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

Wakayama Prefectural Public High School Entrance Examination - Mathematics practice questions

One question at a time. Build your Wakayama 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 84?

  1. 2² × 3² × 7
  2. 2² × 3 × 7 Correct answer.
  3. 2³ × 3 × 7
  4. 2 × 3² × 7

Explanation

84 ÷ 2 = 42; 42 ÷ 2 = 21; 21 = 3 × 7. So 84 = 2² × 3 × 7.

Question 2Numbers and Algebraic Expressions

Find the greatest common divisor (GCD) of 48 and 72.

  1. 12
  2. 36
  3. 24 Correct answer.
  4. 16

Explanation

48 = 2⁴ × 3, 72 = 2³ × 3². GCD = 2³ × 3 = 8 × 3 = 24.

Question 3Numbers and Algebraic Expressions

Solve the simultaneous equations x + y = 10 and 2x + y = 14. What is the value of x?

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

Explanation

Subtract equation 1 from equation 2: (2x + y) − (x + y) = 14 − 10 → x = 4. Then y = 10 − 4 = 6.

Question 4Numbers and Algebraic Expressions

Find the largest integer x satisfying the inequality 2x − 3 < 7.

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

Explanation

2x − 3 < 7 → 2x < 10 → x < 5. The largest integer less than 5 is 4.

Question 5Numbers and Algebraic Expressions

Simplify (2a²b)³.

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

Explanation

(2a²b)³ = 2³ × (a²)³ × b³ = 8 × a⁶ × b³ = 8a⁶b³.

Question 6Numbers and Algebraic Expressions

Compute √12 + √27 and express in simplest form.

  1. √39
  2. 2√3 + 3√3
  3. 3√5
  4. 5√3 Correct answer.

Explanation

√12 = 2√3 and √27 = 3√3. So √12 + √27 = 2√3 + 3√3 = 5√3.

Question 7Numbers and Algebraic Expressions

Expand (x + 3)(x − 5).

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

Explanation

(x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15.

Question 8Numbers and Algebraic Expressions

Factorise x² − 9.

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

Explanation

x² − 9 = (x)² − (3)² = (x + 3)(x − 3), a difference of two squares.

Question 9Numbers and Algebraic Expressions

Factorise x² + 5x + 6.

  1. (x + 1)(x + 6)
  2. (x + 5)(x + 1)
  3. (x − 2)(x − 3)
  4. (x + 2)(x + 3) Correct answer.

Explanation

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

Question 10Numbers and Algebraic Expressions

Solve 2(x − 4) = 3x + 1 for x.

  1. x = −3
  2. x = 9
  3. x = −9 Correct answer.
  4. x = 3

Explanation

2(x − 4) = 3x + 1 → 2x − 8 = 3x + 1 → −8 − 1 = 3x − 2x → −9 = x, so x = −9.

Question 11Numbers and Algebraic Expressions

The sum of two numbers is 15 and their difference is 3. What is the larger number?

  1. 12
  2. 6
  3. 8
  4. 9 Correct answer.

Explanation

Let x + y = 15 and x − y = 3. Adding: 2x = 18 → x = 9. Then y = 6. The larger number is 9.

Question 12Numbers and Algebraic Expressions

Find the smallest integer x satisfying 3x − 2 ≥ 2x + 5.

  1. 8
  2. 5
  3. 6
  4. 7 Correct answer.

Explanation

3x − 2 ≥ 2x + 5 → 3x − 2x ≥ 5 + 2 → x ≥ 7. The smallest integer is 7.

Question 13Numbers and Algebraic Expressions

Compute (3 × 10⁴) × (2 × 10³) and express in scientific notation.

  1. 6 × 10⁷ Correct answer.
  2. 5 × 10⁷
  3. 6 × 10¹
  4. 6 × 10¹²

Explanation

(3 × 10⁴) × (2 × 10³) = (3 × 2) × 10^(4+3) = 6 × 10⁷.

Question 14Numbers and Algebraic Expressions

y is inversely proportional to x. When x = 3, y = 4. Find y when x = 6.

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

Explanation

Inverse proportion: xy = k. With x = 3, y = 4: k = 12. When x = 6: y = 12 ÷ 6 = 2.

Question 15Numbers and Algebraic Expressions

Find the sum of the solutions of x² − 5x + 6 = 0.

