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Gunma Entrance Mathematics study materials
Gunma Entrance Mathematics study resources

Gunma Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate the value of (-12) + (+7) - (-4).

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

Explanation

Subtracting a negative number becomes addition, so -(-4) = +4. The expression is -12 + 7 + 4. Computing left to right: -12 + 7 = -5, then -5 + 4 = -1.

Question 2Numbers and Algebraic Expressions

Calculate the value of (-4)² + (-3) × 2 - 8 ÷ (-2).

  1. -18
  2. 26
  3. 14 Correct answer.
  4. 6

Explanation

Follow the order of operations: exponents and multiplication/division before addition/subtraction. (-4)² = 16, (-3) × 2 = -6, and 8 ÷ (-2) = -4, so the expression is 16 + (-6) - (-4) = 16 - 6 + 4 = 14.

Question 3Numbers and Algebraic Expressions

Calculate 5/8 - 1/6 + 3/4.

  1. 11/24
  2. -11/24
  3. 29/12
  4. 29/24 Correct answer.

Explanation

Use the common denominator 24: 5/8 = 15/24, 1/6 = 4/24, and 3/4 = 18/24. Then 15/24 - 4/24 + 18/24 = 29/24, which is already in simplest form.

Question 4Numbers and Algebraic Expressions

Evaluate the expression 4x - 2y when x = -3 and y = -5.

  1. -2 Correct answer.
  2. 2
  3. 22
  4. -22

Explanation

Substitute the values: 4(-3) - 2(-5) = -12 + 10 = -2. The product -2 times -5 is +10, which is then added to -12.

Question 5Numbers and Algebraic Expressions

Calculate (3/4) ÷ (-5/8).

  1. 6/5
  2. -5/6
  3. -15/32
  4. -6/5 Correct answer.

Explanation

Dividing by a fraction means multiplying by its reciprocal: (3/4) ÷ (-5/8) = (3/4) × (-8/5). Multiplying gives -24/20, which simplifies to -6/5.

Question 6Numbers and Algebraic Expressions

Simplify √72 - √50 + √18.

  1. 4√2 Correct answer.
  2. 2√2
  3. 8√2
  4. 14√2

Explanation

Simplify each radical: √72 = 6√2, √50 = 5√2, and √18 = 3√2. Then 6√2 - 5√2 + 3√2 = (6 - 5 + 3)√2 = 4√2.

Question 7Numbers and Algebraic Expressions

Matchsticks are laid in a row of squares that share sides: 1 square uses 4 matchsticks, 2 squares use 7, and 3 squares use 10. How many matchsticks are needed to make 20 squares?

  1. 60
  2. 80
  3. 64
  4. 61 Correct answer.

Explanation

The first square needs 4 matchsticks, and every square after it adds only 3, because one side is shared with the previous square. For n squares the count is 4 + 3(n − 1) = 3n + 1. Substituting n = 20 gives 3(20) + 1 = 61 matchsticks.

Question 8Numbers and Algebraic Expressions

Expand and simplify (2x + 3)(3x - 2) - (x - 1)².

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

Explanation

First expand (2x + 3)(3x - 2) = 6x² - 4x + 9x - 6 = 6x² + 5x - 6. Then (x - 1)² = x² - 2x + 1. Subtracting: 6x² + 5x - 6 - (x² - 2x + 1) = 6x² + 5x - 6 - x² + 2x - 1 = 5x² + 7x - 7.

Question 9Numbers and Algebraic Expressions

Expand (2x - 1)(3x + 5).

  1. 6x² - 7x - 5
  2. 6x² + 7x - 5 Correct answer.
  3. 6x² + 7x + 5
  4. 6x² + 13x - 5

Explanation

Use FOIL: First 2x·3x = 6x², Outer 2x·5 = 10x, Inner (-1)·3x = -3x, Last (-1)·5 = -5. Combining the middle terms: 10x - 3x = 7x, so the result is 6x² + 7x - 5.

