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

Yamagata Prefectural Public High School Entrance Examination - Mathematics practice questions

One question at a time. Build your Yamagata 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

Solve for x: 3x − 5 = 16.

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

Explanation

Add 5 to both sides: 3x = 21, then divide by 3: x = 7. Check: 3(7)−5 = 21−5 = 16.

Question 2Numbers and Algebraic Expressions

Solve: 2(x + 3) = 4x − 6. What is x?

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

Explanation

Expand: 2x + 6 = 4x − 6 → 6 + 6 = 4x − 2x → 12 = 2x → x = 6. Check: 2(9) = 18 and 4(6)−6 = 18.

Question 3Numbers and Algebraic Expressions

Simplify: (2x + 3) + (x − 5) − (3x + 1).

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

Explanation

Combine like terms: 2x + x − 3x = 0x; 3 − 5 − 1 = −3. The x terms cancel, leaving the constant −3.

Question 4Numbers and Algebraic Expressions

Expand (x + 4)(x − 2).

  1. x² + 2x − 8 Correct answer.
  2. x² − 6x − 8
  3. x² − 2x + 8
  4. x² + 6x − 8

Explanation

FOIL: x·x = x², x·(−2) = −2x, 4·x = 4x, 4·(−2) = −8. Sum: x² + (−2+4)x − 8 = x² + 2x − 8.

Question 5Numbers and Algebraic Expressions

Factor: x² − 9.

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

Explanation

This is a difference of squares: a² − b² = (a−b)(a+b) with a = x, b = 3, giving (x−3)(x+3).

Question 6Numbers and Algebraic Expressions

Factor: x² + 5x + 6.

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

Explanation

Find two numbers whose product is 6 and sum is 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3).

Question 7Numbers and Algebraic Expressions

Solve the simultaneous equations: x + y = 10, x − y = 4. What is x?

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

Explanation

Add the equations: 2x = 14 → x = 7. Then y = 10 − 7 = 3. Check: 7 + 3 = 10 and 7 − 3 = 4.

Question 8Numbers and Algebraic Expressions

Solve: 3x + 2y = 12, x = 2. What is y?

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

Explanation

Substitute x = 2: 3(2) + 2y = 12 → 6 + 2y = 12 → 2y = 6 → y = 3.

Question 9Numbers and Algebraic Expressions

If a:b = 3:4 and b:c = 4:5, what is a:c?

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

Explanation

Since b is common (4 in both ratios), a:b:c = 3:4:5 directly, so a:c = 3:5.

Question 10Numbers and Algebraic Expressions

A number divided by 4 gives quotient 7 and remainder 3. What is the number?

  1. 31 Correct answer.
  2. 28
  3. 34
  4. 25

Explanation

Number = 4 × quotient + remainder = 4 × 7 + 3 = 28 + 3 = 31. Check: 31 ÷ 4 = 7 remainder 3.

Question 11Numbers and Algebraic Expressions

Solve for x: (x − 2)/3 = (x + 1)/4.

  1. −11
  2. 5
  3. 11 Correct answer.
  4. −5

Explanation

Cross-multiply: 4(x − 2) = 3(x + 1) → 4x − 8 = 3x + 3 → x = 11. Check: 9/3 = 3 and 12/4 = 3.

Question 12Numbers and Algebraic Expressions

The sum of two numbers is 20 and their difference is 6. What is the larger number?

  1. 10
  2. 6
  3. 7
  4. 13 Correct answer.

Explanation

Let larger = a, smaller = b. a + b = 20, a − b = 6. Add: 2a = 26 → a = 13 (and b = 7).

Question 13Numbers and Algebraic Expressions

Factor completely: 2x² − 8.

  1. (x − 2)(x + 2)
  2. (2x − 8)
  3. 2(x − 4)(x + 1)
  4. 2(x − 2)(x + 2) Correct answer.

Explanation

First factor out the common factor 2: 2(x² − 4), then difference of squares: 2(x − 2)(x + 2).

