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Yamanashi Entrance Mathematics study materials
Yamanashi Entrance Mathematics study resources

Yamanashi Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate: (−9) + (+5) − (−4).

  1. −18
  2. −8
  3. 8
  4. 0 Correct answer.

Explanation

Removing parentheses gives −9 + 5 + 4. Working left to right: −9 + 5 = −4, then −4 + 4 = 0. Subtracting a negative number is the same as adding its opposite.

Question 2Numbers and Algebraic Expressions

What is the prime factorization of 90?

  1. 2 × 3 × 5²
  2. 3² × 5 × 7
  3. 2² × 3 × 5
  4. 2 × 3² × 5 Correct answer.

Explanation

90 ÷ 2 = 45 and 45 = 9 × 5 = 3² × 5. Therefore 90 = 2 × 3² × 5.

Question 3Numbers and Algebraic Expressions

Find the greatest common divisor (GCD) of 54 and 72.

  1. 36
  2. 18 Correct answer.
  3. 12
  4. 6

Explanation

54 = 2 × 3³ and 72 = 2³ × 3². The GCD uses the minimum powers: 2¹ × 3² = 18. By the Euclidean algorithm, 72 = 1·54 + 18 and 54 = 3·18, so the GCD is 18.

Question 4Numbers and Algebraic Expressions

Find the least common multiple (LCM) of 14 and 21.

  1. 42 Correct answer.
  2. 7
  3. 28
  4. 84

Explanation

14 = 2 × 7 and 21 = 3 × 7. The LCM takes the highest powers of all primes: 2 × 3 × 7 = 42.

Question 5Numbers and Algebraic Expressions

Simplify: 2⁵ × 3² ÷ (2³ × 3).

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

Explanation

Combine like bases: 2^{5−3} × 3^{2−1} = 2² × 3 = 4 × 3 = 12.

Question 6Numbers and Algebraic Expressions

Simplify (√18 − √8) ÷ √2.

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

Explanation

√18 = 3√2 and √8 = 2√2, so the numerator is 3√2 − 2√2 = √2. Then √2 ÷ √2 = 1.

Question 7Numbers and Algebraic Expressions

Expand (x + 4)(x − 6).

  1. x² − 2x + 24
  2. x² − 2x − 24 Correct answer.
  3. x² + 2x − 24
  4. x² − 10x − 24

Explanation

FOIL: x·x + x(−6) + 4·x + 4(−6) = x² − 6x + 4x − 24 = x² − 2x − 24.

Question 8Numbers and Algebraic Expressions

Expand (3x − 2)².

  1. 9x² − 12x + 4 Correct answer.
  2. 9x² + 12x + 4
  3. 9x² − 12x − 4
  4. 9x² − 6x + 4

Explanation

(a − b)² = a² − 2ab + b² with a = 3x and b = 2 gives 9x² − 12x + 4.

Question 9Numbers and Algebraic Expressions

Factor completely: x² − 9x + 20.

  1. (x − 4)(x + 5)
  2. (x − 4)(x − 5) Correct answer.
  3. (x − 2)(x − 10)
  4. (x + 4)(x + 5)

Explanation

Need two integers with product 20 and sum −9: −4 and −5. Hence x² − 9x + 20 = (x − 4)(x − 5).

Question 10Numbers and Algebraic Expressions

Evaluate −3² + (−3)².

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

Explanation

The two expressions look alike but differ in what the exponent applies to. In −3² only the 3 is squared and the minus sign stays outside, giving −9. In (−3)² the whole of −3 is squared, giving +9. So −9 + 9 = 0.

Question 11Numbers and Algebraic Expressions

Simplify 6a²b × (−2ab³) ÷ 3ab.

  1. −4ab³
  2. 4a²b³
  3. −4a²b³ Correct answer.
  4. −12a²b³

Explanation

Multiply first: 6a²b × (−2ab³) = −12a³b⁴. Then divide by 3ab: coefficients −12 ÷ 3 = −4, a³ ÷ a = a², and b⁴ ÷ b = b³. The result is −4a²b³.

Question 12Numbers and Algebraic Expressions

Calculate (−3)⁴ − (−3)³.