Question 10Numbers and Algebraic Expressions

Factor 3x² + 10x + 8.

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

Explanation

Find two numbers whose product is 3·8 = 24 and sum is 10: those are 6 and 4. Split the middle term: 3x² + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2).

Question 11Numbers and Algebraic Expressions

Factor 6x² - 7x - 3.

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

Explanation

Find two numbers whose product is 6·(-3) = -18 and sum is -7: those are -9 and 2. Split the middle term: 6x² - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (3x + 1)(2x - 3).

Question 12Numbers and Algebraic Expressions

Factor x² - 16.

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

Explanation

This is a difference of squares: x² - 16 = x² - 4² = (x + 4)(x - 4). The sum and difference of the same two terms produces the difference of squares.

Question 13Numbers and Algebraic Expressions

Simplify 2(3a - 2b) - 3(a - b) + 4(a + 2b).

  1. 7a + 7b Correct answer.
  2. 7a + 3b
  3. 3a + 7b
  4. 7a - 7b

Explanation

Distribute each term: 6a - 4b - 3a + 3b + 4a + 8b. Collecting a-terms: 6a - 3a + 4a = 7a. Collecting b-terms: -4b + 3b + 8b = 7b. So the result is 7a + 7b.

Question 14Numbers and Algebraic Expressions

Calculate (3.2 × 10⁴) × (2 × 10³) and write the result in scientific notation.

  1. 6.4 × 10¹
  2. 5.2 × 10⁷
  3. 6.4 × 10¹²
  4. 6.4 × 10⁷ Correct answer.

Explanation

Multiply the coefficients: 3.2 × 2 = 6.4. Add the exponents: 10⁴ × 10³ = 10⁷. So the product is 6.4 × 10⁷.

Question 15Numbers and Algebraic Expressions

Find all values of x that satisfy |x - 2| = 5.

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

Explanation

The absolute value equation |x - 2| = 5 gives two cases: x - 2 = 5, so x = 7; or x - 2 = -5, so x = -3. Both values satisfy the original equation.

Question 16Numbers and Algebraic Expressions

Simplify √50 + √8 - √32.

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

Explanation

Simplify each radical: √50 = 5√2, √8 = 2√2, and √32 = 4√2. Then 5√2 + 2√2 - 4√2 = (5 + 2 - 4)√2 = 3√2.

Question 17Numbers and Algebraic Expressions

How many positive divisors does 180 have?

  1. 36
  2. 18 Correct answer.
  3. 9
  4. 12

Explanation

First factor: 180 = 2² × 3² × 5. The number of divisors is (2+1)(2+1)(1+1) = 3 × 3 × 2 = 18, by adding one to each exponent and multiplying.

Question 18Numbers and Algebraic Expressions

Simplify (3x²y)² × (-2xy³).

  1. -18x⁶y⁵
  2. 18x⁵y⁵
  3. -18x⁵y⁶
  4. -18x⁵y⁵ Correct answer.

Explanation

First square: (3x²y)² = 9x⁴y². Then multiply by -2xy³: 9 × (-2) = -18, x⁴ × x = x⁵, y² × y³ = y⁵. The result is -18x⁵y⁵.

Question 19Numbers and Algebraic Expressions

Calculate (-2)³ × (-3)² ÷ (-6)².

  1. -2 Correct answer.
  2. -72
  3. 2
  4. -1/2

Explanation

Evaluate each power: (-2)³ = -8, (-3)² = 9, and (-6)² = 36. Then -8 × 9 ÷ 36 = -72 ÷ 36 = -2.

Question 20Numbers and Algebraic Expressions

Express the ratio 24 : 36 in simplest integer form.

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

Explanation

Divide both terms by their greatest common divisor, 12: 24 ÷ 12 = 2 and 36 ÷ 12 = 3, so the simplest form is 2 : 3.

Question 21Numbers and Algebraic Expressions

Simplify (3x² - 2x + 5) - 2(x² - 3x + 1).

