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Oita Public High School Entrance Examination - Mathematics study materials
Oita Public High School Entrance Examination - Mathematics study resources

Oita Public High School Entrance Examination - Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate: 12 + (−7) − (−5).

  1. 10 Correct answer.
  2. 0
  3. −10
  4. 20

Explanation

12 + (−7) − (−5) = 12 − 7 + 5 = 10. The double negative on the 5 becomes addition.

Question 2Numbers and Algebraic Expressions

Calculate: (−3)² − 8.

  1. 1 Correct answer.
  2. −1
  3. −17
  4. 17

Explanation

(−3)² = 9, so 9 − 8 = 1. The square of a negative number is positive.

Question 3Numbers and Algebraic Expressions

Simplify: −(−2)³.

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

Explanation

(−2)³ = −8, so −(−8) = 8. The outer minus sign flips the sign of the cube.

Question 4Numbers and Algebraic Expressions

Compute: 3/4 + 2/3.

  1. 5/12
  2. 17/12 Correct answer.
  3. 5/7
  4. 1/2

Explanation

Using LCD 12: 3/4 = 9/12 and 2/3 = 8/12, so 9/12 + 8/12 = 17/12.

Question 5Numbers and Algebraic Expressions

Compute: 5/6 − 1/4.

  1. 7/12 Correct answer.
  2. 5/24
  3. 1/12
  4. 4/2

Explanation

Using LCD 12: 5/6 = 10/12 and 1/4 = 3/12, so 10/12 − 3/12 = 7/12.

Question 6Numbers and Algebraic Expressions

Multiply: (−4/7) × (3/8).

  1. −1/14
  2. −12/56
  3. 3/14
  4. −3/14 Correct answer.

Explanation

Multiply numerators and denominators: (−4×3)/(7×8) = −12/56 = −3/14 after dividing top and bottom by 4.

Question 7Numbers and Algebraic Expressions

Solve x² − 4x + 1 = 0 using the quadratic formula (解の公式).

  1. x = 2 ± √3 Correct answer.
  2. x = 4 ± √3
  3. x = 2 ± √5
  4. x = −2 ± √3

Explanation

With a = 1, b = −4, c = 1: x = (4 ± √(16−4))/2 = (4 ± √12)/2 = (4 ± 2√3)/2 = 2 ± √3.

Question 8Numbers and Algebraic Expressions

Solve the simultaneous equations: 2x + y = 7, x − y = 2.

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

Explanation

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

Question 9Numbers and Algebraic Expressions

Expand: 3(x − 2) − 2(x + 5).

  1. 5x − 16
  2. x − 4
  3. x + 4
  4. x − 16 Correct answer.

Explanation

3x − 6 − 2x − 10 = (3x − 2x) + (−6 − 10) = x − 16. Distribute both negatives carefully.

Question 10Numbers and Algebraic Expressions

Factor: x² − 9.

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

Explanation

This is a difference of squares: x² − 9 = x² − 3² = (x − 3)(x + 3).

Question 11Numbers and Algebraic Expressions

Factor: x² + 5x + 6.

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

Explanation

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

Question 12Numbers and Algebraic Expressions

Solve: 3x − 5 = 2x + 4.

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

Explanation

Subtract 2x from both sides: x − 5 = 4. Then add 5: x = 9.

Question 13Numbers and Algebraic Expressions

If a = −2 and b = 3, evaluate a² + 2ab.

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

Explanation

a² = (−2)² = 4, and 2ab = 2(−2)(3) = −12, so a² + 2ab = 4 − 12 = −8.

Question 14Numbers and Algebraic Expressions

Solve the simultaneous equations: 3x + 2y = 12, x − 2y = 4.

  1. x = 4, y = 2
  2. x = 4, y = 0 Correct answer.
  3. x = 2, y = 3
  4. x = 0, y = 6

Explanation

Adding eliminates y: 4x = 16, so x = 4. Substituting: 3(4) + 2y = 12 gives 12 + 2y = 12, so y = 0.

Question 15Numbers and Algebraic Expressions

A shop discounts an item marked 1,800 yen by 20%. What is the sale price?