  1. 54
  2. 90
  3. 108 Correct answer.
  4. −54

Explanation

(−3)⁴ = 81 (even power) and (−3)³ = −27. Then 81 − (−27) = 81 + 27 = 108.

Question 13Numbers and Algebraic Expressions

Evaluate |−12| − |−5| + |3|.

  1. 4
  2. 20
  3. 10 Correct answer.
  4. −10

Explanation

|−12| = 12, |−5| = 5, and |3| = 3, so 12 − 5 + 3 = 10.

Question 14Numbers and Algebraic Expressions

A distance is measured as 62 000 m to two significant figures (有効数字2桁). How should it be written in the form a × 10ⁿ?

  1. 62 × 10³ m
  2. 0.62 × 10⁵ m
  3. 6.2 × 10³ m
  4. 6.2 × 10⁴ m Correct answer.

Explanation

In 有効数字 notation the leading number a must satisfy 1 ≤ a < 10, and the two significant figures here are 6 and 2. Since 62 000 = 6.2 × 10 000 and 10 000 = 10⁴, the measurement is written 6.2 × 10⁴ m.

Question 15Numbers and Algebraic Expressions

Simplify 2√18 + √50 − √8.

  1. 6√2
  2. 7√2
  3. 9√2 Correct answer.
  4. 8√2

Explanation

√18 = 3√2, √50 = 5√2, and √8 = 2√2. Therefore 2·3√2 + 5√2 − 2√2 = 6√2 + 5√2 − 2√2 = 9√2.

Question 16Numbers and Algebraic Expressions

What is 15% of 240?

  1. 30
  2. 40
  3. 24
  4. 36 Correct answer.

Explanation

15% = 0.15, and 0.15 × 240 = 36. Alternatively, 10% of 240 is 24 and 5% is 12, so 24 + 12 = 36.

Question 17Numbers and Algebraic Expressions

A number increased by 20% becomes 84. What was the original number?

  1. 68
  2. 64
  3. 72
  4. 70 Correct answer.

Explanation

If the original number is x, then 1.2x = 84, so x = 84 ÷ 1.2 = 70.

Question 18Numbers and Algebraic Expressions

Simplify (2x + 1)/3 − (x − 4)/4 as a single fraction.

  1. (5x + 16)/7
  2. (5x + 16)/12 Correct answer.
  3. (5x − 16)/12
  4. (x + 16)/12

Explanation

The common denominator of 3 and 4 is 12, so the expression becomes [4(2x + 1) − 3(x − 4)]/12. Expanding gives (8x + 4 − 3x + 12)/12. Note that −3 × (−4) = +12. Collecting terms leaves (5x + 16)/12.

Question 19Numbers and Algebraic Expressions

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

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

Explanation

a² − 2ab + b² = (a − b)². Here a − b = 2 − (−3) = 5, so the value is 5² = 25. Directly: 4 − 2(2)(−3) + 9 = 4 + 12 + 9 = 25.

Question 20Numbers and Algebraic Expressions

Round 3.1462 to three decimal places.

  1. 3.147
  2. 3.146 Correct answer.
  3. 3.150
  4. 3.14

Explanation

The fourth decimal digit is 2, which is less than 5, so the third decimal digit stays 6. The rounded value is 3.146.

Question 21Numbers and Algebraic Expressions

Convert 0.375 to a simplified fraction.

  1. 3/7
  2. 3/8 Correct answer.
  3. 37/100
  4. 5/16

Explanation

0.375 = 375/1000. Divide numerator and denominator by 125: 375 ÷ 125 = 3 and 1000 ÷ 125 = 8, so 3/8. Check: 3 ÷ 8 = 0.375.

Question 22Numbers and Algebraic Expressions

Simplify (√7 + √3)(√7 − √3).

  1. 10
  2. 4 Correct answer.
  3. √21
  4. √7 − √3

Explanation

This is a difference of squares: (√7)² − (√3)² = 7 − 3 = 4.

Question 23Numbers and Algebraic Expressions

What is the value of 2¹⁰?

  1. 1024 Correct answer.
  2. 2048
  3. 256
  4. 512

Explanation

Powers of two double each step: 2⁸ = 256, 2⁹ = 512, and therefore 2¹⁰ = 1024.