  1. 5x² + 4x + 3
  2. x² + 4x + 7
  3. x² + 4x + 3 Correct answer.
  4. x² - 8x + 7

Explanation

Distribute the -2: -2(x² - 3x + 1) = -2x² + 6x - 2. Then (3x² - 2x + 5) + (-2x² + 6x - 2) = (3 - 2)x² + (-2 + 6)x + (5 - 2) = x² + 4x + 3.

Question 22Numbers and Algebraic Expressions

Let a = √3 - 1 and b = √3 + 1. Find the value of a² + b² - ab.

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

Explanation

Compute each piece: a² = (√3 - 1)² = 3 - 2√3 + 1 = 4 - 2√3, b² = (√3 + 1)² = 3 + 2√3 + 1 = 4 + 2√3, and ab = (√3 - 1)(√3 + 1) = 3 - 1 = 2. Adding: a² + b² = (4 - 2√3) + (4 + 2√3) = 8, so a² + b² - ab = 8 - 2 = 6.

Question 23Linear Equations and Functions

Solve for x: 3(x - 2) - 2(x + 1) = 5.

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

Explanation

Distribute: 3x - 6 - 2x - 2 = 5, which simplifies to x - 8 = 5. Adding 8 to both sides gives x = 13.

Question 24Linear Equations and Functions

Solve for x: x/3 + x/4 = 7.

  1. 12 Correct answer.
  2. 1
  3. 7
  4. 21

Explanation

Multiply through by the common denominator 12: 4x + 3x = 84, so 7x = 84, giving x = 12.

Question 25Linear Equations and Functions

Solve the simultaneous equations 2x - y = 5 and 3x + 2y = 4.

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

Explanation

From the first equation, y = 2x - 5. Substitute into the second: 3x + 2(2x - 5) = 4, so 3x + 4x - 10 = 4, giving 7x = 14 and x = 2. Then y = 2(2) - 5 = -1.

Question 26Linear Equations and Functions

Solve the simultaneous equations 2x + y = 7 and x + 2y = 11.

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

Explanation

Double the first equation: 4x + 2y = 14. Subtract the second: (4x + 2y) - (x + 2y) = 14 - 11, so 3x = 3 and x = 1. Substituting back: 2(1) + y = 7, giving y = 5.

Question 27Linear Equations and Functions

Solve x² + 2x - 8 = 0.

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

Explanation

Factor: x² + 2x - 8 = (x + 4)(x - 2) = 0, so x = -4 or x = 2. The two numbers 4 and -2 have product -8 and sum 2.

Question 28Linear Equations and Functions

Solve x² + 3x - 10 = 0.

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

Explanation

Factor: x² + 3x - 10 = (x + 5)(x - 2) = 0, so x = -5 or x = 2. The two numbers 5 and -2 have product -10 and sum 3.

Question 29Linear Equations and Functions

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

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

Explanation

Using x = (4 ± √(16 + 4))/2 = (4 ± √20)/2 = (4 ± 2√5)/2 = 2 ± √5. Both roots are real and irrational.

Question 30Linear Equations and Functions

Solve the inequality 3x - 7 < 2x + 1.

  1. x < 8 Correct answer.
  2. x < 8/5
  3. x > 8
  4. x < -8

Explanation

Subtract 2x from both sides: x - 7 < 1. Add 7: x < 8. The direction of the inequality is preserved because no multiplication or division by a negative occurred.

Question 31Linear Equations and Functions

Solve the inequality -2x + 5 ≥ 11.

  1. x ≤ -3 Correct answer.
  2. x ≥ 3
  3. x ≤ -8
  4. x ≥ -3

Explanation

Subtract 5: -2x ≥ 6. Divide by -2, flipping the inequality: x ≤ -3. Dividing or multiplying by a negative reverses the inequality.

Question 32Linear Equations and Functions

Find the range of x that satisfies both 2x - 1 > 3 and x + 2 ≤ 9 simultaneously.

