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

Shimane Public High School Entrance Examination - Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Compute: -3 + 8 - 2.

  1. -3
  2. 13
  3. 3 Correct answer.
  4. 5

Explanation

Work left to right with signed numbers: -3 + 8 = 5, then 5 - 2 = 3.

Question 2Numbers and Algebraic Expressions

Compute: (-4) × (-3) + (-6).

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

Explanation

(-4) × (-3) = 12 (product of two negatives is positive); 12 + (-6) = 6.

Question 3Numbers and Algebraic Expressions

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

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

Explanation

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

Question 4Numbers and Algebraic Expressions

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

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

Explanation

(x+3)(x+4) = x² + 4x + 3x + 12 = x² + 7x + 12.

Question 5Numbers and Algebraic Expressions

Factorise: x² − 9.

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

Explanation

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

Question 6Numbers 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

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

Question 7Numbers and Algebraic Expressions

Simplify: (x² + 2x + 1) − (x² − 3x + 4).

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

Explanation

Distribute the minus: x² + 2x + 1 − x² + 3x − 4 = (2x + 3x) + (1 − 4) = 5x − 3.

Question 8Numbers and Algebraic Expressions

Solve the simultaneous equations: x + y = 7, x − y = 1.

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

Explanation

Add the equations: 2x = 8 → x = 4. Substitute: 4 + y = 7 → y = 3. Check: 4 − 3 = 1. ✓

Question 9Numbers and Algebraic Expressions

Solve: 3x + 2y = 12, x + 2y = 8.

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

Explanation

Subtract equation 2 from equation 1: (3x − x) + (2y − 2y) = 12 − 8 → 2x = 4 → x = 2. Then 2 + 2y = 8 → 2y = 6 → y = 3.

Question 10Numbers and Algebraic Expressions

A father is 4 times as old as his son. In 12 years he will be twice as old. How old is the son now?

  1. 6 years old Correct answer.
  2. 12 years old
  3. 8 years old
  4. 4 years old

Explanation

Let the son be x; the father is 4x. In 12 years: 4x + 12 = 2(x + 12) → 4x + 12 = 2x + 24 → 2x = 12 → x = 6. Check: father 24, son 6; in 12 years they are 36 and 18, a 2:1 ratio. So the son is 6 years old.

Question 11Numbers and Algebraic Expressions

A bag contains red and blue marbles in the ratio 3:2. If there are 25 marbles in total, how many are blue?

  1. 10 Correct answer.
  2. 5
  3. 12
  4. 15

Explanation

Total parts = 3 + 2 = 5. Blue = (2/5) × 25 = 10.

Question 12Numbers and Algebraic Expressions

Solve: 2(x − 3) ≤ 8. Which is the solution set for x?

  1. x ≤ 5
  2. x ≤ 1
  3. x ≤ 7 Correct answer.
  4. x ≥ 7

Explanation

Divide both sides by 2 (positive, so the inequality keeps direction): x − 3 ≤ 4 → x ≤ 7.

Question 13Numbers and Algebraic Expressions

Divide: (6x² + 4x) ÷ 2x.

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

Explanation

Divide each term by 2x: 6x² ÷ 2x = 3x, 4x ÷ 2x = 2. Result: 3x + 2.

Question 14Numbers and Algebraic Expressions

A shop in Matsue sells pens at 120 yen each. If you buy a set of 5, the total is 550 yen. How much cheaper per pen is the set compared with buying singly?

  1. 20 yen cheaper per pen
  2. 10 yen cheaper per pen Correct answer.
  3. 5 yen cheaper per pen
  4. 0 yen (same price)

Explanation

Set price per pen = 550 ÷ 5 = 110 yen. Single price = 120 yen. Difference = 120 − 110 = 10 yen cheaper per pen.

Question 15Numbers and Algebraic Expressions

Solve: x² − 5x + 6 = 0.

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

Explanation

Factorise: (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Question 16Numbers and Algebraic Expressions

Solve using the quadratic formula: x² − 4x − 5 = 0.

  1. x = 5 or x = -1 Correct answer.
  2. x = -5 or x = -1
  3. x = 1 or x = -5
  4. x = 5 or x = 1

Explanation

a=1, b=-4, c=-5. Discriminant = 16 + 20 = 36; x = (4 ± 6)/2 = 5 or -1.

Question 17Linear Equations and Functions

If y is directly proportional to x and y = 15 when x = 5, find y when x = 8.

  1. 24 Correct answer.
  2. 3
  3. 40
  4. 18

Explanation

y = kx; k = 15/5 = 3. So y = 3 × 8 = 24.

Question 18Quadratic Functions and Equations

If y is directly proportional to x² and y = 12 when x = 2, find y when x = 3.

  1. 6
  2. 27 Correct answer.
  3. 36
  4. 18

Explanation

y = kx²; 12 = k·4 → k = 3. So y = 3·9 = 27.