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

Explanation

Factor: (x − 2)(x − 3) = 0, so x = 2 or x = 3. Sum = 2 + 3 = 5. (Also by Vieta: sum = −(−5)/1 = 5.)

Question 16Numbers and Algebraic Expressions

Simplify (x² − 4)/(x − 2) for x ≠ 2, then evaluate at x = 5.

  1. 3
  2. 7 Correct answer.
  3. 5
  4. 21

Explanation

x² − 4 = (x − 2)(x + 2), so (x² − 4)/(x − 2) = x + 2 (for x ≠ 2). At x = 5: 5 + 2 = 7.

Question 17Numbers and Algebraic Expressions

An item priced at 8,000 yen is discounted by 15%. What is the sale price?

  1. 6,000 yen
  2. 7,000 yen
  3. 7,200 yen
  4. 6,800 yen Correct answer.

Explanation

Discount = 8,000 × 0.15 = 1,200 yen. Sale price = 8,000 − 1,200 = 6,800 yen. (Equivalently 8,000 × 0.85 = 6,800.)

Question 18Linear Equations and Functions

y is directly proportional to x. When x = 3, y = 15. Find y when x = 7.

  1. 45
  2. 21
  3. 35 Correct answer.
  4. 25

Explanation

y = kx; 15 = 3k → k = 5. So y = 5 × 7 = 35.

Question 19Linear Equations and Functions

y is inversely proportional to x. When x = 4, y = 12. Find y when x = 3.

  1. 16 Correct answer.
  2. 9
  3. 4
  4. 36

Explanation

xy = k, so k = 4 × 12 = 48. When x = 3: y = 48 ÷ 3 = 16.

Question 20Quadratic Functions and Equations

For the function y = ax², the average rate of change (変化の割合) as x increases from 1 to 3 is 8. Find the value of a.

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

Explanation

For y = ax², the average rate of change between x = p and x = q is (aq² − ap²)/(q − p) = a(p + q). Here that is a(1 + 3) = 4a, and 4a = 8 gives a = 2. Check: at x = 1, y = 2; at x = 3, y = 18; (18 − 2)/(3 − 1) = 8.

Question 21Quadratic Functions and Equations

For the function y = x², what is the range of values of y when x varies over −2 ≤ x ≤ 1?

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

Explanation

The interval −2 ≤ x ≤ 1 contains x = 0, so the smallest value of y is 0² = 0. The largest value comes from the endpoint furthest from 0, which is x = −2, giving y = (−2)² = 4. The range is therefore 0 ≤ y ≤ 4.

Question 22Quadratic Functions and Equations

Find the x-intercepts of y = x² − 9.

  1. x = 3 and x = −3 Correct answer.
  2. x = 9 and x = −9
  3. x = 0 only
  4. No real x-intercepts

Explanation

Set y = 0: x² − 9 = 0 → x² = 9 → x = ±3. The intercepts are at x = 3 and x = −3.

Question 23Quadratic Functions and Equations

Find the intersection point in the first quadrant of y = x² and y = 2x + 3.

  1. (1, 1)
  2. (3, 9) Correct answer.
  3. (3, 5)
  4. (−1, 1)

Explanation

Set x² = 2x + 3 → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0 → x = 3 or x = −1. In the first quadrant x = 3, y = 9. Point: (3, 9).

Question 24Linear Equations and Functions

Find the equation of the line passing through (1, 2) and (3, 6).

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

Explanation

Slope = (6 − 2)/(3 − 1) = 4/2 = 2. Using (1, 2): y = 2x + b → 2 = 2(1) + b → b = 0. So y = 2x.

Question 25Linear Equations and Functions

What is the y-intercept of the line y = 3x − 4?

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

Explanation

The y-intercept is the constant term when the line is in form y = mx + b. Here b = −4.

Question 26Linear Equations and Functions

Find the equation of the line parallel to y = 2x + 1 that passes through the point (1, 5).

  1. y = 2x + 3 Correct answer.
  2. y = 2x − 1
  3. y = 2x + 1
  4. y = −½x + 5

Explanation

Parallel lines have equal slopes, so the new line has slope 2 and takes the form y = 2x + b. Substituting the point (1, 5) gives 5 = 2(1) + b, so b = 3. The line is y = 2x + 3, which is distinct from y = 2x + 1.