  1. 2 ≤ x < 7
  2. x > 2
  3. x ≤ 7
  4. 2 < x ≤ 7 Correct answer.

Explanation

From 2x - 1 > 3: 2x > 4, so x > 2. From x + 2 ≤ 9: x ≤ 7. Combining: x must be greater than 2 and at most 7, i.e., 2 < x ≤ 7.

Question 33Linear Equations and Functions

A shop sold 80 tickets for a total of 11,200 yen. Adult tickets cost 200 yen and child tickets cost 100 yen. How many adult tickets were sold?

  1. 40
  2. 32 Correct answer.
  3. 48
  4. 56

Explanation

Let a be the number of adult tickets. Then a + (80 - a) = 80, and the revenue is 200a + 100(80 - a) = 11,200. Simplifying: 100a + 8,000 = 11,200, so 100a = 3,200 and a = 32.

Question 34Linear Equations and Functions

A father is 3 times as old as his son. In 10 years, the father will be twice as old as his son. How old is the son now?

  1. 10 Correct answer.
  2. 15
  3. 12
  4. 8

Explanation

Let the son's current age be s. Then the father is 3s now. In 10 years: 3s + 10 = 2(s + 10). Solving: 3s + 10 = 2s + 20, so s = 10.

Question 35Linear Equations and Functions

A boat travels 30 km downstream in 2 hours and 30 km upstream in 3 hours. Find the speed of the current.

  1. 7.5 km/h
  2. 2.5 km/h Correct answer.
  3. 12.5 km/h
  4. 5 km/h

Explanation

Downstream speed = 30/2 = 15 km/h = (boat + current). Upstream speed = 30/3 = 10 km/h = (boat - current). Subtracting: (boat + current) - (boat - current) = 2 × current = 5, so current = 2.5 km/h.

Question 36Linear Equations and Functions

Notebooks cost 120 yen each and pens cost 80 yen each. A student bought 10 items in total for 920 yen. How many notebooks did the student buy?

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

Explanation

Let n be the number of notebooks. Then n + p = 10 and 120n + 80p = 920. Substituting p = 10 - n: 120n + 80(10 - n) = 920, so 40n + 800 = 920, giving 40n = 120 and n = 3.

Question 37Linear Equations and Functions

Find the positive integer x that satisfies 1/x + 1/(x + 1) = 5/6.

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

Explanation

Multiply by 6x(x + 1): 6(x + 1) + 6x = 5x(x + 1), so 12x + 6 = 5x² + 5x. Rearranging: 5x² - 7x - 6 = 0. Factoring: (5x + 3)(x - 2) = 0, so x = 2 or x = -3/5. The positive integer solution is x = 2.

Question 38Linear Equations and Functions

Solve for x in the proportion 3/4 = x/20.

  1. 20
  2. 15 Correct answer.
  3. 18
  4. 12

Explanation

Cross-multiply: 3 × 20 = 4 × x, so 60 = 4x, giving x = 15.

Question 39Linear Equations and Functions

Find all values of x that satisfy |2x + 1| = 7.

  1. x = -3 or x = 4
  2. x = 3 or x = 4
  3. x = 3 or x = -4 Correct answer.
  4. x = 7 or x = -7

Explanation

The absolute value equation gives two cases: 2x + 1 = 7, so 2x = 6 and x = 3; or 2x + 1 = -7, so 2x = -8 and x = -4.

Question 40Linear Equations and Functions

The length of a rectangle is 4 cm more than its width. The area of the rectangle is 96 cm². Find the width.

  1. 8 cm Correct answer.
  2. 12 cm
  3. 10 cm
  4. 6 cm

Explanation

Let the width be w; then the length is w + 4. The area is w(w + 4) = 96, so w² + 4w - 96 = 0. Factoring: (w + 12)(w - 8) = 0, giving w = 8 (the positive root, since width must be positive).

