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Kanagawa Entrance Mathematics study materials
Kanagawa Entrance Mathematics study resources

Kanagawa Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Evaluate (−7) + 4 − (−2).

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

Explanation

Work left to right: (−7) + 4 = −3, then −3 − (−2) = −3 + 2 = −1. Subtracting a negative number is the same as adding its opposite.

Question 2Numbers and Algebraic Expressions

Evaluate (−3) × (−4) − 2 × 5.

  1. −22
  2. 2 Correct answer.
  3. 12
  4. −2

Explanation

(−3) × (−4) = 12, and 2 × 5 = 10, so 12 − 10 = 2. The product of two negatives is positive.

Question 3Numbers and Algebraic Expressions

Evaluate (−2)² × (−3).

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

Explanation

(−2)² = 4, so 4 × (−3) = −12. A negative base squared is positive; multiplying that by a negative gives a negative product.

Question 4Numbers and Algebraic Expressions

Simplify √72 − √50.

  1. √22
  2. √2 Correct answer.
  3. 11√2
  4. 2√2

Explanation

√72 = √(36·2) = 6√2 and √50 = √(25·2) = 5√2, so 6√2 − 5√2 = √2. Simplify each radical before combining like terms.

Question 5Numbers and Algebraic Expressions

Evaluate √18 × √2.

  1. 12
  2. 6 Correct answer.
  3. 2√5
  4. √20

Explanation

Combine under one radical: √18 × √2 = √(18 × 2) = √36 = 6. Multiplication of square roots merges the radicands.

Question 6Numbers and Algebraic Expressions

Rationalize and simplify 6/√3.

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

Explanation

Multiply numerator and denominator by √3: 6/√3 = (6√3)/3 = 2√3. Rationalizing removes the radical from the denominator.

Question 7Numbers and Algebraic Expressions

Evaluate (√3 + √2)(√3 − √2).

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

Explanation

This is a difference-of-squares product: (√3)² − (√2)² = 3 − 2 = 1. The cross terms cancel in (a+b)(a−b) = a² − b².

Question 8Numbers and Algebraic Expressions

Evaluate (−2)³ × (1/2).

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

Explanation

(−2)³ = −8, so −8 × (1/2) = −4. An odd power keeps the negative sign.

Question 9Numbers and Algebraic Expressions

Express 0.375 as a fraction in lowest terms.

  1. 3/80
  2. 8/3
  3. 375/1000
  4. 3/8 Correct answer.

Explanation

0.375 = 375/1000; dividing numerator and denominator by 125 gives 3/8. Reduce by the greatest common divisor.

Question 10Numbers and Algebraic Expressions

Expand (x + 4)(x − 4).

  1. x² − 16 Correct answer.
  2. x² − 8x − 16
  3. x² + 16
  4. x² − 8x + 16

Explanation

This is a difference of squares: (x + 4)(x − 4) = x² − 16. The middle terms cancel because the constants are opposites.

Question 11Numbers and Algebraic Expressions

Expand (2x + 3)(x − 2).

  1. 2x² − x − 6 Correct answer.
  2. 2x² − 5x − 6
  3. 2x² + x − 6
  4. 2x² − 7x + 6

Explanation

FOIL: 2x·x = 2x², 2x·(−2) = −4x, 3·x = 3x, 3·(−2) = −6. The middle terms combine to −4x + 3x = −x, giving 2x² − x − 6.

Question 12Numbers and Algebraic Expressions

Factor x² − 5x − 14.

  1. (x − 7)(x + 2) Correct answer.
  2. (x − 2)(x + 7)
  3. (x − 14)(x + 1)
  4. (x + 7)(x + 2)

Explanation

We need two numbers whose product is −14 and sum is −5: −7 and +2. So x² − 5x − 14 = (x − 7)(x + 2).

Question 13Numbers and Algebraic Expressions

If a = 2 and b = −3, evaluate a² − 2ab + b².