Question 27Linear Equations and Functions

A straight line passes through the points (0, 5) and (4, −3). What is its equation?

  1. y = −2x + 5 Correct answer.
  2. y = −½x + 5
  3. y = −2x − 3
  4. y = 2x + 5

Explanation

The slope is the change in y divided by the change in x: (−3 − 5)/(4 − 0) = −8/4 = −2. The line crosses the y-axis at (0, 5), so the intercept is 5, giving y = −2x + 5. Check at x = 4: −2(4) + 5 = −3.

Question 28Quadratic Functions and Equations

For the parabola y = −2x², what is the maximum value of y?

  1. −2
  2. 0 Correct answer.
  3. 2
  4. No maximum

Explanation

Since a = −2 < 0, the parabola opens downward and the vertex at x = 0 gives the maximum y = 0.

Question 29Linear Equations and Functions

A rectangle has width x and length (10 − x). For what value of x is the area A = x(10 − x) maximised, and what is that maximum area?

  1. x = 5, A = 25 Correct answer.
  2. x = 5, A = 10
  3. x = 3, A = 21
  4. x = 4, A = 24

Explanation

A = x(10 − x) = 10x − x² = −(x² − 10x) = −(x − 5)² + 25. The maximum is at x = 5 with A = 25.

Question 30Linear Equations and Functions

In an inverse proportion, p = 4 when q = 6. Find q when p = 8.

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

Explanation

pq = k, so k = 4 × 6 = 24. When p = 8: q = 24 ÷ 8 = 3.

Question 31Linear Equations and Functions

What are the coordinates of the point (2, 3) reflected across the x-axis?

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

Explanation

Reflection across the x-axis negates the y-coordinate: (x, y) → (x, −y). So (2, 3) → (2, −3).

Question 32Quadratic Functions and Equations

The graph of y = x² is transformed to y = (x − 1)². How does the graph shift?

  1. Right by 1 unit Correct answer.
  2. Up by 1 unit
  3. Left by 1 unit
  4. Down by 1 unit

Explanation

Replacing x with (x − 1) shifts the graph right by 1 unit. The vertex moves from (0, 0) to (1, 0).

Question 33Quadratic Functions and Equations

For the function y = −3x², find the average rate of change (変化の割合) as x increases from 2 to 4.

  1. −3
  2. −36
  3. −18 Correct answer.
  4. 18

Explanation

At x = 2, y = −3(4) = −12; at x = 4, y = −3(16) = −48. The average rate of change is (−48 − (−12))/(4 − 2) = −36/2 = −18. For y = ax² this always equals a(p + q), here −3(2 + 4) = −18.

Question 34Quadratic Functions and Equations

Points A and B lie on the parabola y = x², where A has x-coordinate −1 and B has x-coordinate 2. Find the equation of the straight line AB.

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

Explanation

Because both points lie on y = x², A is (−1, 1) and B is (2, 4). The slope of AB is (4 − 1)/(2 − (−1)) = 3/3 = 1. Substituting B into y = x + b gives 4 = 2 + b, so b = 2 and the line is y = x + 2.

Question 35Quadratic Functions and Equations

For y = x² − 3x, what is the product of the roots?

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

Explanation

Factor: y = x(x − 3), roots x = 0 and x = 3. Product = 0 × 3 = 0. (By Vieta: product = c/a = 0/1 = 0.)

Question 36Geometry: Congruence, Similarity and Measurement

A right triangle has legs of length 3 cm and 4 cm. What is the length of the hypotenuse (三平方の定理)?

  1. 7 cm
  2. 25 cm
  3. 12 cm
  4. 5 cm Correct answer.

Explanation

By Pythagoras: c² = 3² + 4² = 9 + 16 = 25, so c = 5 cm.

Question 37Geometry: Plane Figures and Angles

A right triangle has legs of length 5 cm and 12 cm. Find the hypotenuse.

  1. 15 cm
  2. 13 cm Correct answer.
  3. 17 cm
  4. 60 cm

Explanation

c² = 5² + 12² = 25 + 144 = 169, so c = 13 cm.

Question 38Geometry: Plane Figures and Angles

In a right triangle, the hypotenuse is 10 cm and one leg is 6 cm. Find the length of the other leg.

  1. 6 cm
  2. √136 cm
  3. 8 cm Correct answer.
  4. 4 cm

Explanation

a² + 6² = 10² → a² = 100 − 36 = 64 → a = 8 cm.