  1. 1,600 yen
  2. 1,440 yen Correct answer.
  3. 1,560 yen
  4. 1,360 yen

Explanation

A 20% discount removes 0.20 × 1,800 = 360 yen, so the sale price is 1,800 − 360 = 1,440 yen.

Question 16Numbers and Algebraic Expressions

Solve: 5(x − 3) = 2x + 6.

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

Explanation

Expand: 5x − 15 = 2x + 6. Subtract 2x: 3x − 15 = 6. Add 15: 3x = 21, so x = 7.

Question 17Numbers and Algebraic Expressions

Which expression is equivalent to (x + 2)(x − 2) − x²?

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

Explanation

(x + 2)(x − 2) = x² − 4 by difference of squares. So x² − 4 − x² = −4.

Question 18Numbers and Algebraic Expressions

Evaluate √72 / √2.

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

Explanation

√72/√2 = √(72/2) = √36 = 6. Equivalently, √72 = 6√2, so 6√2/√2 = 6.

Question 19Numbers and Algebraic Expressions

Solve x²/4 = 9.

  1. x = ±6 Correct answer.
  2. x = 6
  3. x = ±12
  4. x = ±3

Explanation

Multiply both sides by 4: x² = 36, so x = ±6. Both signs satisfy the original equation.

Question 20Numbers and Algebraic Expressions

Find the value of x satisfying 2(x + 3) = 3(x − 1) + 1.

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

Explanation

Expand: 2x + 6 = 3x − 3 + 1 = 3x − 2. Subtract 2x: 6 = x − 2. Add 2: x = 8.

Question 21Numbers and Algebraic Expressions

Which is the largest? 3/5, 5/8, 2/3, 7/12.

  1. 3/5
  2. 5/8
  3. 7/12
  4. 2/3 Correct answer.

Explanation

Using LCD 120: 3/5 = 72/120, 5/8 = 75/120, 2/3 = 80/120, 7/12 = 70/120. The largest is 2/3.

Question 22Linear Equations and Functions

A function is given by y = 3x − 4. What is y when x = −2?

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

Explanation

Substitute x = −2: y = 3(−2) − 4 = −6 − 4 = −10.

Question 23Linear Equations and Functions

Which point lies on the line y = 2x + 1?

  1. (2, 5) Correct answer.
  2. (1, 2)
  3. (2, 4)
  4. (0, 2)

Explanation

Substitute x = 2: y = 2(2) + 1 = 5. The point (2, 5) lies on the line.

Question 24Linear Equations and Functions

The line y = ax + b passes through (1, 5) and (3, 11). Find a and b.

  1. a = 6, b = −1
  2. a = 3, b = 2 Correct answer.
  3. a = 2, b = 3
  4. a = 3, b = 8

Explanation

Slope a = (11 − 5)/(3 − 1) = 6/2 = 3. Then b = 5 − 3(1) = 2. So a = 3, b = 2.

Question 25Linear Equations and Functions

If y is directly proportional to x and y = 12 when x = 4, what is y when x = 7?

  1. 28
  2. 3
  3. 16
  4. 21 Correct answer.

Explanation

y = kx, so 12 = 4k gives k = 3. Then y = 3(7) = 21.

Question 26Linear Equations and Functions

If y is inversely proportional to x and y = 6 when x = 2, what is y when x = 4?

  1. 12
  2. 3 Correct answer.
  3. 1.5
  4. 8

Explanation

y = k/x, so 6 = k/2 gives k = 12. Then y = 12/4 = 3.

Question 27Quadratic Functions and Equations

For the function y = x², find the range of y when the domain is −2 ≤ x ≤ 3.

  1. −2 ≤ y ≤ 3
  2. 0 ≤ y ≤ 9 Correct answer.
  3. 0 ≤ y ≤ 4
  4. 4 ≤ y ≤ 9

Explanation

Because x = 0 lies inside the domain and the graph passes through the origin, the smallest value of y is 0. The largest value comes from the endpoint furthest from 0, which is x = 3, giving y = 9. So the range is 0 ≤ y ≤ 9.