Question 14Numbers and Algebraic Expressions

If 3(x − 4) = 2x + 5, what is x?

  1. 17 Correct answer.
  2. 7
  3. −7
  4. 1

Explanation

Expand: 3x − 12 = 2x + 5 → 3x − 2x = 5 + 12 → x = 17. Check: 3(13) = 39 and 2(17)+5 = 39.

Question 15Numbers and Algebraic Expressions

A shop gives a 20% discount on an item marked ¥3,000. What is the sale price?

  1. ¥2,800
  2. ¥2,600
  3. ¥2,400 Correct answer.
  4. ¥600

Explanation

Discount = 20% of 3,000 = 0.2 × 3,000 = 600. Sale price = 3,000 − 600 = ¥2,400. (Equivalently 0.8 × 3,000 = 2,400.)

Question 16Numbers and Algebraic Expressions

Simplify: (x² − 4)/(x − 2) for x ≠ 2.

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

Explanation

Factor the numerator: x² − 4 = (x − 2)(x + 2). Cancel (x − 2) with the denominator, leaving x + 2 (valid for x ≠ 2).

Question 17Numbers and Algebraic Expressions

Solve: x² − 5x + 6 = 0. What are the solutions?

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

Explanation

Factor: (x − 2)(x − 3) = 0, so x = 2 or x = 3. Check: 2·3 = 6 and 2+3 = 5.

Question 18Numbers and Algebraic Expressions

Solve: x² = 49.

  1. x = −7 only
  2. x = ±7 Correct answer.
  3. x = 7 only
  4. x = ±49

Explanation

Taking the square root of both sides gives x = 7 or x = −7, i.e. x = ±7, since both (7)² and (−7)² equal 49.

Question 19Numbers and Algebraic Expressions

If a + b = 5 and ab = 6, what is a² + b²?

  1. 25
  2. 13 Correct answer.
  3. 37
  4. 11

Explanation

Use (a + b)² = a² + 2ab + b², so a² + b² = (a + b)² − 2ab = 25 − 12 = 13.

Question 20Numbers and Algebraic Expressions

Expand (x + 3)².

  1. x² + 3x + 9
  2. x² + 6x + 9 Correct answer.
  3. x² + 6x + 6
  4. x² + 9

Explanation

(a + b)² = a² + 2ab + b² = x² + 2·x·3 + 9 = x² + 6x + 9.

Question 21Numbers and Algebraic Expressions

Solve: 4x − 7 < 2x + 3. Which inequality gives the solution?

  1. x < 5 Correct answer.
  2. x > 5
  3. x < −5
  4. x < 2

Explanation

Subtract 2x: 2x − 7 < 3. Add 7: 2x < 10. Divide by 2: x < 5. (Inequality direction unchanged since dividing by positive 2.)

Question 22Numbers and Algebraic Expressions

Which value of x satisfies 2(x + 1) ≤ 8?

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

Explanation

Solve: 2x + 2 ≤ 8 → 2x ≤ 6 → x ≤ 3. Of the choices, only x = 3 satisfies the inequality (2(3+1) = 8 ≤ 8).

Question 23Numbers and Algebraic Expressions

Simplify: √72 (leave in simplest radical form).

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

Explanation

72 = 36 × 2, and √72 = √(36·2) = √36·√2 = 6√2. This is fully simplified.

Question 24Numbers and Algebraic Expressions

Solve: (x − 3)(x + 4) = 0. What are the solutions?

  1. x = 12 or x = −1
  2. x = −3 or x = 4
  3. x = 3 or x = 4
  4. x = 3 or x = −4 Correct answer.

Explanation

A product is zero when a factor is zero: x − 3 = 0 → x = 3, or x + 4 = 0 → x = −4.

Question 25Numbers and Algebraic Expressions

The average of 4, 7, 9, x is 8. What is x?

  1. 6
  2. 8
  3. 12 Correct answer.
  4. 10

Explanation

Average = (4 + 7 + 9 + x)/4 = 8, so 20 + x = 32 → x = 12.

