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Kumamoto Entrance Mathematics study materials
Kumamoto Entrance Mathematics study resources

Kumamoto Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Evaluate (-3) + 5 - (-2).

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

Explanation

Rewrite -(-2) as +2, so the expression becomes -3 + 5 + 2 = 4. Working left to right, -3 + 5 = 2, then 2 + 2 = 4.

Question 2Numbers and Algebraic Expressions

Evaluate (-2) × 3 × (-1).

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

Explanation

There are two negative factors, so the product is positive: (-2) × 3 = -6, then -6 × (-1) = 6.

Question 3Numbers and Algebraic Expressions

Evaluate (-4)².

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

Explanation

The exponent applies to -4 as a whole: (-4) × (-4) = 16. A square is always non-negative.

Question 4Numbers and Algebraic Expressions

Simplify 5x - 2x + 3x.

  1. 10x
  2. 4x
  3. 6x Correct answer.
  4. 6

Explanation

Combine the coefficients of like terms: 5 - 2 + 3 = 6, so the expression is 6x.

Question 5Numbers and Algebraic Expressions

Expand -3(x - 4).

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

Explanation

Distribute -3 to both terms: -3 × x = -3x and -3 × (-4) = +12, giving -3x + 12.

Question 6Numbers and Algebraic Expressions

Expand (x + 3)(x - 2).

  1. x² + x + 6
  2. x² + x - 6 Correct answer.
  3. x² + 5x - 6
  4. x² - x - 6

Explanation

Using FOIL: x·x = x², x·(-2) = -2x, 3·x = 3x, 3·(-2) = -6. Combine -2x + 3x = x, giving x² + x - 6.

Question 7Numbers and Algebraic Expressions

Factor x² - 16.

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

Explanation

This is a difference of squares: a² - b² = (a + b)(a - b) with a = x and b = 4, so x² - 16 = (x + 4)(x - 4).

Question 8Numbers and Algebraic Expressions

Factor x² + 7x + 12.

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

Explanation

Find two numbers that multiply to 12 and add to 7: those are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).

Question 9Numbers and Algebraic Expressions

Factor 3x² - 12 completely.

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

Explanation

First factor out 3: 3(x² - 4). Then x² - 4 is a difference of squares: 3(x + 2)(x - 2).

Question 10Numbers and Algebraic Expressions

Simplify √8 + √18.

  1. 5
  2. 3√2
  3. √26
  4. 5√2 Correct answer.

Explanation

Simplify each radical: √8 = √(4·2) = 2√2 and √18 = √(9·2) = 3√2. Then 2√2 + 3√2 = 5√2.

Question 11Numbers and Algebraic Expressions

Evaluate √27 × √3.

  1. 27
  2. 9√3
  3. 9 Correct answer.
  4. 3

Explanation

Combine under one root: √(27 × 3) = √81 = 9. Equivalently, √27 = 3√3, so 3√3 × √3 = 3 × 3 = 9.

Question 12Numbers and Algebraic Expressions

If x = 2, evaluate x³ - 3x² + 2x.

  1. 0 Correct answer.
  2. -4
  3. -2
  4. 12

Explanation

Substitute x = 2: 2³ - 3(2²) + 2(2) = 8 - 12 + 4 = 0. Note that x = 2 is a root of x(x - 1)(x - 2), the factored form of this expression.

Question 13Linear Equations and Functions

Solve 3x - 7 = 8.

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

Explanation

Add 7 to both sides: 3x = 15, then divide by 3: x = 5.

Question 14Linear Equations and Functions

Solve 4(x - 2) = 12.

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

Explanation

Divide both sides by 4: x - 2 = 3, then add 2: x = 5.

Question 15Linear Equations and Functions

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

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

Explanation

Cross-multiply: 3(x + 1) = 2(2x - 1), so 3x + 3 = 4x - 2. Subtract 3x and add 2: 5 = x, so x = 5.

Question 16Linear Equations and Functions

Solve 5x - 3 = 2x + 9.

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

Explanation

Subtract 2x from both sides: 3x - 3 = 9. Add 3: 3x = 12, then divide by 3: x = 4.

Question 17Linear Equations and Functions

The system x + y = 7 and x - y = 1 is given. Find x.

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

Explanation

Add the two equations: (x + y) + (x - y) = 7 + 1, so 2x = 8 and x = 4.

Question 18Linear Equations and Functions

The system x + y = 10 and 2x + y = 14 is given. Find y.

  1. 4
  2. 6 Correct answer.
  3. 10
  4. 14

Explanation

Subtract the first equation from the second: (2x + y) - (x + y) = 14 - 10, so x = 4. Then 4 + y = 10 gives y = 6.

