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Kagoshima Entrance Mathematics study materials
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Kagoshima Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Evaluate (−5) + 8 − (−3).

  1. −6
  2. 6 Correct answer.
  3. 10
  4. 0

Explanation

Work left to right: (−5) + 8 = 3, then 3 − (−3) = 3 + 3 = 6. Subtracting a negative number is the same as adding its opposite.

Question 2Numbers and Algebraic Expressions

Compute (−6) × 4 ÷ (−3).

  1. −24
  2. −8
  3. 24
  4. 8 Correct answer.

Explanation

First (−6) × 4 = −24. Then −24 ÷ (−3) = 8. A negative divided by a negative is positive.

Question 3Numbers and Algebraic Expressions

Simplify 4x − 3(x − 2) − 5.

  1. x + 1 Correct answer.
  2. x − 1
  3. x − 11
  4. 7x − 11

Explanation

Distribute first: −3(x − 2) = −3x + 6. Then 4x − 3x + 6 − 5 = x + 1, combining the like x-terms (4x − 3x = x) and the constants (6 − 5 = 1).

Question 4Numbers and Algebraic Expressions

Simplify −(a − b).

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

Explanation

Distribute the negative sign across both terms inside the parentheses: −(a − b) = −a + b = b − a. The order of the terms is reversed and the signs swap.

Question 5Numbers and Algebraic Expressions

Expand (x + 3)(x − 4).

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

Explanation

Using FOIL: x·x = x², x·(−4) = −4x, 3·x = 3x, 3·(−4) = −12. Combine −4x + 3x = −x, giving x² − x − 12.

Question 6Numbers and Algebraic Expressions

Factor x² − 5x − 24.

  1. (x − 6)(x + 4)
  2. (x − 12)(x + 2)
  3. (x − 8)(x + 3) Correct answer.
  4. (x + 8)(x − 3)

Explanation

We need two numbers whose product is −24 and sum is −5. The numbers −8 and +3 give (−8)(3) = −24 and −8 + 3 = −5, so x² − 5x − 24 = (x − 8)(x + 3).

Question 7Numbers and Algebraic Expressions

Compute the value of √72 − √50 + √18, simplified.

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

Explanation

Simplify each: √72 = √(36·2) = 6√2, √50 = √(25·2) = 5√2, √18 = √(9·2) = 3√2. Then 6√2 − 5√2 + 3√2 = (6 − 5 + 3)√2 = 4√2.

Question 8Numbers and Algebraic Expressions

Evaluate (2³ − 3²) ÷ (−1) + 4.

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

Explanation

2³ = 8 and 3² = 9, so 2³ − 3² = 8 − 9 = −1. Then (−1) ÷ (−1) = 1, and 1 + 4 = 5.

Question 9Numbers and Algebraic Expressions

If a = −2 and b = 5, evaluate a² − 2ab + b².

  1. −9
  2. 49 Correct answer.
  3. 9
  4. 25

Explanation

a² = (−2)² = 4, 2ab = 2(−2)(5) = −20, b² = 25. Then a² − 2ab + b² = 4 − (−20) + 25 = 4 + 20 + 25 = 49. Equivalently, (a − b)² = (−2 − 5)² = (−7)² = 49.

Question 10Linear Equations and Functions

Solve for x: 3x − 7 = 2x + 5.

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

Explanation

Subtract 2x: x − 7 = 5. Add 7: x = 12. Checking, 3(12) − 7 = 29 and 2(12) + 5 = 29.

Question 11Linear Equations and Functions

Solve the simultaneous equations: x + y = 9, 2x − y = 6.

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

Explanation

Add the two equations: (x + y) + (2x − y) = 9 + 6, giving 3x = 15, so x = 5. Substitute back: 5 + y = 9, y = 4. Check: 2(5) − 4 = 6.

Question 12Linear Equations and Functions

Solve 2(x − 3) = 4x + 6.

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

Explanation

Distribute: 2x − 6 = 4x + 6. Subtract 2x: −6 = 2x + 6. Subtract 6: −12 = 2x, so x = −6. Check: 2(−6 − 3) = −18 and 4(−6) + 6 = −18.

Question 13Linear Equations and Functions

Solve the inequality 3x − 5 < x + 7 and express the solution for x.

