x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16. The middle term is 2 × x × 4 = 8x.

Question 12Numbers and Algebraic Expressions

Expand (2x - 1)(3x + 2).

  1. 6x² + 5x - 2
  2. 6x² + x + 2
  3. 6x² - x - 2
  4. 6x² + x - 2 Correct answer.

Explanation

FOIL: 2x·3x = 6x², 2x·2 = 4x, (-1)·3x = -3x, (-1)·2 = -2. The middle terms combine to 4x - 3x = x, giving 6x² + x - 2.

Question 13Numbers and Algebraic Expressions

Factorise x² + 5x + 6.

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

Explanation

Find two numbers that multiply to 6 and add to 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3).

Question 14Numbers and Algebraic Expressions

Factorise x² - 7x + 12.

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

Explanation

Two numbers that multiply to +12 and add to -7 are -3 and -4. So x² - 7x + 12 = (x - 3)(x - 4).

Question 15Numbers and Algebraic Expressions

Factorise 2x² + 5x - 3.

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

Explanation

Check (2x - 1)(x + 3): 2x² + 6x - x - 3 = 2x² + 5x - 3. The cross terms 6x and -x combine to 5x, confirming the factorisation.

Question 16Numbers and Algebraic Expressions

Solve for x: 2x - 7 = 9.

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

Explanation

Add 7 to both sides: 2x = 16, then divide by 2: x = 8. Checking: 2(8) - 7 = 9.

Question 17Numbers and Algebraic Expressions

Solve for x: 5x + 2 = 3x + 10.

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

Explanation

Subtract 3x: 2x + 2 = 10. Subtract 2: 2x = 8. Divide by 2: x = 4. Check: 5(4)+2 = 22 = 3(4)+10.

Question 18Numbers and Algebraic Expressions

Solve the simultaneous equations x + y = 9 and 2x - y = 6. What is x?

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

Explanation

Adding the equations eliminates y: (x + y) + (2x - y) = 9 + 6, so 3x = 15 and x = 5. Then y = 9 - 5 = 4.

Question 19Numbers and Algebraic Expressions

Solve the simultaneous equations 3x + 2y = 16 and x - y = 2. What is y?

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

Explanation

From x - y = 2, x = y + 2. Substitute: 3(y + 2) + 2y = 16, so 5y + 6 = 16, 5y = 10, and y = 2 (with x = 4).

Question 20Numbers and Algebraic Expressions

Solve x² - 5x + 6 = 0.

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

Explanation

Factorise: (x - 2)(x - 3) = 0, so x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.

Question 21Numbers and Algebraic Expressions

Solve x² + 4x - 5 = 0.

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

Explanation

Factorise: (x + 5)(x - 1) = 0, so x = -5 or x = 1. The product is -5 and the sum is 4.

Question 22Numbers and Algebraic Expressions

Using the quadratic formula, solve 2x² - 3x - 2 = 0.

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

Explanation

x = (3 ± √(9 + 16))/4 = (3 ± √25)/4 = (3 ± 5)/4, giving x = 8/4 = 2 or x = -2/4 = -1/2.

Question 23Numbers and Algebraic Expressions

Rearrange a = (2b + c)/3 to make b the subject.

  1. b = (3a + c)/2
  2. b = (a - c)/2
  3. b = (2a - c)/3
  4. b = (3a - c)/2 Correct answer.

Explanation

Multiply by 3: 3a = 2b + c. Subtract c: 3a - c = 2b. Divide by 2: b = (3a - c)/2.

Question 24Linear Equations and Functions

In the direct proportion y = 3x, what is y when x = 5?

  1. 8
  2. -15
  3. 15 Correct answer.
  4. 12

Explanation

Direct proportion means y = kx with k = 3, so y = 3 × 5 = 15.

Question 25Linear Equations and Functions

Which of these is a linear function (一次関数)?

  1. y = x²
  2. y = √x
  3. y = 2x + 1 Correct answer.
  4. y = 1/x

Explanation

A linear function has the form y = ax + b with x to the first power. y = 2x + 1 fits; the others are quadratic, inverse, and square-root relations.

Question 26Linear Equations and Functions

What is the slope (傾き) of the line y = 3x - 2?

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

Explanation

In y = ax + b, the coefficient a is the slope. Here a = 3, so the slope is 3.

Question 27Linear Equations and Functions

What is the y-intercept (切片) of y = -2x + 5?

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

Explanation

In y = ax + b, the constant b is the y-intercept. Here b = 5, so the line crosses the y-axis at (0, 5).

Question 28Linear Equations and Functions

A direct proportion y = kx passes through (2, 10). What is k?

  1. 5 Correct answer.
  2. 10
  3. 2
  4. 20

Explanation

Substitute x = 2, y = 10: 10 = k × 2, so k = 5.

Question 29Linear Equations and Functions

Find the linear function passing through (1, 5) and (3, 11).

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

Explanation

Slope = (11 - 5)/(3 - 1) = 6/2 = 3. Using (1, 5): 5 = 3(1) + b, so b = 2, giving y = 3x + 2.

Question 30Linear Equations and Functions

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

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

Explanation

Slope = (change in y)/(change in x) = (11 - 3)/(6 - 2) = 8/4 = 2.

Question 31Linear Equations and Functions

Evaluate y = -2x + 7 at x = -3.

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

Explanation

y = -2(-3) + 7 = 6 + 7 = 13. The product of two negatives is positive.

Question 32Linear Equations and Functions

An inverse proportion y = k/x passes through (4, 3). What is y when x = 6?

  1. 2 Correct answer.
  2. 8
  3. 6
  4. 18

Explanation

First find k: k = xy = 4 × 3 = 12. Then at x = 6, y = 12/6 = 2.