Question 39Geometry: Congruence, Similarity and Measurement

Two similar triangles have a side-length ratio of 2 : 3. If the smaller triangle has area 8 cm², what is the area of the larger triangle?

  1. 12 cm²
  2. 27 cm²
  3. 18 cm² Correct answer.
  4. 16 cm²

Explanation

Area ratio = (side ratio)² = (2/3)² = 4/9. So larger area = 8 × (9/4) = 18 cm².

Question 40Geometry: Plane Figures and Angles

Two angles of a triangle measure 50° and 60°. What is the measure of the third angle?

  1. 60°
  2. 80°
  3. 70° Correct answer.
  4. 110°

Explanation

Angles in a triangle sum to 180°. Third angle = 180 − 50 − 60 = 70°.

Question 41Geometry: Plane Figures and Angles

Two parallel lines are cut by a transversal. If a corresponding angle measures 65°, what is the measure of the alternate interior angle?

  1. 25°
  2. 65° Correct answer.
  3. 130°
  4. 115°

Explanation

When parallel lines are cut by a transversal, alternate interior angles are equal, and corresponding angles are equal. So the alternate interior angle is also 65°.

Question 42Geometry: Plane Figures and Angles

A circle has diameter 12 cm. What is its circumference (use π)?

  1. 6π cm
  2. 144π cm
  3. 24π cm
  4. 12π cm Correct answer.

Explanation

Circumference = πd = π × 12 = 12π cm. (Or 2πr = 2π(6) = 12π.)

Question 43Geometry: Plane Figures and Angles

A sector of a circle has central angle 90° and radius 4 cm. Find the sector's area (use π).

  1. 4π cm² Correct answer.
  2. 2π cm²
  3. 8π cm²
  4. 16π cm²

Explanation

Sector area = (θ/360) × πr² = (90/360) × π × 16 = (1/4) × 16π = 4π cm².

Question 44Geometry: Plane Figures and Angles

A sector of a circle has central angle 60° and radius 6 cm. Find the arc length (use π).

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

Explanation

Arc length = (θ/360) × 2πr = (60/360) × 2π × 6 = (1/6) × 12π = 2π cm.

Question 45Geometry: Congruence, Similarity and Measurement

A cylinder has radius 2 cm and height 4 cm. Find its total surface area (use π).

  1. 32π cm²
  2. 16π cm²
  3. 24π cm² Correct answer.
  4. 20π cm²

Explanation

Total SA = 2πr² + 2πrh = 2π(4) + 2π(2)(4) = 8π + 16π = 24π cm².

Question 46Geometry: Plane Figures and Angles

In a circle, a central angle measures 80°. What is the measure of the inscribed angle (円周角) subtending the same arc?

  1. 80°
  2. 20°
  3. 160°
  4. 40° Correct answer.

Explanation

By the inscribed angle theorem (円周角の定理), the inscribed angle is half the central angle subtending the same arc: 80° ÷ 2 = 40°.

Question 47Geometry: Plane Figures and Angles

A tangent line touches a circle at point P, and O is the centre. What is the angle between the tangent and the radius OP?

  1. 45°
  2. 0°
  3. 90° Correct answer.
  4. 60°

Explanation

A tangent to a circle is perpendicular to the radius at the point of contact, so the angle is 90°.

Question 48Geometry: Plane Figures and Angles

A trapezoid has parallel sides of 6 cm and 10 cm and height 4 cm. Find its area.

  1. 40 cm²
  2. 32 cm² Correct answer.
  3. 24 cm²
  4. 16 cm²

Explanation

Area of trapezoid = (1/2)(a + b)h = (1/2)(6 + 10)(4) = (1/2)(16)(4) = 32 cm².

Question 49Geometry: Plane Figures and Angles

A triangle has base 10 cm and height 6 cm. Find its area.

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

Explanation

Area = (1/2) × base × height = (1/2) × 10 × 6 = 30 cm².

Question 50Geometry: Plane Figures and Angles

A rhombus has diagonals of length 6 cm and 8 cm. Find its area.

  1. 48 cm²
  2. 24 cm² Correct answer.
  3. 12 cm²
  4. 14 cm²

Explanation

Area of rhombus = (1/2) × d₁ × d₂ = (1/2) × 6 × 8 = 24 cm².