Question 24Numbers and Algebraic Expressions

Calculate 7/12 + 5/18.

  1. 12/30
  2. 2/3
  3. 35/36
  4. 31/36 Correct answer.

Explanation

LCM of 12 and 18 is 36. Convert: 7/12 = 21/36 and 5/18 = 10/36. Sum: 21/36 + 10/36 = 31/36.

Question 25Linear Equations and Functions

Solve for x: 3x − 7 = 8.

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

Explanation

Add 7 to both sides of 3x − 7 = 8 to obtain 3x = 15. Divide both sides by 3 to get x = 5.

Question 26Linear Equations and Functions

Solve for x: 2(x − 3) = 5x + 6.

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

Explanation

Expand: 2x − 6 = 5x + 6. Subtract 2x: −6 = 3x + 6. Subtract 6: −12 = 3x. Divide by 3: x = −4.

Question 27Linear Equations and Functions

Solve for x: x/3 − x/4 = 1.

  1. 12 Correct answer.
  2. 24
  3. 7
  4. 1

Explanation

Common denominator 12: (4x − 3x)/12 = 1 → x/12 = 1 → x = 12.

Question 28Linear Equations and Functions

Solve the simultaneous equations (連立方程式): 2x + 3y = 7 and 3x − y = 5. What is the ordered pair (x, y)?

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

Explanation

From 3x − y = 5, y = 3x − 5. Substitute: 2x + 3(3x − 5) = 7 → 2x + 9x − 15 = 7 → 11x = 22 → x = 2. Then y = 6 − 5 = 1. So (x, y) = (2, 1).

Question 29Linear Equations and Functions

Solve x² = 49 for real x.

  1. x = 24.5
  2. x = ±7 Correct answer.
  3. x = 7 only
  4. x = ±49

Explanation

Taking square roots: x = ±√49 = ±7. Both 7 and −7 square to 49.

Question 30Linear Equations and Functions

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 = 1 or x = 6

Explanation

Factor: (x − 2)(x − 3) = 0, so x = 2 or x = 3. Check: 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0.

Question 31Linear Equations and Functions

Solve x² − 4x − 12 = 0.

  1. x = 4 or x = −3
  2. x = 12 or x = 1
  3. x = 3 or x = −4
  4. x = 6 or x = −2 Correct answer.

Explanation

Factor x² − 4x − 12 as (x − 6)(x + 2) because (−6)×2 = −12 and −6 + 2 = −4. Therefore x = 6 or x = −2.

Question 32Linear Equations and Functions

Solve the inequality 3x − 5 < 7.

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

Explanation

Add 5: 3x < 12. Divide by 3: x < 4. The inequality direction is unchanged because 3 > 0.

Question 33Linear Equations and Functions

Solve −2x + 6 ≥ 10.

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

Explanation

Subtract 6: −2x ≥ 4. Divide by −2 and reverse the inequality: x ≤ −2.

Question 34Linear Equations and Functions

Two numbers have sum 10 and product 21. What are the numbers?

  1. 2 and 8
  2. 1 and 9
  3. 3 and 7 Correct answer.
  4. 4 and 6

Explanation

They are roots of t² − 10t + 21 = 0. Factor: (t − 3)(t − 7) = 0, so the numbers are 3 and 7.

Question 35Linear Equations and Functions

Solve 5x = 2x + 15.

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

Explanation

Subtract 2x from both sides of 5x = 2x + 15 to get 3x = 15. Divide by 3 to obtain x = 5.

Question 36Linear Equations and Functions

Solve (x + 1)/2 = (x − 3)/4.

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

Explanation

Cross-multiply (or multiply both sides by 4): 2(x + 1) = x − 3 → 2x + 2 = x − 3 → x = −5.

Question 37Linear Equations and Functions

Solve x² + 2x − 4 = 0 using the quadratic formula. Which is correct?

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

Explanation

For ax² + bx + c = 0 with a = 1, b = 2, c = −4: x = [−2 ± √(4 + 16)] / 2 = (−2 ± √20)/2 = (−2 ± 2√5)/2 = −1 ± √5.

Question 38Linear Equations and Functions

Solve the system x + y = 5 and 2x − 3y = 1. Find x.