  1. x < 6 Correct answer.
  2. x > 6
  3. x < 4
  4. x < 12

Explanation

Subtract x: 2x − 5 < 7. Add 5: 2x < 12. Divide by 2: x < 6. The inequality sign is preserved because we divided by a positive number.

Question 14Linear Equations and Functions

Solve the inequality −2x + 4 ≥ 8.

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

Explanation

Subtract 4: −2x ≥ 4. Divide by −2 and flip the inequality sign (dividing by a negative): x ≤ −2.

Question 15Linear Equations and Functions

Solve x² − 6x + 8 = 0 by factoring.

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

Explanation

Find two numbers with product 8 and sum 6: 2 and 4. So x² − 6x + 8 = (x − 2)(x − 4) = 0, giving x = 2 or x = 4.

Question 16Linear Equations and Functions

Solve x² + 4x − 5 = 0.

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

Explanation

Find two numbers with product −5 and sum 4: 5 and −1. So (x + 5)(x − 1) = 0, giving x = 1 or x = −5.

Question 17Linear Equations and Functions

Using the quadratic formula, solve x² − 4x + 1 = 0.

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

Explanation

Here a = 1, b = −4, c = 1. The discriminant is (−4)² − 4(1)(1) = 16 − 4 = 12, and √12 = 2√3. So x = (4 ± 2√3)/2 = 2 ± √3.

Question 18Linear Equations and Functions

A number is 3 less than twice another number; their sum is 21. What is the larger number?

  1. 8
  2. 15
  3. 11
  4. 13 Correct answer.

Explanation

Let the two numbers be x and y with x = 2y − 3 and x + y = 21. Substitute: (2y − 3) + y = 21, so 3y = 24, y = 8. Then x = 2(8) − 3 = 13. The larger number is 13.

Question 19Linear Equations and Functions

A shop sells pens at ¥120 each. If you buy a set of 5, the total is ¥500. How much do you save per pen in the set?

  1. ¥25
  2. ¥40
  3. ¥20 Correct answer.
  4. ¥100

Explanation

Single price per pen is ¥120. In the set, each pen costs 500 ÷ 5 = ¥100. The saving per pen is 120 − 100 = ¥20.

Question 20Linear Equations and Functions

Solve the simultaneous equations 3x + 2y = 16 and x − y = 2.

  1. x = 3, y = 1
  2. x = 5, y = 3
  3. x = 4, y = 2 Correct answer.
  4. x = 2, y = 0

Explanation

From x − y = 2, x = y + 2. Substitute into 3x + 2y = 16: 3(y + 2) + 2y = 16, so 5y + 6 = 16, 5y = 10, y = 2, and x = 4. Check: 3(4) + 2(2) = 16.

Question 21Linear Equations and Functions

For what value of k does x² − kx + 9 have a repeated (double) root?

  1. k = 6 only
  2. k = 6 or k = −6 Correct answer.
  3. k = 9
  4. k = 3

Explanation

A repeated root requires the discriminant to be 0: (−k)² − 4(1)(9) = 0, so k² = 36, giving k = 6 or k = −6. Both yield a perfect-square trinomial.

Question 22Linear Equations and Functions

Which ordered pair satisfies both y = 2x + 1 and y = −x + 7?

  1. (2, 6)
  2. (3, 7)
  3. (2, 5) Correct answer.
  4. (1, 3)

Explanation

Set the equations equal: 2x + 1 = −x + 7, so 3x = 6, x = 2. Then y = 2(2) + 1 = 5. The point (2, 5) lies on both lines.

Question 23Linear Equations and Functions

The sum of three consecutive even integers is 54. What is the largest?

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

Explanation

Let the integers be n, n+2, n+4. Their sum 3n + 6 = 54, so 3n = 48, n = 16. The three are 16, 18, 20, and the largest is 20.

Question 24Linear Equations and Functions

A father is 4 times as old as his son. In 6 years he will be 3 times as old. How old is the son now?

  1. 8
  2. 12 Correct answer.
  3. 9
  4. 6

Explanation

Let the son be s and the father 4s. In 6 years: 4s + 6 = 3(s + 6). So 4s + 6 = 3s + 18, giving s = 12. The son is 12 now (father 48; in 6 years, 54 = 3 × 18).