Question 26Linear Equations and Functions

For y = 3x − 2, what is the y-intercept?

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

Explanation

In slope-intercept form y = mx + b, the y-intercept is b. Here b = −2, so the line crosses the y-axis at (0, −2).

Question 27Linear Equations and Functions

The line y = ax + 5 passes through (2, 11). What is a?

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

Explanation

Substitute (2, 11): 11 = 2a + 5 → 2a = 6 → a = 3.

Question 28Linear Equations and Functions

Find the x-intercept of y = 2x − 6.

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

Explanation

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

Question 29Quadratic Functions and Equations

For the quadratic y = x² − 4x + 3, what is the value of y at x = 0?

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

Explanation

At x = 0, y = 0 − 0 + 3 = 3. This is the y-intercept.

Question 30Quadratic Functions and Equations

For y = 2x², what is the rate of change (変化の割合) as x increases from 1 to 3?

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

Explanation

The rate of change is the increase in y divided by the increase in x. When x = 1, y = 2; when x = 3, y = 18. So the rate is (18 − 2)/(3 − 1) = 16/2 = 8. A useful shortcut for y = ax² is a(p + q) = 2(1 + 3) = 8.

Question 31Quadratic Functions and Equations

For y = x², what is the range of y (yの変域) when −1 ≤ x ≤ 3?

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

Explanation

Because x = 0 lies inside the interval −1 ≤ x ≤ 3, y reaches its smallest value 0 there. The largest value comes from the endpoint furthest from 0, x = 3, giving y = 9. So 0 ≤ y ≤ 9. Always check whether x = 0 is included before using the endpoints.

Question 32Quadratic Functions and Equations

The graph of y = −x² opens in which direction?

  1. To the right
  2. Upward (minimum at vertex)
  3. To the left
  4. Downward (maximum at vertex) Correct answer.

Explanation

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

Question 33Quadratic Functions and Equations

Find the roots (x-intercepts) of y = x² − 5x + 6.

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

Explanation

Set y = 0: x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 and x = 3.

Question 34Linear Equations and Functions

If y is directly proportional to x and y = 15 when x = 5, what is y when x = 8?

  1. 3
  2. 15
  3. 24 Correct answer.
  4. 40

Explanation

y = kx; 15 = 5k → k = 3. So y = 3·8 = 24.

Question 35Linear Equations and Functions

If y is inversely proportional to x and y = 6 when x = 2, what is y when x = 3?

  1. 2
  2. 9
  3. 18
  4. 4 Correct answer.

Explanation

y = k/x; 6 = k/2 → k = 12. So y = 12/3 = 4.

Question 36Linear Equations and Functions

A straight line has slope −2 and passes through the point (3, 1). What is its equation?

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

Explanation

Write the line as y = −2x + b, then substitute the point: 1 = −2(3) + b = −6 + b, so b = 7. The equation is y = −2x + 7. Check: at x = 3, y = −6 + 7 = 1.

Question 37Quadratic Functions and Equations

Which graph opens downward and is narrower than the graph of y = x²?

  1. y = 3x²
  2. y = −3x² Correct answer.
  3. y = (1/3)x²
  4. y = −(1/3)x²

Explanation

For y = ax², the sign of a decides the direction: a < 0 opens downward. The size of a decides the width: the larger |a| is, the narrower the curve. y = −3x² has a = −3, so it opens downward and, since |−3| = 3 > 1, it is narrower than y = x².

Question 38Linear Equations and Functions

At which point do the lines y = 2x − 1 and y = −x + 5 intersect?

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

Explanation

At the intersection both equations give the same y, so set them equal: 2x − 1 = −x + 5. Then 3x = 6 and x = 2. Substituting back, y = 2(2) − 1 = 3. The lines meet at (2, 3), which satisfies both equations.

Question 39Linear Equations and Functions

Given the table of values (x, y) = (1, 5), (2, 7), (3, 9), (4, 11), which formula fits?