  1. 25 Correct answer.
  2. −5
  3. 1
  4. 13

Explanation

a² − 2ab + b² = (a − b)² = (2 − (−3))² = 5² = 25. Recognizing the perfect-square form shortens the arithmetic.

Question 14Linear Equations and Functions

Solve 2x − 5 = 7.

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

Explanation

Add 5 to both sides: 2x = 12, then divide by 2: x = 6. Isolate the variable with inverse operations.

Question 15Linear Equations and Functions

Solve 3(x + 1) = 12.

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

Explanation

Divide both sides by 3 first: x + 1 = 4, so x = 3. You may also distribute first; both orders agree.

Question 16Linear Equations and Functions

Solve x/2 + 3 = 7.

  1. 4
  2. 2
  3. 20
  4. 8 Correct answer.

Explanation

Subtract 3: x/2 = 4, then multiply by 2: x = 8. Undo the addition before clearing the denominator.

Question 17Linear Equations and Functions

Solve 2x − (x + 4) = −3x + 8.

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

Explanation

Distribute the minus: 2x − x − 4 = −3x + 8, so x − 4 = −3x + 8. Add 3x and add 4: 4x = 12, x = 3.

Question 18Linear Equations and Functions

Solve 5x − 3 = 2x + 9.

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

Explanation

Subtract 2x and add 3: 3x = 12, so x = 4. Collect the variable terms on one side and constants on the other.

Question 19Linear Equations and Functions

Solve x − 7 < 3.

  1. x > 10
  2. x < 10 Correct answer.
  3. x < 4
  4. x ≤ 10

Explanation

Add 7 to both sides: x < 10. Inequalities with addition behave like equations; no sign flip is needed.

Question 20Linear Equations and Functions

Solve 2(x − 3) ≥ 4.

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

Explanation

Divide by 2: x − 3 ≥ 2, then add 3: x ≥ 5. Splitting the division first keeps the arithmetic clean.

Question 21Linear Equations and Functions

Solve the simultaneous equations x + y = 10 and x − y = 4.

  1. x = 6, y = 4
  2. x = 4, y = 6
  3. x = 8, y = 2
  4. x = 7, y = 3 Correct answer.

Explanation

Add the two equations: 2x = 14, so x = 7. Substitute back: 7 + y = 10, so y = 3. The elimination method removes y when the equations are summed.

Question 22Linear Equations and Functions

Solve the simultaneous equations 2x + 3y = 12 and x − y = 1.

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

Explanation

From x − y = 1, x = y + 1. Substitute: 2(y + 1) + 3y = 12, so 5y + 2 = 12, y = 2, and x = 3. Substitution turns a system into one equation in one variable.

Question 23Linear Equations and Functions

Solve the simultaneous equations 3x + 2y = 16 and 2x + 3y = 14.

  1. x = 4, y = 2 Correct answer.
  2. x = 5, y = 0.5
  3. x = 4, y = 3
  4. x = 2, y = 5

Explanation

Multiply the first by 3 (9x + 6y = 48) and the second by 2 (4x + 6y = 28); subtract: 5x = 20, x = 4, then 2(4) + 3y = 14 gives y = 2.

Question 24Linear Equations and Functions

Solve x² − 3x − 10 = 0.

  1. x = 5 or x = −2 Correct answer.
  2. x = 5 or x = 2
  3. x = 10 or x = −1
  4. x = 2 or x = −5

Explanation

Factor: (x − 5)(x + 2) = 0, so x = 5 or x = −2. The pair −5 and 2 has product −10 and sum −3.

Question 25Linear Equations and Functions

Solve x² + 5x = 0.

  1. x = 25
  2. x = 5
  3. x = −5
  4. x = 0 or x = −5 Correct answer.

Explanation

Factor out x: x(x + 5) = 0, so x = 0 or x = −5. A common-factor quadratic always has 0 as one root.