Question 25Linear Equations and Functions

Which of the following is a proportional (direct-variation) relationship?

  1. y = 3x Correct answer.
  2. y = 3x + 1
  3. y = x²
  4. y = 6/x

Explanation

Direct proportion means y = kx for a constant k, so the graph passes through the origin. Only y = 3x has the form y = kx with no constant term.

Question 26Linear Equations and Functions

Which relationship represents inverse proportion?

  1. y = x + 12
  2. y = 12x
  3. y = 12 − x
  4. y = 12/x Correct answer.

Explanation

Inverse proportion has the form xy = k (constant), i.e. y = k/x. Only y = 12/x fits; as x increases, y decreases so their product stays 12.

Question 27Linear Equations and Functions

What is the slope of the line passing through (1, 2) and (4, 11)?

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

Explanation

Slope = (change in y)/(change in x) = (11 − 2)/(4 − 1) = 9/3 = 3.

Question 28Linear Equations and Functions

Give the equation of the line with slope −2 passing through (3, 1).

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

Explanation

Using y = mx + b with m = −2: 1 = −2(3) + b, so 1 = −6 + b, b = 7. The equation is y = −2x + 7.

Question 29Linear Equations and Functions

For the linear function y = −3x + 6, what is the y-intercept?

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

Explanation

In y = mx + b, the y-intercept is b. Here b = 6, so the line crosses the y-axis at (0, 6).

Question 30Linear Equations and Functions

Find the x-intercept of y = 2x − 8.

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

Explanation

Set y = 0: 0 = 2x − 8, so 2x = 8, x = 4. The x-intercept is (4, 0).

Question 31Linear Equations and Functions

Two points (0, 3) and (2, 7) lie on a line. Find its equation in slope-intercept form.

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

Explanation

Slope = (7 − 3)/(2 − 0) = 4/2 = 2. The y-intercept is the y-value at x = 0, which is 3. So y = 2x + 3.

Question 32Quadratic Functions and Equations

For y = ax² (a > 0), which statement is always true?

  1. The domain is x ≥ 0
  2. The vertex is below the x-axis
  3. The graph opens downward
  4. The axis of symmetry is the y-axis Correct answer.

Explanation

y = ax² has no linear x-term, so its axis of symmetry is x = 0 (the y-axis), and with a > 0 the parabola opens upward with vertex at the origin.

Question 33Quadratic Functions and Equations

For y = 2x², what is the y-value when x = −3?

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

Explanation

Substitute x = −3: y = 2(−3)² = 2(9) = 18. The square of −3 is +9, so the result is positive.

Question 34Quadratic Functions and Equations

Find the axis of symmetry of y = x² − 6x + 5.

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

Explanation

The axis of symmetry is x = −b/(2a) = −(−6)/(2·1) = 6/2 = 3. The parabola is symmetric about x = 3.

Question 35Quadratic Functions and Equations

Find the minimum value of y = x² − 4x + 7.

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

Explanation

The minimum occurs at x = −b/(2a) = 4/2 = 2. Substitute: y = (2)² − 4(2) + 7 = 4 − 8 + 7 = 3.

Question 36Quadratic Functions and Equations

How does the graph of y = −x² compare to y = x²?

  1. It is shifted up by 1
  2. It is shifted right
  3. It is wider
  4. It is reflected across the x-axis Correct answer.

Explanation

Negating the function reflects every point across the x-axis: y = −x² opens downward, the mirror image of y = x² which opens upward.

Question 37Linear Equations and Functions

A quantity is inversely proportional to x and equals 24 when x = 2. Find its value when x = 6.

  1. 8 Correct answer.
  2. 72
  3. 4
  4. 12

Explanation

Inverse proportion: y = k/x. With y = 24 at x = 2, k = 24·2 = 48. So y = 48/6 = 8.

Question 38Linear Equations and Functions

A car travels at a constant speed; distance d (km) is proportional to time t (h), with d = 30 when t = 0.5. Find d when t = 2.5.

  1. 30
  2. 75
  3. 60
  4. 150 Correct answer.

Explanation

Direct proportion: d = kt. With d = 30 at t = 0.5, k = 30/0.5 = 60. So d = 60·2.5 = 150 km.

