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

Tokushima Public High School Entrance Examination - Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Compute: -9 + 4 - (-3).

  1. -8
  2. -10
  3. 16
  4. -2 Correct answer.

Explanation

Work left to right: -9 + 4 = -5, then -5 - (-3) = -5 + 3 = -2.

Question 2Numbers and Algebraic Expressions

Compute 5/6 - 1/4 and write the answer as a reduced fraction.

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

Explanation

Common denominator 12: 5/6 = 10/12 and 1/4 = 3/12; 10/12 - 3/12 = 7/12.

Question 3Numbers and Algebraic Expressions

Find the prime factorisation of 84.

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

Explanation

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

Question 4Numbers and Algebraic Expressions

Solve for x: 3x - 7 = 2x + 5.

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

Explanation

Subtract 2x and add 7: 3x - 2x = 5 + 7, so x = 12.

Question 5Numbers and Algebraic Expressions

Solve the simultaneous equations: x + y = 10, x - y = 4.

  1. x = 14, y = -4
  2. x = 3, y = 7
  3. x = 7, y = 3 Correct answer.
  4. x = 6, y = 4

Explanation

Add the equations: 2x = 14, so x = 7; then y = 10 - 7 = 3.

Question 6Numbers and Algebraic Expressions

Solve: 2(x - 3) ≤ 4x + 2. Which set describes x?

  1. x ≥ -4 Correct answer.
  2. x ≤ -4
  3. x ≥ 4
  4. x ≤ 4

Explanation

Expand: 2x - 6 ≤ 4x + 2, so -8 ≤ 2x, giving x ≥ -4.

Question 7Numbers and Algebraic Expressions

Factorise: x^2 - 7x + 12.

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

Explanation

We need two numbers that multiply to 12 and add to -7: -3 and -4, so (x - 3)(x - 4).

Question 8Numbers and Algebraic Expressions

Solve x^2 - 5x + 6 = 0.

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

Explanation

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

Question 9Numbers and Algebraic Expressions

A shop discounts an item marked ¥3,000 by 20%. What is the sale price?

  1. ¥2,800
  2. ¥2,400 Correct answer.
  3. ¥2,500
  4. ¥600

Explanation

The discount is 20% of 3,000 = 600, so the sale price is 3,000 - 600 = ¥2,400.

Question 10Numbers and Algebraic Expressions

Simplify √72 + √50 - √32.

  1. 7√2 Correct answer.
  2. 11√2
  3. 5√2
  4. 9√2

Explanation

√72 = 6√2, √50 = 5√2, √32 = 4√2; 6√2 + 5√2 - 4√2 = 7√2.

Question 11Numbers and Algebraic Expressions

If 3 pens and 2 notebooks cost ¥1,400, and 1 pen and 3 notebooks cost ¥1,050, what is the price of one notebook?

  1. ¥200
  2. ¥250 Correct answer.
  3. ¥300
  4. ¥350

Explanation

Let pen = p, notebook = n. From 3p + 2n = 1400 and p + 3n = 1050, multiply the second equation by 3 to get 3p + 9n = 3150, then subtract: (3p + 2n) - (3p + 9n) = 1400 - 3150, so -7n = -1750 and n = ¥250. Substituting back gives p = 1050 - 750 = ¥300.

Question 12Numbers and Algebraic Expressions

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

  1. x^2 - 16 Correct answer.
  2. x^2 - 8
  3. x^2 + 16
  4. x^2 - 8x - 16

Explanation

This is a difference of squares: (x + 4)(x - 4) = x^2 - 16.

Question 13Numbers and Algebraic Expressions

Use the identity (a + b)^2 = a^2 + 2ab + b^2 to compute 23^2.

  1. 461
  2. 529 Correct answer.
  3. 576
  4. 518

Explanation

23 = 20 + 3, so 23^2 = 400 + 2·20·3 + 9 = 400 + 120 + 9 = 529.

Question 14Numbers and Algebraic Expressions

A train travels 180 km in 2 hours 15 minutes. What is its average speed in km/h?

  1. 120
  2. 90
  3. 75
  4. 80 Correct answer.

Explanation

2 h 15 min = 2.25 h, so speed = 180 / 2.25 = 80 km/h.