  1. 14/5
  2. 16/5 Correct answer.
  3. 2
  4. 3

Explanation

From x + y = 5, y = 5 − x. Substitute: 2x − 3(5 − x) = 1 → 2x − 15 + 3x = 1 → 5x = 16 → x = 16/5.

Question 39Linear Equations and Functions

Solve (x − 3)² = 25.

  1. x = 8 only
  2. x = 8 or x = −2 Correct answer.
  3. x = 2 or x = −8
  4. x = 5 or x = −5

Explanation

Take the square root of both sides, remembering both signs: x − 3 = 5 or x − 3 = −5. Adding 3 to each gives x = 8 or x = −2. Checking, (8 − 3)² = 25 and (−2 − 3)² = 25.

Question 40Linear Equations and Functions

For the linear function y = 2x + 3, what is y when x = 4?

  1. 8
  2. 11 Correct answer.
  3. 14
  4. 5

Explanation

Substitute x = 4 into y = 2x + 3: y = 2·4 + 3 = 8 + 3 = 11.

Question 41Linear Equations and Functions

What is the slope of the line through (1, 2) and (5, 10)?

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

Explanation

Slope m = (y₂ − y₁)/(x₂ − x₁) = (10 − 2)/(5 − 1) = 8/4 = 2.

Question 42Linear Equations and Functions

What is the y-intercept of y = −3x + 7?

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

Explanation

In slope-intercept form y = mx + b, the y-intercept is b. Here b = 7, so the graph crosses the y-axis at (0, 7).

Question 43Linear Equations and Functions

A line y = ax + b passes through (2, 5) and (4, 11). Find the equation.

  1. y = 2x + 1
  2. y = −3x + 11
  3. y = 3x − 1 Correct answer.
  4. y = 3x + 1

Explanation

Slope a = (11 − 5)/(4 − 2) = 6/2 = 3. Using (2, 5): 5 = 3·2 + b → b = −1. So y = 3x − 1.

Question 44Quadratic Functions and Equations

For 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. 16
  4. 8 Correct answer.

Explanation

The rate of change is (change in y) ÷ (change in x). When x = 1, y = 2; when x = 3, y = 18. So the rate is (18 − 2) ÷ (3 − 1) = 16 ÷ 2 = 8. Unlike a linear function, y = ax² has a rate of change that depends on the interval chosen.

Question 45Quadratic Functions and Equations

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

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

Explanation

Substitute: 12 = a·(2)² = 4a, so a = 12/4 = 3. The function is y = 3x².

Question 46Linear Equations and Functions

For y = 3x − 5, x takes values from 2 to 6. What is the corresponding range of y (yの変域)?

  1. −5 ≤ y ≤ 13
  2. 1 ≤ y ≤ 13 Correct answer.
  3. 2 ≤ y ≤ 6
  4. 1 ≤ y ≤ 18

Explanation

Because the coefficient of x is positive, y increases as x increases, so the endpoints of the x-range give the endpoints of the y-range. When x = 2, y = 6 − 5 = 1; when x = 6, y = 18 − 5 = 13. The range is therefore 1 ≤ y ≤ 13.

Question 47Linear Equations and Functions

What is the rate of change (傾き) of y = 5x − 1?

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

Explanation

For y = mx + b the constant rate of change is the slope m. Here m = 5.

Question 48Linear Equations and Functions

Find the x-intercept of y = −2x + 8.

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

Explanation

Set y = 0: 0 = −2x + 8 → 2x = 8 → x = 4. The graph crosses the x-axis at (4, 0).

Question 49Linear Equations and Functions

Find the intersection point of y = 2x + 1 and y = −x + 7.

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

Explanation

Set equal: 2x + 1 = −x + 7 → 3x = 6 → x = 2. Then y = 2·2 + 1 = 5. Intersection: (2, 5).

Question 50Linear Equations and Functions

y is proportional to x, and y = 12 when x = 4. Find y when x = 7.

  1. 21 Correct answer.
  2. 24
  3. 16
  4. 28

Explanation

y = kx with 12 = 4k ⇒ k = 3. When x = 7, y = 3·7 = 21. Equivalently y/x is constant: 12/4 = y/7 ⇒ y = 21.