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

Shizuoka Public High School Entrance Examination - Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Compute: -5 + 9 - 3.

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

Explanation

Work left to right with signed numbers: -5 + 9 = 4, then 4 - 3 = 1.

Question 2Numbers and Algebraic Expressions

Compute: (-6) × (-2) + (-4).

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

Explanation

(-6) × (-2) = 12 (product of two negatives is positive); 12 + (-4) = 8.

Question 3Numbers and Algebraic Expressions

Compute (1/2) + (1/3) and write the answer as a single reduced fraction.

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

Explanation

Common denominator 6: 1/2 = 3/6 and 1/3 = 2/6; 3/6 + 2/6 = 5/6, already reduced.

Question 4Numbers and Algebraic Expressions

Compute (3/5) × (10/9) and write the answer as a reduced fraction.

  1. 5/9
  2. 2/3 Correct answer.
  3. 6/5
  4. 1/3

Explanation

Multiply numerators and denominators: (3×10)/(5×9) = 30/45; divide top and bottom by 15 to get 2/3.

Question 5Numbers and Algebraic Expressions

What is the prime factorisation of 84?

  1. 2² × 3 × 7²
  2. 2² × 3 × 7 Correct answer.
  3. 2 × 3² × 7
  4. 2 × 3 × 7

Explanation

84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7.

Question 6Numbers and Algebraic Expressions

Find the greatest common divisor of 48 and 36.

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

Explanation

48 = 2⁴ × 3 and 36 = 2² × 3²; the GCD takes the smaller power of each prime: 2² × 3 = 12.

Question 7Numbers and Algebraic Expressions

Find the least common multiple of 12 and 18.

  1. 24
  2. 6
  3. 36 Correct answer.
  4. 72

Explanation

12 = 2² × 3 and 18 = 2 × 3²; the LCM takes the larger power of each prime: 2² × 3² = 36.

Question 8Numbers and Algebraic Expressions

Solve for x: 2x + 3 = 11.

  1. 14
  2. 4 Correct answer.
  3. -4
  4. 7

Explanation

Subtract 3 from both sides: 2x = 8; divide by 2 to get x = 4.

Question 9Numbers and Algebraic Expressions

Simplify: 3(x - 2) - 2(x + 1).

  1. x - 4
  2. 5x - 8
  3. x + 4
  4. x - 8 Correct answer.

Explanation

Expand: 3x - 6 - 2x - 2 = (3x - 2x) + (-6 - 2) = x - 8.

Question 10Numbers and Algebraic Expressions

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

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

Explanation

Subtract 3x from both sides: 2x - 2 = 10; add 2: 2x = 12; divide by 2 to get x = 6.

Question 11Numbers and Algebraic Expressions

Solve the inequality: 3x - 5 < 10.

  1. x < 15
  2. x > 5
  3. x < 5 Correct answer.
  4. x ≤ 5

Explanation

Add 5 to both sides: 3x < 15; divide by 3 (positive, so the sign stays) to get x < 5.

Question 12Numbers and Algebraic Expressions

Solve the simultaneous equations x + y = 12, x - y = 4. What is x?

  1. 8 Correct answer.
  2. 16
  3. 4
  4. 6

Explanation

Add the two equations: 2x = 16, so x = 8; then y = 12 - 8 = 4.

Question 13Numbers and Algebraic Expressions

Solve 2x + 3y = 19 and x + 2y = 11. What is y?

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

Explanation

From the second equation x = 11 - 2y; substitute: 2(11 - 2y) + 3y = 22 - 4y + 3y = 22 - y = 19, so y = 3 (and x = 5).

Question 14Numbers and Algebraic Expressions

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

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

Explanation

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

Question 15Numbers and Algebraic Expressions

Solve x² + 3x - 10 = 0. What are the solutions?

  1. -5 and 2 Correct answer.
  2. 5 and -2
  3. -5 and -2
  4. -10 and 1

Explanation

Factor: (x + 5)(x - 2) = 0, so x = -5 or x = 2.

Question 16Numbers and Algebraic Expressions

Expand (x + 3)(x - 4).

  1. x² - x - 12 Correct answer.
  2. x² + x - 12
  3. x² - x + 12
  4. x² - 7x - 12

Explanation

Use FOIL: x² - 4x + 3x - 12 = x² - x - 12.

Question 17Numbers and Algebraic Expressions

Factorise x² - 7x + 12.

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

Explanation

Find two numbers that multiply to 12 and add to -7: -3 and -4, giving (x - 3)(x - 4).

Question 18Numbers and Algebraic Expressions

Expand (x + 2)².

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

Explanation

A perfect-square trinomial: (x + 2)² = x² + 2·x·2 + 2² = x² + 4x + 4.