Question 15Numbers and Algebraic Expressions

Solve for x: (x - 2)/3 = (x + 1)/4.

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

Explanation

Cross-multiply: 4(x - 2) = 3(x + 1), so 4x - 8 = 3x + 3, giving x = 11.

Question 16Numbers and Algebraic Expressions

How many integer values of x satisfy -3 < 2x + 1 ≤ 9?

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

Explanation

Subtract 1: -4 < 2x ≤ 8, divide by 2: -2 < x ≤ 4. The integers are -1, 0, 1, 2, 3, 4 - six values.

Question 17Numbers and Algebraic Expressions

If x + 1/x = 5, what is the value of x^2 + 1/x^2?

  1. 25
  2. 23 Correct answer.
  3. 3
  4. 27

Explanation

Square both sides: (x + 1/x)^2 = 25 = x^2 + 2 + 1/x^2, so x^2 + 1/x^2 = 23.

Question 18Numbers and Algebraic Expressions

Three consecutive even integers have a sum of 54. What is the largest?

  1. 22
  2. 16
  3. 18
  4. 20 Correct answer.

Explanation

Let the integers be n, n+2, n+4. Then 3n + 6 = 54, so n = 16 and the largest is 20.

Question 19Linear Equations and Functions

A quantity y is directly proportional to x. When x = 4, y = 12. Find y when x = 9.

  1. 27 Correct answer.
  2. 18
  3. 3
  4. 36

Explanation

y = kx with k = 12/4 = 3, so when x = 9, y = 3 × 9 = 27.

Question 20Linear Equations and Functions

y is inversely proportional to x. When x = 6, y = 4. Find y when x = 8.

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

Explanation

xy = k = 24, so when x = 8, y = 24/8 = 3.

Question 21Linear Equations and Functions

Two lines y = 2x + 1 and y = -x + 7 intersect at point P. Find the x-coordinate of P.

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

Explanation

Set 2x + 1 = -x + 7, so 3x = 6 and x = 2 (then y = 5).

Question 22Quadratic Functions and Equations

For the quadratic y = 2x^2, what are the coordinates of the vertex and the axis of symmetry?

  1. (0, 0); y = 0
  2. (0, 0); x = 0 Correct answer.
  3. (0, 2); y = 0
  4. (2, 0); x = 2

Explanation

y = 2x^2 has its vertex at the origin (0, 0) with axis of symmetry x = 0; it opens upward.

Question 23Quadratic Functions and Equations

For y = -3x^2, compared with y = x^2, the parabola is:

  1. wider and opens upward
  2. wider and opens downward
  3. narrower and opens downward Correct answer.
  4. narrower and opens upward

Explanation

The negative sign makes it open downward, and |a| = 3 > 1 makes it narrower than y = x^2.

Question 24Quadratic Functions and Equations

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

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

Explanation

Set x^2 = x + 2, so x^2 - x - 2 = 0 = (x - 2)(x + 1); x = 2 gives y = 4, x = -1 gives y = 1.

Question 25Linear Equations and Functions

A car travels at speed v km/h for t hours, covering 120 km. If v = 60, find t; then state how t changes when v doubles.

  1. t = 0.5; t halves
  2. t = 2; t halves to 1 Correct answer.
  3. t = 2; t is unchanged
  4. t = 2; t doubles to 4

Explanation

t = 120/60 = 2 h. Distance fixed means t is inversely proportional to v, so doubling v halves t to 1 h.

Question 26Linear Equations and Functions

Given y = -2x + 5, what is the rate of change of y as x increases by 1?

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

Explanation

The slope -2 is the rate of change, so y decreases by 2 for each unit increase in x.

Question 27Linear Equations and Functions

Which point lies on the line y = (1/2)x - 3?

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

Explanation

Substitute x = 4: y = 2 - 3 = -1, so (4, -1) lies on the line.

Question 28Linear Equations and Functions

The line y = ax + b is parallel to y = 4x - 1 and passes through (0, 5). Find a and b.

  1. a = -1, b = 5
  2. a = 4, b = -1
  3. a = 5, b = 4
  4. a = 4, b = 5 Correct answer.

Explanation

Parallel lines share the slope, so a = 4; the y-intercept through (0, 5) gives b = 5.