Question 19Linear Equations and Functions

The graph of y = -2x + 3 has what y-intercept?

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

Explanation

Set x = 0: y = -2(0) + 3 = 3, so the y-intercept is 3.

Question 20Linear Equations and Functions

Find the equation of the line with slope 2 passing through (3, 5).

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

Explanation

y − 5 = 2(x − 3) → y = 2x − 6 + 5 = 2x − 1.

Question 21Linear Equations and Functions

Find the intersection point of y = 2x − 3 and y = -x + 6.

  1. (1, -1)
  2. (3, 9)
  3. (3, 3) Correct answer.
  4. (0, 6)

Explanation

Set equal: 2x − 3 = -x + 6 → 3x = 9 → x = 3; y = 2(3) − 3 = 3. Point (3, 3).

Question 22Linear Equations and Functions

Two linear functions y = ax + 1 and y = 2x + b intersect on the y-axis. Which relation between a and b must hold?

  1. b = 1 Correct answer.
  2. a = 2
  3. a + b = 1
  4. a = b

Explanation

On the y-axis x = 0; the first line gives y = 1, so the intersection is (0,1). The second line at x = 0 gives y = b, so for (0,1) we need b = 1. (a can be any value - the lines meet on the y-axis at the first line's intercept.)

Question 23Quadratic Functions and Equations

For the quadratic function y = x² − 4x + 3, what are the x-intercepts (roots)?

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

Explanation

Set y = 0: x² − 4x + 3 = (x − 1)(x − 3) = 0 → x = 1 or x = 3.

Question 24Quadratic Functions and Equations

What is the minimum value of y = x² + 2x + 5?

  1. 0
  2. 4 Correct answer.
  3. 2
  4. 5

Explanation

Vertex at x = -b/2a = -1; y = (-1)² + 2(-1) + 5 = 1 − 2 + 5 = 4. Since a > 0, this is the minimum.

Question 25Quadratic Functions and Equations

How many real solutions does the equation x² − 2x + 5 = 0 have?

  1. infinitely many
  2. 2
  3. 1 (repeated)
  4. 0 (no real solutions) Correct answer.

Explanation

Discriminant = (-2)² − 4(1)(5) = 4 − 20 = -16 < 0, so there are no real solutions.

Question 26Quadratic Functions and Equations

The parabola y = a(x − 2)² + 1 passes through (0, 9). Find a.

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

Explanation

Substitute (0,9): 9 = a(0 − 2)² + 1 = 4a + 1 → 4a = 8 → a = 2.

Question 27Linear Equations and Functions

Which is the axis of symmetry of y = (x − 3)² + 4?

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

Explanation

Vertex form y = a(x − h)² + k has axis x = h; here h = 3, so x = 3.

Question 28Linear Equations and Functions

A ball is thrown so its height (metres) after t seconds is y = -5t² + 20t. What is the maximum height?

  1. 40 metres
  2. 20 metres Correct answer.
  3. 5 metres
  4. 10 metres

Explanation

Vertex at t = -b/2a = -20/(-10) = 2 s; y = -5(4) + 20(2) = -20 + 40 = 20 m.

Question 29Quadratic Functions and Equations

Solve the system from the previous intersection: y = x + 2 and y = x². What are the intersection points?

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

Explanation

x² − x − 2 = (x − 2)(x + 1) = 0 → x = 2 or x = -1. Points: (-1, 1) and (2, 4).

Question 30Quadratic Functions and Equations

As x grows large, the function y = -x² + 4x + 1 tends toward which value?

  1. 1
  2. Positive infinity (+∞)
  3. Negative infinity (-∞) Correct answer.
  4. 0

Explanation

With a < 0, the parabola opens downward, so y → -∞ as x → ±∞.

Question 31Linear Equations and Functions

Which is the range (possible y-values) of y = (x + 1)² − 3?

  1. y ≥ 0
  2. y ≥ -3 Correct answer.
  3. y ≤ -3
  4. all real y

Explanation

The squared term is ≥ 0, so y ≥ 0 − 3 = -3; the minimum is -3.

Question 32Linear Equations and Functions

If y = kx and y = 18 when x = 6, what is the constant of proportionality k?

  1. 3 Correct answer.
  2. 1/3
  3. 12
  4. 108

Explanation

In direct proportion y = kx, the constant k is the ratio y/x, which stays the same for every matching pair of values. Substituting the given pair gives k = 18/6 = 3, so the relationship is y = 3x.

Question 33Linear Equations and Functions

The graph of an inversely proportional relationship y = k/x (k > 0) is what kind of curve?

  1. A circle centred at the origin
  2. A parabola opening upward
  3. A rectangular hyperbola in the 1st and 3rd quadrants Correct answer.
  4. A straight line through the origin

Explanation

y = k/x with k > 0 has two branches in the 1st and 3rd quadrants, a rectangular hyperbola; it never passes through the origin.