Question 19Linear Equations and Functions

The system 3x + 2y = 14 and x - y = 3 is given. Find x.

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

Explanation

From x - y = 3, x = y + 3. Substitute: 3(y + 3) + 2y = 14, so 5y + 9 = 14, y = 1, and x = 4.

Question 20Linear Equations and Functions

The system 2x + 3y = 18 and 3x - y = 5 is given. Find x.

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

Explanation

From 3x - y = 5, y = 3x - 5. Substitute: 2x + 3(3x - 5) = 18, so 11x - 15 = 18, 11x = 33, x = 3.

Question 21Linear Equations and Functions

Solve x² - 9 = 0.

  1. x = 3
  2. x = 3 or x = -3 Correct answer.
  3. x = 9
  4. x = -3

Explanation

Rewrite as x² = 9, so x = ±√9 = ±3. Both 3 and -3 satisfy the equation.

Question 22Linear Equations and Functions

Solve x² + 6x + 9 = 0.

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

Explanation

Factor as (x + 3)² = 0, so x + 3 = 0 and x = -3. This is a perfect-square trinomial with a double root.

Question 23Linear Equations and Functions

Solve x² - 5x + 6 = 0.

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

Explanation

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

Question 24Linear Equations and Functions

Solve 2x² - 5x - 3 = 0.

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

Explanation

Factor: (2x + 1)(x - 3) = 0. Set each factor to zero: 2x + 1 = 0 gives x = -1/2, and x - 3 = 0 gives x = 3.

Question 25Linear Equations and Functions

Solve the inequality -3x + 1 > 7.

  1. x > -2
  2. x < 2
  3. x > 2
  4. x < -2 Correct answer.

Explanation

Subtract 1: -3x > 6. Divide by -3 and reverse the inequality: x < -2.

Question 26Linear Equations and Functions

Solve the inequality (2x - 1)/3 ≤ 5.

  1. x ≤ 16
  2. x ≤ 5
  3. x ≤ 8 Correct answer.
  4. x ≤ 7

Explanation

Multiply both sides by 3: 2x - 1 ≤ 15. Add 1: 2x ≤ 16. Divide by 2: x ≤ 8.

Question 27Linear Equations and Functions

The sum of two numbers is 12 and their difference is 4. Find the larger number.

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

Explanation

Let the numbers be x (larger) and y. Then x + y = 12 and x - y = 4. Adding gives 2x = 16, so x = 8.

Question 28Linear Equations and Functions

Twice a number, increased by 5, gives 23. Find the number.

  1. 14
  2. 28
  3. 9 Correct answer.
  4. 18

Explanation

Let the number be x. Then 2x + 5 = 23, so 2x = 18 and x = 9. Check: 2 × 9 + 5 = 23.

Question 29Linear Equations and Functions

A father is 3 times as old as his son. In 12 years, he will be twice as old as his son. Find the son's current age.

  1. 36
  2. 12 Correct answer.
  3. 6
  4. 24

Explanation

Let the son be s years old; the father is 3s. In 12 years: 3s + 12 = 2(s + 12), so 3s + 12 = 2s + 24, giving s = 12.

Question 30Linear Equations and Functions

y is proportional to x. When x = 3, y = 12. Find y when x = 5.

  1. 20 Correct answer.
  2. 9
  3. 4
  4. 15

Explanation

Since y = kx, substitute: 12 = 3k, so k = 4. Then when x = 5, y = 4 × 5 = 20.

Question 31Linear Equations and Functions

y is proportional to x. When x = 4, y = 6. Find the constant of proportionality.

  1. 3/2 Correct answer.
  2. 10
  3. 24
  4. 2/3

Explanation

With y = kx, substitute x = 4 and y = 6: 6 = 4k, so k = 6/4 = 3/2.

Question 32Linear Equations and Functions

y is proportional to x. When x = 2, y = -6. Find y when x = -5.

  1. -10
  2. 10
  3. -15
  4. 15 Correct answer.

Explanation

Find k: -6 = 2k, so k = -3. Then y = -3 × (-5) = 15.

Question 33Linear Equations and Functions

y is inversely proportional to x. When x = 3, y = 4. Find y when x = 6.

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

Explanation

Since y = k/x, substitute: 4 = k/3, so k = 12. Then y = 12/6 = 2. As x doubles, y halves.

Question 34Linear Equations and Functions

y is inversely proportional to x. When x = 2, y = 5. Find the constant of proportionality.

  1. 2/5
  2. 10 Correct answer.
  3. 5/2
  4. 7

Explanation

With y = k/x, substitute x = 2 and y = 5: 5 = k/2, so k = xy = 10.