Question 41Linear Equations and Functions

Tiles are laid in a growing pattern: the 1st figure uses 5 tiles, the 2nd uses 8, the 3rd uses 11, and each figure after that uses 3 more tiles than the one before. Which figure number uses exactly 92 tiles?

  1. The 31st
  2. The 29th
  3. The 30th Correct answer.
  4. The 45th

Explanation

The tile counts go up by 3 each time and start at 5, so the n-th figure uses 5 + 3(n − 1) = 3n + 2 tiles. Setting 3n + 2 = 92 gives 3n = 90, so n = 30. Checking: 3(30) + 2 = 92.

Question 42Linear Equations and Functions

Twice a number plus 5 is at most 17. What is the largest integer value the number can take?

  1. 7
  2. 6 Correct answer.
  3. 12
  4. 5

Explanation

Let n be the number. The inequality is 2n + 5 ≤ 17. Subtracting 5: 2n ≤ 12. Dividing by 2: n ≤ 6. The largest integer satisfying this is n = 6.

Question 43Linear Equations and Functions

For the proportional relationship y = 3x, find y when x = -4.

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

Explanation

Substitute x = -4: y = 3(-4) = -12. In a proportional relationship, a negative input produces a negative output.

Question 44Linear Equations and Functions

In an inverse proportion y = k/x, when x = 2 the value of y is 6. Find y when x = -4.

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

Explanation

First find k: k = xy = 2 × 6 = 12. So y = 12/x. When x = -4: y = 12/(-4) = -3. A negative x gives a negative y in an inverse proportion with positive k.

Question 45Linear Equations and Functions

For the linear function y = 2x - 3, find y when x = 4.

  1. 11
  2. 5 Correct answer.
  3. 1
  4. -5

Explanation

Substitute x = 4: y = 2(4) - 3 = 8 - 3 = 5. The slope 2 multiplies x first, then the intercept -3 is subtracted.

Question 46Linear Equations and Functions

Find the slope of the line passing through the points (-2, 3) and (4, 15).

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

Explanation

The slope is (y₂ - y₁)/(x₂ - x₁) = (15 - 3)/(4 - (-2)) = 12/6 = 2. The denominator uses the difference including the sign of -2.

Question 47Linear Equations and Functions

Find the equation of the line with slope 2 that passes through the point (2, 5).

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

Explanation

Using point-slope form: y - 5 = 2(x - 2), so y = 2x - 4 + 5 = 2x + 1. The intercept is b = 5 - 2(2) = 1.

Question 48Linear Equations and Functions

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

  1. (2, 8)
  2. (4, 2)
  3. (2, 4) Correct answer.
  4. (4, 6)

Explanation

Set the expressions equal: x + 2 = -x + 6, so 2x = 4 and x = 2. Substituting back: y = 2 + 2 = 4. The intersection is (2, 4).

Question 49Quadratic Functions and Equations

A quadratic function y = ax² passes through the point (3, 18). Find the value of y when x = -2.

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

Explanation

Substitute (3, 18): 18 = a(3²) = 9a, so a = 2 and the function is y = 2x². When x = -2: y = 2(-2)² = 2(4) = 8. Squaring removes the sign, so y is positive.

Question 50Quadratic Functions and Equations

For the function y = 2x², find the range of values that y takes when x varies over -1 ≤ x ≤ 3.

  1. 0 ≤ y ≤ 2
  2. -2 ≤ y ≤ 18
  3. 0 ≤ y ≤ 18 Correct answer.
  4. 2 ≤ y ≤ 18

Explanation

Because a = 2 is positive, the graph is a parabola opening upward with its lowest point at the origin. The value x = 0 lies inside the interval -1 ≤ x ≤ 3, so the smallest value of y is 2(0)² = 0. The largest value comes from whichever endpoint is furthest from 0, namely x = 3, giving y = 2(3)² = 18. The range is therefore 0 ≤ y ≤ 18.