Question 29Quadratic Functions and Equations

A function is defined by y = x^2 for -2 ≤ x ≤ 2. What is the range of y?

  1. -4 ≤ y ≤ 4
  2. 0 ≤ y ≤ 2
  3. 0 ≤ y ≤ 16
  4. 0 ≤ y ≤ 4 Correct answer.

Explanation

x^2 is non-negative and largest at |x| = 2, where y = 4; at x = 0, y = 0. So 0 ≤ y ≤ 4.

Question 30Linear Equations and Functions

Water fills a tank at a constant rate. After 3 minutes the depth is 15 cm. Assuming depth is proportional to time, what is the depth after 8 minutes?

  1. 24 cm
  2. 40 cm Correct answer.
  3. 45 cm
  4. 120 cm

Explanation

Rate = 15/3 = 5 cm/min, so after 8 min the depth is 5 × 8 = 40 cm.

Question 31Quadratic Functions and Equations

The graph of y = ax^2 + c passes through (0, -3) and opens downward. Which could be a and c?

  1. a = 2, c = -3
  2. a = -2, c = -3 Correct answer.
  3. a = -2, c = 3
  4. a = 2, c = 3

Explanation

The y-intercept (0, -3) gives c = -3; opening downward requires a < 0, so a = -2 fits.

Question 32Linear Equations and Functions

Two quantities x and y satisfy y = k/x. If x is increased by 50%, by what factor does y change?

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

Explanation

Increasing x by 50% gives x' = 1.5x, so y' = k/(1.5x) = (2/3)·(k/x) = (2/3)y.

Question 33Linear Equations and Functions

A rectangle has width x and area 24. Its height h as a function of x is h = 24/x. What graph best represents this?

  1. A straight line through the origin
  2. An upward-opening parabola
  3. A hyperbola in the first quadrant Correct answer.
  4. A horizontal line

Explanation

h = 24/x is an inverse-proportion relation, whose graph is a hyperbola; for positive x it lies in the first quadrant.

Question 34Quadratic Functions and Equations

The parabola y = ax^2 has its vertex at the origin. When x = 3, y = 18. Find the value of y when x = -3.

  1. 9
  2. 18 Correct answer.
  3. -18
  4. 0

Explanation

From 18 = a·9, a = 2, so y = 2x^2 is even; at x = -3, y = 2·9 = 18.

Question 35Linear Equations and Functions

A worker completes a job alone in 6 hours; a second worker alone in 3 hours. Working together, how long do they take?

  1. 9 hours
  2. 2 hours Correct answer.
  3. 1.5 hours
  4. 4.5 hours

Explanation

Rates add: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 job per hour, so together they take 2 hours.

Question 36Quadratic Functions and Equations

For y = -x^2 + 4, what is the maximum value of y and where does it occur?

  1. 4 at x = 0 Correct answer.
  2. 0 at x = 2
  3. 4 at x = 2
  4. -4 at x = 0

Explanation

The parabola opens downward with vertex at (0, 4), so the maximum y is 4 at x = 0.

Question 37Quadratic Functions and Equations

Two functions are f(x) = 2x + 1 and g(x) = x^2. Find g(f(1)).

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

Explanation

f(1) = 2(1) + 1 = 3, so g(f(1)) = g(3) = 3^2 = 9.

Question 38Geometry: Plane Figures and Angles

Two angles of a triangle measure 38° and 71°. What is the third angle?

  1. 109°
  2. 90°
  3. 71° Correct answer.
  4. 81°

Explanation

Angles of a triangle sum to 180°, so the third angle is 180 - 38 - 71 = 71°.

Question 39Geometry: Plane Figures and Angles

In triangle ABC, angles A and B are equal and angle C = 50°. Find angle A.

  1. 130°
  2. 80°
  3. 65° Correct answer.
  4. 50°

Explanation

A + B + C = 180 with A = B, so 2A + 50 = 180, giving A = 65°.

Question 40Geometry: Plane Figures and Angles

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

  1. 12 cm
  2. 10 cm Correct answer.
  3. 5 cm
  4. 14 cm

Explanation

By the Pythagorean theorem, c = √(6^2 + 8^2) = √(36 + 64) = √100 = 10 cm.