Question 28Quadratic Functions and Equations

For y = x², what is the rate of change (変化の割合) between x = 1 and x = 3?

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

Explanation

Rate of change = (y₂ − y₁)/(x₂ − x₁) = (9 − 1)/(3 − 1) = 8/2 = 4.

Question 29Quadratic Functions and Equations

A ball rolls down a slope so that the distance y metres it has travelled after x seconds is y = 3x². How many seconds does it take to travel 48 m?

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

Explanation

Substitute y = 48 into y = 3x²: 3x² = 48, so x² = 16 and x = ±4. Elapsed time cannot be negative, so the ball takes 4 seconds.

Question 30Linear Equations and Functions

What is the y-intercept of the line 2x − 3y = 6?

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

Explanation

Set x = 0: −3y = 6, so y = −2. The y-intercept is (0, −2).

Question 31Quadratic Functions and Equations

For the parabola y = −2x², which statement is true?

  1. It opens right with vertex at the origin
  2. It opens upward with vertex at the origin
  3. It opens downward with vertex at the origin Correct answer.
  4. It opens downward with vertex at (0, −2)

Explanation

The coefficient −2 is negative, so the parabola opens downward. With no horizontal or vertical shift, the vertex is at (0, 0).

Question 32Linear Equations and Functions

Find the x-coordinate where the lines y = 2x − 3 and y = −x + 6 intersect.

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

Explanation

Set equal: 2x − 3 = −x + 6. Add x: 3x − 3 = 6. Add 3: 3x = 9, so x = 3.

Question 33Linear Equations and Functions

If the rate of change of a linear function is −2 and it passes through (1, 4), what is the equation?

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

Explanation

Slope a = −2, so y = −2x + b. Substitute (1, 4): 4 = −2(1) + b, so b = 6. The equation is y = −2x + 6.

Question 34Linear Equations and Functions

A candle burns at a constant rate, so its height h cm after t minutes is h = 20 − 0.4t. What is its height after 25 minutes?

  1. 19.6 cm
  2. 0 cm
  3. 10 cm Correct answer.
  4. 30 cm

Explanation

Substitute t = 25: h = 20 − 0.4 × 25 = 20 − 10 = 10 cm. The coefficient −0.4 is the rate at which the candle shortens each minute, and 20 is its starting height.

Question 35Quadratic Functions and Equations

At how many points do y = x² and y = x + 2 intersect?

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

Explanation

Set x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0. Two distinct real roots, so two intersection points.

Question 36Linear Equations and Functions

For y = 3x − 2, what is the change in y when x increases from 2 to 5?

  1. 3
  2. 6
  3. 15
  4. 9 Correct answer.

Explanation

Linear function: Δy = 3·Δx = 3 × (5 − 2) = 3 × 3 = 9.

Question 37Linear Equations and Functions

Which graph represents y = −x + 4?

  1. A line with slope 1 and y-intercept 4
  2. A line with slope −1 and y-intercept −4
  3. A line with slope −1 and y-intercept 4 Correct answer.
  4. A parabola with vertex (0, 4)

Explanation

In y = ax + b form, a = −1 is the slope and b = 4 is the y-intercept. The graph is a line, not a parabola.

Question 38Linear Equations and Functions

When water drains at 6 L/min, the volume V (L) after t minutes is V = 60 − 6t. When is the tank empty?

  1. t = 54 min
  2. t = 60 min
  3. t = 10 min Correct answer.
  4. t = 6 min

Explanation

Set V = 0: 60 − 6t = 0 → 6t = 60 → t = 10 minutes.

Question 39Quadratic Functions and Equations

The quadratic y = 2x² passes through which point?

  1. (2, 4)
  2. (3, 18) Correct answer.
  3. (3, 6)
  4. (3, 12)

Explanation

Substitute x = 3: y = 2(3²) = 2(9) = 18. The point (3, 18) lies on the parabola.

Question 40Quadratic Functions and Equations

The parabola y = x² and the line y = 2x + 3 meet at two points. What are the x-coordinates of those points?