Question 19Numbers and Algebraic Expressions

Adult tickets cost ¥1200 and child tickets ¥700. Eight tickets in total cost ¥7600. How many adult tickets were sold?

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

Explanation

Let a be adults: a + c = 8 and 1200a + 700c = 7600. Substitute c = 8 - a: 1200a + 700(8 - a) = 500a + 5600 = 7600, so 500a = 2000 and a = 4.

Question 20Numbers and Algebraic Expressions

A taxi charges ¥720 initial fee plus ¥320 per kilometre. With a budget of ¥3000, what is the maximum whole number of kilometres you can travel?

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

Explanation

Solve 720 + 320k ≤ 3000: 320k ≤ 2280, k ≤ 7.125, so the largest whole number is 7.

Question 21Numbers and Algebraic Expressions

Solve x² - 6x + 5 = 0 by completing the square. What are the solutions?

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

Explanation

Complete the square: x² - 6x + 9 = 4, so (x - 3)² = 4, x - 3 = ±2, giving x = 1 or x = 5.

Question 22Numbers and Algebraic Expressions

Given x² - y² = 21 and x + y = 7, find x - y.

  1. 3 Correct answer.
  2. 14
  3. 21
  4. 7

Explanation

Factor the difference of squares: (x + y)(x - y) = 21. Substitute x + y = 7: 7(x - y) = 21, so x - y = 3.

Question 23Numbers and Algebraic Expressions

The sum of two numbers is 14 and their product is 45. What are the two numbers?

  1. 9 and 14
  2. 3 and 15
  3. 5 and 14
  4. 5 and 9 Correct answer.

Explanation

The numbers are roots of t² - 14t + 45 = 0; factor (t - 5)(t - 9) = 0, giving 5 and 9 (sum 14, product 45).

Question 24Linear Equations and Functions

In a direct proportion y = kx, y = 15 when x = 5. Find k.

  1. 3 Correct answer.
  2. 15
  3. 5
  4. 75

Explanation

In a direct proportion y = kx, so k = y/x = 15/5 = 3.

Question 25Linear Equations and Functions

In an inverse proportion y = k/x, y = 6 when x = 4. Find k.

  1. 48
  2. 10
  3. 1.5
  4. 24 Correct answer.

Explanation

In an inverse proportion y = k/x, so k = xy = 4 × 6 = 24.

Question 26Linear Equations and Functions

For the linear function y = 2x + 3, find y when x = -1.

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

Explanation

Substitute x = -1: y = 2(-1) + 3 = -2 + 3 = 1.

Question 27Linear Equations and Functions

Find the slope of the line passing through (1, 2) and (3, 8).

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

Explanation

Slope = (y₂ - y₁)/(x₂ - x₁) = (8 - 2)/(3 - 1) = 6/2 = 3.

Question 28Linear Equations and Functions

Find the x-intercept of the line y = -x + 4.

  1. 4 Correct answer.
  2. -4
  3. 8
  4. 0

Explanation

Set y = 0: 0 = -x + 4, so x = 4; the intercept is the point (4, 0).

Question 29Linear Equations and Functions

A line passes through (0, 2) and (4, 10). Find its equation in the form y = ax + b.

  1. y = 8x + 2
  2. y = 2x + 2 Correct answer.
  3. y = 2x
  4. y = 2x + 4

Explanation

Slope a = (10 - 2)/(4 - 0) = 2; the y-intercept b is the y-value at x = 0, which is 2, so y = 2x + 2.

Question 30Linear Equations and Functions

In a direct proportion y = kx, y = 12 when x = 3. Find y when x = 7.

  1. 21
  2. 28 Correct answer.
  3. 36
  4. 16

Explanation

k = 12/3 = 4, so y = 4x; at x = 7, y = 4 × 7 = 28.

Question 31Linear Equations and Functions

In an inverse proportion y = k/x, y = 9 when x = 2. Find y when x = 6.

  1. 27
  2. 6
  3. 18
  4. 3 Correct answer.

Explanation

k = xy = 2 × 9 = 18, so y = 18/x; at x = 6, y = 18/6 = 3.

Question 32Quadratic Functions and Equations

Find the vertex of the parabola y = x² - 2x.

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

Explanation

Complete the square (平方完成): y = x² - 2x = (x - 1)² - 1. The squared term is smallest (zero) at x = 1, where y = -1, so the vertex is (1, -1).

Question 33Quadratic Functions and Equations

What is the maximum value of y = -x² + 4?

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

Explanation

The parabola opens downward (a = -1), so the vertex at x = 0 gives the maximum; y = 0 + 4 = 4.