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

Explanation

The y-values increase by 2 as x increases by 1, so slope = 2. Using (1, 5): 5 = 2(1) + b → b = 3. So y = 2x + 3.

Question 40Linear Equations and Functions

A tank holds 80 litres of water and drains at a steady 5 litres per minute. Taking y as the litres left after x minutes, which statement is correct?

  1. y = 80 + 5x, and the tank is empty after 16 minutes
  2. y = 80 − 5x, and the tank is empty after 16 minutes Correct answer.
  3. y = 5x, and the tank is empty after 16 minutes
  4. y = 80 − 5x, and the tank is empty after 400 minutes

Explanation

The tank starts at 80 litres, so 80 is the y-intercept, and it loses 5 litres each minute, so the rate of change is −5: y = 80 − 5x. It is empty when y = 0, that is 80 − 5x = 0, giving x = 16 minutes.

Question 41Linear Equations and Functions

A rectangle is 3 cm longer than it is wide, and its area is 40 cm². What is its width?

  1. 10 cm
  2. 8 cm
  3. 4 cm
  4. 5 cm Correct answer.

Explanation

Let the width be x cm, so the length is x + 3. Then x(x + 3) = 40, which rearranges to x² + 3x − 40 = 0 and factorises as (x + 8)(x − 5) = 0. The roots are x = −8 and x = 5, and a width cannot be negative, so the width is 5 cm (length 8 cm).

Question 42Linear Equations and Functions

For the linear function y = −3x + 2, what is the range of y when −1 ≤ x ≤ 2?

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

Explanation

Substitute the endpoints: at x = −1, y = 3 + 2 = 5; at x = 2, y = −6 + 2 = −4. Because the slope is negative, y decreases as x increases, so the smallest value is −4 and the largest is 5, written −4 ≤ y ≤ 5.

Question 43Linear Equations and Functions

The line y = 2x is reflected across the x-axis. The new line is:

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

Explanation

Reflecting across the x-axis sends (x, y) → (x, −y), so y = 2x becomes −y = 2x, i.e. y = −2x.

Question 44Geometry: Plane Figures and Angles

A triangle has angles of 50° and 60°. What is the third angle?

  1. 60°
  2. 70° Correct answer.
  3. 50°
  4. 110°

Explanation

Angles in a triangle sum to 180°, so the third angle = 180 − 50 − 60 = 70°.

Question 45Geometry: Plane Figures and Angles

In a right triangle with legs 6 and 8, what is the hypotenuse?

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

Explanation

By the Pythagorean theorem, c² = 6² + 8² = 36 + 64 = 100, so c = 10.

Question 46Geometry: Plane Figures and Angles

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

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

Explanation

b² = 13² − 5² = 169 − 25 = 144, so b = 12. (This is the 5-12-13 Pythagorean triple.)

Question 47Geometry: Plane Figures and Angles

In a circle, an inscribed angle that intercepts an arc of 80° has measure:

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

Explanation

An inscribed angle is half the central angle (and half its intercepted arc), so 80°/2 = 40°.

Question 48Geometry: Plane Figures and Angles

Two parallel lines are cut by a transversal. A pair of corresponding angles is 65° and x°. What is x?

  1. 90
  2. 65 Correct answer.
  3. 115
  4. 25

Explanation

Corresponding angles formed by a transversal cutting parallel lines are equal, so x = 65.

Question 49Geometry: Plane Figures and Angles

In triangle ABC, AB = AC. If angle B = 40°, what is angle A?

  1. 40°
  2. 140°
  3. 100° Correct answer.
  4. 70°

Explanation

Since AB = AC, the base angles are equal: angle C = angle B = 40°. So angle A = 180 − 40 − 40 = 100°.

Question 50Geometry: Plane Figures and Angles

A square has diagonal of length 6. What is its area?

  1. 9
  2. 18 Correct answer.
  3. 36
  4. 12

Explanation

For a square, diagonal d = s√2, so s = 6/√2 = 3√2, and area = s² = (3√2)² = 18. (Alternatively, area = d²/2 = 36/2 = 18.)