  1. x = 3 only
  2. x = 1 and x = −3
  3. x = −1 and x = 3 Correct answer.
  4. x = 0 and x = 2

Explanation

At an intersection both equations give the same y, so x² = 2x + 3. Rearranging gives x² − 2x − 3 = 0, which factorises as (x − 3)(x + 1) = 0. Hence x = 3 or x = −1, the two x-coordinates where the graphs meet.

Question 41Geometry: Plane Figures and Angles

A right triangle has legs of 6 cm and 8 cm. What is the length of the hypotenuse?

  1. 100 cm
  2. 10 cm Correct answer.
  3. 12 cm
  4. 14 cm

Explanation

By the Pythagorean theorem (三平方の定理): c² = 6² + 8² = 36 + 64 = 100, so c = 10 cm.

Question 42Geometry: Plane Figures and Angles

A triangle has sides 5, 12, 13. Is it a right triangle?

  1. Yes, because 5 + 12 = 17
  2. Cannot be determined
  3. Yes, because 5² + 12² = 13² Correct answer.
  4. No, because 5 + 12 ≠ 13

Explanation

Check: 5² + 12² = 25 + 144 = 169 = 13². The converse of the Pythagorean theorem confirms it is a right triangle.

Question 43Geometry: Congruence, Similarity and Measurement

Two triangles are congruent by SSS. Which must be true?

  1. All three pairs of corresponding sides are equal Correct answer.
  2. Two angles and a side are equal
  3. Two sides and the included angle are equal
  4. All three angles are equal

Explanation

SSS (三辺相等) congruence requires all three pairs of corresponding sides to be equal in length.

Question 44Geometry: Congruence, Similarity and Measurement

Two triangles are similar with a scale factor of 3. If the smaller triangle has area 12 cm², what is the area of the larger triangle?

  1. 36 cm²
  2. 48 cm²
  3. 108 cm² Correct answer.
  4. 9 cm²

Explanation

Area scales as the square of the linear scale: 3² × 12 = 9 × 12 = 108 cm².

Question 45Geometry: Plane Figures and Angles

A circle has radius 5 cm. What is the length of an arc subtended by a central angle of 72°?

  1. 72π cm
  2. 5π cm
  3. 2π cm Correct answer.
  4. 10π cm

Explanation

Arc length = (θ/360) × 2πr = (72/360) × 2π(5) = (1/5) × 10π = 2π cm.

Question 46Geometry: Plane Figures and Angles

Find the area of a trapezoid (台形) with bases 8 cm and 14 cm and height 5 cm.

  1. 70 cm²
  2. 22 cm²
  3. 55 cm² Correct answer.
  4. 110 cm²

Explanation

Area = (上底 + 下底) × 高さ / 2 = (8 + 14) × 5 / 2 = 22 × 5 / 2 = 55 cm².

Question 47Geometry: Congruence, Similarity and Measurement

A cube has edge length 4 cm. What is its total surface area?

  1. 16 cm²
  2. 64 cm²
  3. 48 cm²
  4. 96 cm² Correct answer.

Explanation

A cube has 6 faces, each 4 × 4 = 16 cm². Total = 6 × 16 = 96 cm².

Question 48Geometry: Plane Figures and Angles

A rectangle has perimeter 36 cm and width 7 cm. What is its length?

  1. 18 cm
  2. 29 cm
  3. 11 cm Correct answer.
  4. 22 cm

Explanation

Perimeter = 2(length + width), so 36 = 2(L + 7), giving L + 7 = 18, so L = 11 cm.

Question 49Geometry: Plane Figures and Angles

The tangent (接線) from an external point P touches a circle of radius 6 at T. If OP = 10, find PT.

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

Explanation

The radius OT is perpendicular to the tangent at T, so triangle OTP is right. PT = √(OP² − OT²) = √(100 − 36) = √64 = 8.

Question 50Geometry: Plane Figures and Angles

A regular hexagon (正六角形) has side length 4 cm. What is its area?

  1. 16√3 cm²
  2. 24√3 cm² Correct answer.
  3. 96 cm²
  4. 48 cm²

Explanation

A regular hexagon splits into 6 equilateral triangles of side 4. Each has area (√3/4)(4²) = 4√3. Total = 6 × 4√3 = 24√3 cm².