Question 34Quadratic Functions and Equations

The parabola y = ax² passes through the point (2, 12). Find a.

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

Explanation

Substitute x = 2, y = 12: 12 = a × 2² = 4a, so a = 3.

Question 35Quadratic Functions and Equations

Find the axis of symmetry of y = x² - 6x + 8.

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

Explanation

Complete the square (平方完成): y = x² - 6x + 8 = (x - 3)² - 1, so the graph is symmetric about the line x = 3.

Question 36Quadratic Functions and Equations

How many intersection points do y = x² and y = x + 2 have?

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

Explanation

Set x² = x + 2, giving x² - x - 2 = 0. Factor it as (x - 2)(x + 1) = 0, so x = 2 and x = -1 - two distinct solutions, hence 2 intersection points, at (2, 4) and (-1, 1).

Question 37Quadratic Functions and Equations

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

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

Explanation

The parabola opens downward (a = -2), so the vertex at x = 0 gives the maximum value y = 0.

Question 38Quadratic Functions and Equations

Find the x-intercepts of y = x² + 2x.

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

Explanation

Set y = 0: x² + 2x = x(x + 2) = 0, so x = 0 or x = -2.

Question 39Quadratic Functions and Equations

Find the x-intercepts of y = x² - 4x + 3.

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

Explanation

Set x² - 4x + 3 = 0; factor (x - 1)(x - 3) = 0, giving x = 1 and x = 3, so the intercepts are (1, 0) and (3, 0).

Question 40Quadratic Functions and Equations

Find the y-coordinate of the vertex of y = x² - 2x - 3.

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

Explanation

Complete the square (平方完成): y = x² - 2x - 3 = (x - 1)² - 4. The squared term is zero at x = 1, so the vertex is (1, -4) and its y-coordinate is -4.

Question 41Quadratic Functions and Equations

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

  1. (2, 0) only
  2. no intersection
  3. (0, 2) and (2, 2)
  4. (0, 0) and (2, 4) Correct answer.

Explanation

Set x² = 2x, giving x² - 2x = x(x - 2) = 0; x = 0 or x = 2, with points (0, 0) and (2, 4).

Question 42Linear Equations and Functions

The height of a ball follows h = -5t² + 20t (metres, seconds). What is the maximum height?

  1. 10
  2. 20 Correct answer.
  3. 40
  4. 25

Explanation

Complete the square (平方完成): h = -5t² + 20t = -5(t - 2)² + 20. Because -5(t - 2)² is never positive, the greatest height is 20 metres, reached at t = 2 seconds.

Question 43Linear Equations and Functions

Find the equation of the line parallel to y = 2x + 5 passing through (1, 3).

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

Explanation

Parallel lines share the slope, so a = 2; substitute (1, 3): 3 = 2(1) + b, giving b = 1, so y = 2x + 1.

Question 44Quadratic Functions and Equations

Find the minimum value of y = x² + 4x + 5.

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

Explanation

Complete the square (平方完成): y = x² + 4x + 5 = (x + 2)² + 1. Since (x + 2)² is never negative, the smallest value of y is 1, reached at x = -2.

Question 45Geometry: Plane Figures and Angles

What is the sum of the interior angles of a quadrilateral?

  1. 180°
  2. 720°
  3. 540°
  4. 360° Correct answer.

Explanation

A quadrilateral splits into two triangles, so the interior angles sum to 2 × 180° = 360°.

Question 46Geometry: Plane Figures and Angles

A right triangle has legs of length 3 and 4. Find the hypotenuse.

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

Explanation

By the Pythagorean theorem (ピタゴラスの定理), 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. Find the other leg.

  1. 18
  2. 12 Correct answer.
  3. 25
  4. 8

Explanation

By the Pythagorean theorem, leg² = 13² - 5² = 169 - 25 = 144, so the leg is 12.

Question 48Geometry: Plane Figures and Angles

Find the area of a triangle with base 8 and height 5.

  1. 13
  2. 20 Correct answer.
  3. 26
  4. 40

Explanation

Area = (1/2) × base × height = (1/2) × 8 × 5 = 20.

Question 49Geometry: Congruence, Similarity and Measurement

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

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

Explanation

Areas of similar figures scale as the square of the side ratio: (3/2)² × 8 = (9/4) × 8 = 18.

Question 50Geometry: Plane Figures and Angles

Two parallel lines are cut by a transversal. If one of the alternate interior angles is 70°, what is the other alternate interior angle?

  1. 70° Correct answer.
  2. 20°
  3. 110°
  4. 90°

Explanation

When parallel lines are cut by a transversal, alternate interior angles are equal, so the other angle is also 70°.