Question 39Linear Equations and Functions

Lines y = 2x + 3 and y = 2x − 5 have the same slope. How are they related?

  1. Perpendicular
  2. Identical
  3. Parallel Correct answer.
  4. Intersecting on the y-axis

Explanation

Both have slope 2 but different y-intercepts (3 and −5), so they never meet; they are parallel.

Question 40Quadratic Functions and Equations

A point moves so that y = 5x². By what factor does y change when x is doubled?

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

Explanation

If x becomes 2x, then y becomes 5(2x)² = 5·4x² = 4·(5x²). So y is multiplied by 4 - the square law.

Question 41Linear Equations and Functions

Which is the slope of a line perpendicular to y = (1/2)x + 4?

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

Explanation

Perpendicular slopes multiply to −1. The given slope is 1/2, so the perpendicular slope is −1/(1/2) = −2.

Question 42Geometry: Plane Figures and Angles

When two parallel lines are cut by a transversal, which pair of angles is always equal?

  1. Adjacent angles on one line
  2. Supplementary pairs on the same side
  3. Corresponding angles Correct answer.
  4. Same-side interior angles

Explanation

With parallel lines and a transversal, corresponding angles are equal. Same-side interior angles are supplementary, not equal.

Question 43Geometry: Plane Figures and Angles

In triangle ABC, two angles are 50° and 60°. What is the third angle?

  1. 60°
  2. 90°
  3. 70° Correct answer.
  4. 80°

Explanation

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

Question 44Geometry: Congruence, Similarity and Measurement

Which condition guarantees that two triangles are congruent?

  1. SSA (two sides and a non-included angle)
  2. SAS (two sides and the included angle equal) Correct answer.
  3. Two sides equal only
  4. AAA (three angles equal)

Explanation

SAS - two sides and the included angle - is a valid congruence condition. SSA is not generally valid, and AAA only guarantees similarity, not congruence.

Question 45Geometry: Congruence, Similarity and Measurement

Two similar triangles have side lengths in ratio 2:3. The smaller has area 16 cm². What is the area of the larger?

  1. 48 cm²
  2. 64 cm²
  3. 36 cm² Correct answer.
  4. 24 cm²

Explanation

For similar figures, areas are in the ratio of the square of the side ratio: (3/2)² = 9/4. So the larger area = 16 × 9/4 = 36 cm².

Question 46Geometry: Plane Figures and Angles

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

  1. √28
  2. 14
  3. 12
  4. 10 Correct answer.

Explanation

By the Pythagorean theorem, c² = 6² + 8² = 36 + 64 = 100, so c = 10. This is the classic 3-4-5 triangle doubled.

Question 47Geometry: Plane Figures and Angles

A ladder 13 m long leans against a wall with its foot 5 m from the wall. How high up the wall does it reach?

  1. 12 m Correct answer.
  2. 8 m
  3. 10 m
  4. √194 m

Explanation

The wall, ground, and ladder form a right triangle with hypotenuse 13 and one leg 5. Height = √(13² − 5²) = √(169 − 25) = √144 = 12 m.

Question 48Geometry: Plane Figures and Angles

In a circle, an inscribed angle is 30°. What is the measure of the arc it intercepts?

  1. 30°
  2. 60° Correct answer.
  3. 15°
  4. 90°

Explanation

An inscribed angle is half the central angle (and half its intercepted arc), so the arc measures 2 × 30° = 60°.

Question 49Geometry: Plane Figures and Angles

A tangent line and the radius to the point of tangency form what angle?

  1. 180°
  2. 90° Correct answer.
  3. 60°
  4. 45°

Explanation

A tangent to a circle is perpendicular to the radius drawn to the point of tangency, so the angle is 90°.

Question 50Geometry: Plane Figures and Angles

In a circle of radius 5 cm, what is the length of an arc that subtends a central angle of 72°? (Use π ≈ 3.14.)

  1. 1.57 cm
  2. 12.56 cm
  3. 6.28 cm Correct answer.
  4. 3.14 cm

Explanation

Arc length = (central angle/360) × 2πr = (72/360) × 2(3.14)(5) = (1/5) × 31.4 = 6.28 cm.