Question 34Geometry: Plane Figures and Angles

In a right triangle, the legs are 6 and 8. What is the hypotenuse?

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

Explanation

By Pythagoras: 6² + 8² = 36 + 64 = 100; hypotenuse = √100 = 10.

Question 35Geometry: Plane Figures and Angles

An arc of a circle of radius 6 subtends a central angle of 60°. What is the arc length (leave π symbolic)?

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

Explanation

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

Question 36Geometry: Plane Figures and Angles

A chord of a circle of radius 10 is 12 long. What is the perpendicular distance from the centre to the chord?

  1. √244
  2. 8 Correct answer.
  3. 5
  4. 6

Explanation

Half the chord is 6; radius 10 to the endpoint forms a right triangle: distance = √(10² − 6²) = √64 = 8.

Question 37Geometry: Plane Figures and Angles

A triangle has base 10 and height 6. What is its area?

  1. 15
  2. 60
  3. 16
  4. 30 Correct answer.

Explanation

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

Question 38Geometry: Plane Figures and Angles

A trapezoid has parallel sides 6 and 10, and height 4. What is its area?

  1. 32 Correct answer.
  2. 16
  3. 64
  4. 60

Explanation

Area = (1/2) × (6 + 10) × 4 = (1/2) × 16 × 4 = 32.

Question 39Geometry: Plane Figures and Angles

The angles of a regular pentagon's interior each measure how many degrees?

  1. 72°
  2. 120°
  3. 108° Correct answer.
  4. 135°

Explanation

Sum of interior angles = (5 − 2) × 180 = 540°; each = 540/5 = 108°.

Question 40Geometry: Plane Figures and Angles

The sum of the interior angles of a hexagon (6 sides) is?

  1. 900°
  2. 360°
  3. 540°
  4. 720° Correct answer.

Explanation

Sum = (n − 2) × 180 = (6 − 2) × 180 = 720°.

Question 41Geometry: Congruence, Similarity and Measurement

A cube has edge 4. What is its volume?

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

Explanation

A cube has all three dimensions equal, so its volume is edge × edge × edge = edge³. With an edge of 4, the volume is 4³ = 4 × 4 × 4 = 64 cubic units.

Question 42Geometry: Congruence, Similarity and Measurement

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

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

Explanation

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

Question 43Geometry: Plane Figures and Angles

Two angles of a triangle are equal (each 40°) and the third is 100°. This triangle is best classified as?

  1. Isosceles and right
  2. Scalene and right
  3. Equilateral
  4. Isosceles and obtuse Correct answer.

Explanation

Two equal angles → two equal sides → isosceles; one angle 100° > 90° → obtuse.

Question 44Geometry: Plane Figures and Angles

In triangle ABC, point D on AB and E on AC such that DE ∥ BC, AD = 3, DB = 6, AE = 4. Find EC.

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

Explanation

By the basic proportionality (midpoint/intercept) theorem, AD/DB = AE/EC: 3/6 = 4/EC → EC = 8.

Question 45Geometry: Plane Figures and Angles

The diagonal of a square is 8√2. What is its area?

  1. 8
  2. 128
  3. 32
  4. 64 Correct answer.

Explanation

For a square of side s, diagonal = s√2; so s = 8, area = 64. (Also area = (diagonal²)/2 = 128/2 = 64.)

Question 46Geometry: Plane Figures and Angles

A circle is inscribed in a square of side 10. What is the area of the circle (take π ≈ 3.14)?

  1. 31.4
  2. 100
  3. 157
  4. 78.5 Correct answer.

Explanation

The circle's diameter equals the square's side 10, so radius = 5; area = 3.14 × 25 = 78.5.

Question 47Geometry: Plane Figures and Angles

A triangle has sides 7, 24, 25. Is it a right triangle?

  1. Only if scaled
  2. Yes, because 7² + 24² = 25² Correct answer.
  3. Cannot be determined
  4. No

Explanation

7² + 24² = 49 + 576 = 625 = 25², so by the converse of Pythagoras it is a right triangle.

Question 48Data Handling and Probability

Find the mean of the data set: 4, 8, 6, 10, 7.

  1. 7 Correct answer.
  2. 8
  3. 5
  4. 35

Explanation

Sum = 4 + 8 + 6 + 10 + 7 = 35; mean = 35/5 = 7.

Question 49Data Handling and Probability

Find the median of: 2, 8, 4, 10 (even number of values).

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

Explanation

Sort: 2, 4, 8, 10. Median = average of the two middle values (4 and 8) = 6.

Question 50Data Handling and Probability

A die is rolled once. What is the probability of rolling a number greater than 4?

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

Explanation

Numbers greater than 4 are {5, 6} - 2 outcomes out of 6: 2/6 = 1/3.