Question 33Quadratic Functions and Equations

Evaluate y = 2x² at x = -3.

  1. 18 Correct answer.
  2. 12
  3. -18
  4. 36

Explanation

y = 2(-3)² = 2 × 9 = 18. The square removes the negative sign before multiplying by 2.

Question 34Quadratic Functions and Equations

What are the coordinates of the vertex (頂点) of y = 3x²?

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

Explanation

y = ax² has its vertex at the origin (0, 0), the minimum point when a > 0.

Question 35Quadratic Functions and Equations

Which direction does the parabola y = -x² open?

  1. Downward (opens down) Correct answer.
  2. To the right
  3. To the left
  4. Upward (opens up)

Explanation

When the coefficient of x² is negative, the parabola opens downward and has a maximum at its vertex.

Question 36Quadratic Functions and Equations

What is the axis of symmetry (軸) of y = 2x²?

  1. y = 0
  2. x = 2
  3. y = 2
  4. x = 0 Correct answer.

Explanation

y = ax² is symmetric about the y-axis, so the axis of symmetry is the line x = 0.

Question 37Quadratic Functions and Equations

Find the intersection points of y = x² and y = x + 2.

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

Explanation

Set equal: x² = x + 2, so x² - x - 2 = 0, (x - 2)(x + 1) = 0, giving x = 2 or x = -1. The points are (2, 4) and (-1, 1).

Question 38Quadratic Functions and Equations

At x = 3, which is larger: y = x² or y = 2x + 1?

  1. They are equal
  2. y = x² (value 9) Correct answer.
  3. y = 2x + 1 (value 7)
  4. Cannot be determined

Explanation

At x = 3, y = x² = 9 and y = 2x + 1 = 7. Since 9 > 7, the parabola is higher there.

Question 39Quadratic Functions and Equations

A parabola y = ax² passes through (2, 12). What is a?

  1. 3 Correct answer.
  2. 48
  3. 6
  4. 12

Explanation

Substitute x = 2, y = 12: 12 = a(2)² = 4a, so a = 3.

Question 40Quadratic Functions and Equations

What is the maximum value of y = -2x²?

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

Explanation

Since a = -2 < 0, the parabola opens downward with its maximum at the vertex (0, 0), so the maximum value is 0.

Question 41Linear Equations and Functions

A linear function y = ax + b satisfies y = 1 when x = 2 and y = 7 when x = 4. Find the function.

  1. y = 7x - 13
  2. y = 3x - 5 Correct answer.
  3. y = 2x - 3
  4. y = 3x + 5

Explanation

Slope = (7 - 1)/(4 - 2) = 6/2 = 3. Using (2, 1): 1 = 3(2) + b, so b = -5, giving y = 3x - 5.

Question 42Geometry: Congruence, Similarity and Measurement

What is the volume of a sphere of radius 3?

  1. 27π
  2. 108π
  3. 36π Correct answer.
  4. 12π

Explanation

The volume of a sphere is (4/3)πr³. With r = 3, r³ = 27, so the volume is (4/3) × π × 27 = 36π. Remember the pair: surface area 4πr² and volume (4/3)πr³.

Question 43Quadratic Functions and Equations

How many points do y = x² and y = 2x - 1 share, and what is the shared point?

  1. No intersection
  2. (0, -1)
  3. One point, (1, 1) Correct answer.
  4. Two points

Explanation

Set equal: x² = 2x - 1, so x² - 2x + 1 = 0, (x - 1)² = 0, giving x = 1 only. The line is tangent to the parabola at (1, 1).

Question 44Linear Equations and Functions

The sequence 2, 6, 12, 20, 30, ... follows aₙ = n(n + 1). What is the 10th term?

  1. 90
  2. 100
  3. 110 Correct answer.
  4. 120

Explanation

Using aₙ = n(n + 1), the 10th term is 10 × 11 = 110.

Question 45Quadratic Functions and Equations

The line y = 2x and the parabola y = x² meet at (0, 0) and (2, 4). What is the vertical gap between them at x = 1?

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

Explanation

At x = 1, the line gives y = 2 and the parabola gives y = 1, so the vertical gap is 2 - 1 = 1.

Question 46Geometry: Plane Figures and Angles

A right triangle has legs of length 3 and 4. What is the length of the hypotenuse?

  1. 5 Correct answer.
  2. 6
  3. 7
  4. 12

Explanation

By the Pythagorean theorem (三平方の定理), c² = 3² + 4² = 9 + 16 = 25, so c = 5.

Question 47Geometry: Plane Figures and Angles

A right triangle has hypotenuse 13 and one leg 5. What is the other leg?

  1. 12 Correct answer.
  2. √194
  3. 8
  4. 18

Explanation

b² = 13² - 5² = 169 - 25 = 144, so b = 12.

Question 48Geometry: Plane Figures and Angles

An angle inscribed in a semicircle (diameter as the base) measures:

  1. 60°
  2. 90° Correct answer.
  3. 180°
  4. 45°

Explanation

By the inscribed-angle theorem (円周角の定理), an angle subtending the diameter is half the central angle of 180°, so it is 90°.

Question 49Geometry: Plane Figures and Angles

The sum of the interior angles of a quadrilateral is:

  1. 540°
  2. 360° Correct answer.
  3. 270°
  4. 180°

Explanation

A quadrilateral splits into two triangles, so its interior angles total 2 × 180° = 360°.

Question 50Geometry: Plane Figures and Angles

A rectangle measures 6 by 8. What is its perimeter?

  1. 28 Correct answer.
  2. 48
  3. 14
  4. 20

Explanation

Perimeter = 2 × (length + width) = 2 × (6 + 8) = 2 × 14 = 28.