Question 35Linear Equations and Functions

y is inversely proportional to x. When x = 4, y = 3. Find y when x = -2.

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

Explanation

Find k: k = xy = 4 × 3 = 12. Then y = 12/(-2) = -6.

Question 36Linear Equations and Functions

y is inversely proportional to x. If x doubles, what happens to y?

  1. y doubles
  2. y becomes a quarter
  3. y quadruples
  4. y becomes half Correct answer.

Explanation

Since xy = k is constant, when x becomes 2x, y must become k/(2x) = y/2 to keep the product fixed. So y halves.

Question 37Linear Equations and Functions

z is inversely proportional to x and directly proportional to y. When x = 2 and y = 3, z = 12. Find z when x = 4 and y = 6.

  1. 12 Correct answer.
  2. 18
  3. 24
  4. 6

Explanation

Write z = k·(y/x). Substitute the first point: 12 = k·(3/2), so k = 8. Then z = 8·(6/4) = 8·(3/2) = 12.

Question 38Linear Equations and Functions

A line passes through (0, 2) and (1, 5). Find its slope.

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

Explanation

Slope m = (5 - 2)/(1 - 0) = 3/1 = 3. The y-intercept is 2 but the slope is the rise over run.

Question 39Linear Equations and Functions

The equation of a line is y = 2x + 1. Find y when x = 3.

  1. 5
  2. 7 Correct answer.
  3. 6
  4. 9

Explanation

Substitute x = 3: y = 2(3) + 1 = 6 + 1 = 7.

Question 40Linear Equations and Functions

A line has slope -2 and passes through (3, 4). Find the y-intercept.

  1. 10 Correct answer.
  2. -10
  3. 6
  4. -2

Explanation

Use y = mx + b with m = -2 and the point (3, 4): 4 = -2(3) + b, so 4 = -6 + b and b = 10.

Question 41Linear Equations and Functions

Find the equation of the line passing through (1, 3) and (-1, 7).

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

Explanation

Slope m = (7 - 3)/(-1 - 1) = 4/(-2) = -2. Using (1, 3): 3 = -2(1) + b, so b = 5. The line is y = -2x + 5.

Question 42Linear Equations and Functions

The lines y = 2x + 1 and y = -x + 4 intersect. Find the x-coordinate of the intersection point.

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

Explanation

Set the expressions equal: 2x + 1 = -x + 4, so 3x = 3 and x = 1. (The y-value is 3.)

Question 43Linear Equations and Functions

A line passes through (2, 5) and is parallel to y = 3x - 2. Find its equation.

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

Explanation

Parallel lines share the same slope, so m = 3. Using (2, 5): 5 = 3(2) + b, so 5 = 6 + b and b = -1. The line is y = 3x - 1.

Question 44Linear Equations and Functions

A line is perpendicular to y = (1/2)x + 3 and passes through (4, 1). Find its equation.

  1. y = 2x + 9
  2. y = (1/2)x - 1
  3. y = -2x + 9 Correct answer.
  4. y = -2x - 7

Explanation

Perpendicular slope is the negative reciprocal of 1/2, which is -2. Using (4, 1): 1 = -2(4) + b, so 1 = -8 + b and b = 9. The line is y = -2x + 9.

Question 45Quadratic Functions and Equations

For y = x² + 2x + 1, find y when x = -1.

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

Explanation

Substitute x = -1: y = (-1)² + 2(-1) + 1 = 1 - 2 + 1 = 0. This is the vertex value of the perfect square (x + 1)².

Question 46Quadratic Functions and Equations

For y = x² - 3x, find y when x = 4.

  1. 12
  2. 0
  3. 4 Correct answer.
  4. -4

Explanation

Substitute x = 4: y = 4² - 3(4) = 16 - 12 = 4.

Question 47Quadratic Functions and Equations

Find the vertex of y = x² - 4x + 3.

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

Explanation

The x-coordinate is -b/(2a) = 4/2 = 2. Substitute: y = 4 - 8 + 3 = -1. The vertex is (2, -1).

Question 48Quadratic Functions and Equations

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

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

Explanation

The axis of symmetry is x = -b/(2a) = -6/2 = -3, a vertical line through the vertex.

Question 49Quadratic Functions and Equations

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

  1. 6
  2. 4
  3. 48
  4. 3 Correct answer.

Explanation

Substitute x = 2 and y = 12: 12 = a(2²) = 4a, so a = 3.

Question 50Quadratic Functions and Equations

The parabola y = x² - 2x - 3 intersects the x-axis at two points. Find the distance between them.

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

Explanation

Set y = 0: x² - 2x - 3 = (x - 3)(x + 1) = 0, so x = 3 and x = -1. The distance is 3 - (-1) = 4.