Question 26Linear Equations and Functions

Solve 2x² − 7x + 3 = 0.

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

Explanation

When the coefficient of x² is not 1, use the quadratic formula x = (−b ± √(b² − 4ac)) / (2a). With a = 2, b = −7, c = 3: x = (7 ± √(49 − 24)) / 4 = (7 ± 5) / 4, giving x = 12/4 = 3 and x = 2/4 = 1/2. Substituting either value back into the original equation confirms both roots.

Question 27Linear Equations and Functions

Solve x² − 4x + 1 = 0 using the quadratic formula.

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

Explanation

Using x = (−b ± √(b² − 4ac))/(2a) with a = 1, b = −4, c = 1: x = (4 ± √(16 − 4))/2 = (4 ± √12)/2 = (4 ± 2√3)/2 = 2 ± √3.

Question 28Linear Equations and Functions

The sum of two consecutive even integers is 26. Find the smaller integer.

  1. 13
  2. 12 Correct answer.
  3. 14
  4. 24

Explanation

Let the integers be n and n + 2. Then n + (n + 2) = 26, so 2n = 24 and n = 12. The smaller even integer is 12; the pair is 12 and 14.

Question 29Linear Equations and Functions

A father is 3 times as old as his son. In 12 years he will be twice as old. How old is the son now?

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

Explanation

Let the son's age be s; the father is 3s. In 12 years: 3s + 12 = 2(s + 12) = 2s + 24, so s = 12. The son is 12 now; the father is 36.

Question 30Linear Equations and Functions

In a proportional relationship y = kx, y = 15 when x = 5. Find k.

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

Explanation

Substitute: 15 = k·5, so k = 3. The constant of proportionality is the ratio y/x for any point on the line through the origin.

Question 31Linear Equations and Functions

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

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

Explanation

Substitute: 6 = k/2, so k = 12. For inverse proportion the product xy is constant.

Question 32Linear Equations and Functions

In an inverse proportion xy = 24, find y when x = 8.

  1. 3 Correct answer.
  2. 16
  3. 192
  4. 4

Explanation

Substitute x = 8: 8y = 24, so y = 3. Inverse proportion means the product is fixed; divide the constant by the given value.

Question 33Linear Equations and Functions

For y = 2x + 3, find y when x = −1.

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

Explanation

Substitute: y = 2(−1) + 3 = −2 + 3 = 1. Take care with the sign of the substituted value.

Question 34Linear Equations and Functions

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

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

Explanation

Slope = (y₂ − y₁)/(x₂ − x₁) = (8 − 2)/(3 − 1) = 6/2 = 3. Rise over run between the two points.

Question 35Linear Equations and Functions

What is the y-intercept of y = 3x − 4?

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

Explanation

In slope-intercept form y = mx + b, the constant b is the y-intercept. Here b = −4, so the line crosses the y-axis at (0, −4).

Question 36Linear Equations and Functions

Find the equation of the line with y-intercept 2 and slope −3.

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

Explanation

Slope-intercept form is y = mx + b. With m = −3 and b = 2, the equation is y = −3x + 2.

Question 37Linear Equations and Functions

What is the slope of the line y = 5 − 2x?

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

Explanation

Rewrite as y = −2x + 5. The coefficient of x is the slope, so m = −2. Order the terms in slope-intercept form before reading off m.

Question 38Linear Equations and Functions

Find the equation of the line passing through (2, 1) and (4, 7).

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

Explanation

Slope = (7 − 1)/(4 − 2) = 3. Using point-slope with (2, 1): y − 1 = 3(x − 2), so y = 3x − 6 + 1 = 3x − 5.

Question 39Linear Equations and Functions

Find the equation of the line parallel to y = 2x + 1 that passes through (3, 4).

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

Explanation

Parallel lines share the same slope, so m = 2. Using (3, 4): 4 = 2·3 + b, so b = −2 and the equation is y = 2x − 2.