Question 41Geometry: Plane Figures and Angles

An arc subtends a central angle of 72° in a circle of radius 10 cm. What is the arc length (use π ≈ 3.14)?

  1. 31.4 cm
  2. 12.56 cm Correct answer.
  3. 62.8 cm
  4. 6.28 cm

Explanation

Arc length = (72/360) × 2πr = (1/5) × 2 × 3.14 × 10 = (1/5) × 62.8 = 12.56 cm.

Question 42Geometry: Congruence, Similarity and Measurement

The ratio of the areas of two similar triangles is 9 : 25. What is the ratio of their corresponding sides?

  1. 81 : 625
  2. 3 : 5 Correct answer.
  3. 9 : 25
  4. 3 : √5

Explanation

Area ratio is the square of the side ratio, so the side ratio is √9 : √25 = 3 : 5.

Question 43Geometry: Plane Figures and Angles

In triangle ABC, point D lies on AB and point E lies on AC, with DE ∥ BC. If AD = 4, DB = 6, and BC = 15, find DE.

  1. 10
  2. 6 Correct answer.
  3. 9
  4. 22.5

Explanation

By similarity, DE/BC = AD/AB = 4/(4+6) = 4/10, so DE = (4/10) × 15 = 6.

Question 44Geometry: Plane Figures and Angles

A tangent from an external point P touches a circle at T. If O is the centre and OT = 8 cm, OP = 10 cm, find PT.

  1. 18 cm
  2. 6 cm Correct answer.
  3. 2 cm
  4. √(164) cm

Explanation

The radius OT is perpendicular to tangent PT, so PT = √(OP² - OT²) = √(100 - 64) = √36 = 6 cm.

Question 45Geometry: Plane Figures and Angles

An inscribed angle subtends an arc of 110°. What is the measure of the inscribed angle?

  1. 220°
  2. 70°
  3. 110°
  4. 55° Correct answer.

Explanation

An inscribed angle is half the central angle (arc), so it is 110/2 = 55°.

Question 46Geometry: Plane Figures and Angles

An equilateral triangle has sides of 8 cm. What is its area?

  1. 8√3 cm²
  2. 32 cm²
  3. 16 cm²
  4. 16√3 cm² Correct answer.

Explanation

Drop a perpendicular from one vertex to the opposite side; it bisects that side into two 4 cm halves. By the Pythagorean theorem the height is √(8² - 4²) = √(64 - 16) = √48 = 4√3 cm. Area = (1/2) × base × height = (1/2) × 8 × 4√3 = 16√3 cm².

Question 47Geometry: Plane Figures and Angles

In a right triangle, the side opposite the 30° angle is 5 cm. What is the hypotenuse?

  1. 10 cm Correct answer.
  2. 2.5 cm
  3. 5√3 cm
  4. 5 cm

Explanation

In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse, so the hypotenuse is 2 × 5 = 10 cm.

Question 48Geometry: Plane Figures and Angles

In triangle ABC, M is the midpoint of side AB and N is the midpoint of side AC. If BC = 14 cm, what is the length of MN?

  1. 7 cm Correct answer.
  2. 14 cm
  3. 10.5 cm
  4. 28 cm

Explanation

By the midpoint connector theorem (中点連結定理), the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. So MN = 14 ÷ 2 = 7 cm.

Question 49Geometry: Congruence, Similarity and Measurement

A rectangular prism measures 3 cm × 4 cm × 12 cm. What is the length of its space diagonal (the diagonal joining opposite corners through the solid)?

  1. √19 cm
  2. 19 cm
  3. 13 cm Correct answer.
  4. 12.65 cm

Explanation

Apply the Pythagorean theorem twice. The diagonal of the 3 × 4 base is √(3² + 4²) = 5 cm. That base diagonal and the 12 cm height form a right triangle, so the space diagonal is √(5² + 12²) = √(25 + 144) = √169 = 13 cm.

Question 50Geometry: Congruence, Similarity and Measurement

A rectangular prism measures 5 cm × 4 cm × 3 cm. What is its total surface area?

  1. 94 cm² Correct answer.
  2. 60 cm²
  3. 120 cm²
  4. 47 cm²

Explanation

Surface area = 2(5·4 + 5·3 + 4·3) = 2(20 + 15 + 12) = 2 × 47 = 94 cm².