Question 40Quadratic Functions and Equations

For the function y = x², x varies over the range −1 ≤ x ≤ 3. What is the corresponding range of y?

  1. 1 ≤ y ≤ 3
  2. 0 ≤ y ≤ 9 Correct answer.
  3. 1 ≤ y ≤ 9
  4. −1 ≤ y ≤ 9

Explanation

Because the range of x includes 0, the graph passes through its lowest point, so the smallest value of y is 0 rather than the value at either endpoint. The largest value comes from whichever endpoint is farther from 0: x = 3 gives y = 9, while x = −1 gives only y = 1. The range is therefore 0 ≤ y ≤ 9.

Question 41Quadratic Functions and Equations

For the function y = 2x², what is the rate of change (変化の割合) as x increases from 1 to 3?

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

Explanation

The rate of change is the increase in y divided by the increase in x. Here y goes from 2 × 1² = 2 to 2 × 3² = 18, an increase of 16, while x increases by 2. So the rate of change is 16 ÷ 2 = 8. Unlike a straight line, this value changes depending on which interval of x you choose.

Question 42Quadratic Functions and Equations

For y = 2x², find y when x = 3.

  1. 8
  2. 9
  3. 18 Correct answer.
  4. 12

Explanation

y = 2(3²) = 2·9 = 18. Square the input first, then multiply by the coefficient.

Question 43Quadratic Functions and Equations

Find the values of x for which y = x² − 2x − 3 equals 0.

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

Explanation

Factor: (x − 3)(x + 1) = 0, so x = 3 or x = −1. The pair −3 and +1 has product −3 and sum −2.

Question 44Linear Equations and Functions

For y = 3x − 2, what is the average rate of change between x = 1 and x = 4?

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

Explanation

y(1) = 1 and y(4) = 10, so the rate of change is (10 − 1)/(4 − 1) = 9/3 = 3. For a linear function this equals the slope.

Question 45Linear Equations and Functions

Which point lies on the line y = −x + 5?

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

Explanation

Substitute (2, 3): 3 = −2 + 5 = 3 ✓. The point's coordinates must satisfy the equation.

Question 46Quadratic Functions and Equations

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

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

Explanation

Substitute: 12 = a·2² = 4a, so a = 3. Divide by the squared input to isolate the coefficient.

Question 47Linear Equations and Functions

The line y = mx + 4 passes through (2, 10). Find m.

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

Explanation

Substitute (2, 10): 10 = 2m + 4, so 2m = 6 and m = 3. Isolate m by subtracting the intercept and dividing by x.

Question 48Quadratic Functions and Equations

Which statement correctly describes the graph of y = −2x² compared with the graph of y = x²?

  1. It opens downward and has its lowest point at the origin
  2. It opens downward and is narrower than y = x² Correct answer.
  3. It opens downward and is wider than y = x²
  4. It opens upward and is narrower than y = x²

Explanation

In y = ax², the sign of a decides which way the curve opens: a is negative here, so it opens downward. The size of a decides the width: the larger |a| is, the more steeply the curve rises or falls, so it looks narrower. Since |−2| = 2 is greater than 1, y = −2x² is narrower than y = x².

Question 49Linear Equations and Functions

At what point do the lines y = 2x − 1 and y = −x + 5 intersect?

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

Explanation

The point of intersection lies on both lines, so its coordinates satisfy both equations. Setting them equal: 2x − 1 = −x + 5, so 3x = 6 and x = 2. Substituting into either equation gives y = 2(2) − 1 = 3, so the lines meet at (2, 3). This is why solving a pair of simultaneous equations is the same as finding where two lines cross.

Question 50Geometry: Plane Figures and Angles

Two angles of a triangle measure 50° and 60°. What is the measure of the third angle?

  1. 110°
  2. 130°
  3. 70° Correct answer.
  4. 90°

Explanation

The angles of a triangle sum to 180°, so the third angle is 180 − 50 − 60 = 70°.