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Fukui Entrance Mathematics study materials
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Fukui Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate the value of (-8) + (+15) - (-4).

  1. 27
  2. -11
  3. 11 Correct answer.
  4. 3

Explanation

Subtracting a negative number becomes addition, so -(-4) = +4. The expression becomes -8 + 15 + 4. Computing from the left: -8 + 15 = 7, then 7 + 4 = 11.

Question 2Numbers and Algebraic Expressions

Calculate the value of (-5)² × (-2) + 24 ÷ (-6).

  1. -54 Correct answer.
  2. 46
  3. -46
  4. 54

Explanation

Follow the order of operations. First (-5)² = 25, then 25 × (-2) = -50, and 24 ÷ (-6) = -4. Finally, -50 + (-4) = -54.

Question 3Numbers and Algebraic Expressions

Calculate 3/4 - 5/6.

  1. 1/12
  2. -1/12 Correct answer.
  3. 19/12
  4. -1/2

Explanation

Use the common denominator 12: 3/4 = 9/12 and 5/6 = 10/12. Then 9/12 - 10/12 = -1/12. The result is negative because 10/12 is larger than 9/12.

Question 4Numbers and Algebraic Expressions

Calculate 2/3 ÷ (-4/9) + 1/2.

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

Explanation

Division comes first: 2/3 ÷ (-4/9) = 2/3 × (-9/4) = -18/12 = -3/2. Then add: -3/2 + 1/2 = -2/2 = -1.

Question 5Numbers and Algebraic Expressions

Evaluate the expression 2x - 3y when x = 4 and y = -1.

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

Explanation

Substitute the values: 2(4) - 3(-1) = 8 - (-3). Subtracting a negative becomes addition, so 8 + 3 = 11.

Question 6Numbers and Algebraic Expressions

Simplify the expression 3(2a - b) - 2(a - 4b).

  1. 4a - 5b
  2. 4a - 11b
  3. 8a + 5b
  4. 4a + 5b Correct answer.

Explanation

Distribute: 3(2a - b) = 6a - 3b and -2(a - 4b) = -2a + 8b. Combine like terms: (6a - 2a) + (-3b + 8b) = 4a + 5b.

Question 7Numbers and Algebraic Expressions

Simplify (3x + y)/2 - (x - 2y)/3.

  1. (7x - y)/6
  2. (7x + 7y)/6 Correct answer.
  3. (11x + 7y)/6
  4. (7x + 7y)/5

Explanation

Use the common denominator 6: [3(3x + y) - 2(x - 2y)]/6. Expand the numerator: 9x + 3y - 2x + 4y. Combine: 7x + 7y, so the result is (7x + 7y)/6.

Question 8Numbers and Algebraic Expressions

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

  1. x² - 8x - 15
  2. x² + 2x - 15 Correct answer.
  3. x² - 2x - 15
  4. x² + 2x + 15

Explanation

Use the expansion pattern (x + a)(x + b) = x² + (a + b)x + ab. Here a = 5 and b = -3, so a + b = 2 and ab = -15. The result is x² + 2x - 15.

Question 9Numbers and Algebraic Expressions

Expand (2x - 3)².

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

Explanation

Use (a - b)² = a² - 2ab + b² with a = 2x and b = 3. Then (2x)² = 4x², 2(2x)(3) = 12x, and 3² = 9. The result is 4x² - 12x + 9.

Question 10Numbers and Algebraic Expressions

Factor x² - 7x + 12.

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

Explanation

Find two numbers whose sum is -7 and product is +12: those are -3 and -4. So x² - 7x + 12 = (x - 3)(x - 4). Expanding confirms x² - 4x - 3x + 12 = x² - 7x + 12.

Question 11Numbers and Algebraic Expressions

Factor 3x² - 12 completely.

  1. 3(x + 2)(x - 2) Correct answer.
  2. 3(x² - 2)
  3. 3(x + 2)²
  4. 3(x - 2)²

Explanation

First factor out the common factor 3: 3(x² - 4). Then factor the difference of squares: x² - 4 = (x + 2)(x - 2). The complete factorization is 3(x + 2)(x - 2).

Question 12Numbers and Algebraic Expressions

Solve the equation 5x - 7 = 3x + 9.

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

Explanation

Move the x terms to one side: 5x - 3x = 9 + 7, so 2x = 16. Dividing by 2 gives x = 8. Check: 5(8) - 7 = 33 and 3(8) + 9 = 33.

Question 13Numbers and Algebraic Expressions

Solve the equation 0.4x - 1.2 = 0.1x + 0.6.

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

Explanation

Multiply both sides by 10 to clear decimals: 4x - 12 = x + 6. Then 4x - x = 6 + 12, so 3x = 18 and x = 6. Check: 0.4(6) - 1.2 = 1.2 and 0.1(6) + 0.6 = 1.2.

Question 14Numbers and Algebraic Expressions

Solve the simultaneous equations x + y = 9 and 2x - y = 3. Give the answer as (x, y).

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

Explanation

Add the two equations to eliminate y: 3x = 12, so x = 4. Substitute into x + y = 9: y = 5. Check in the second equation: 2(4) - 5 = 3, which is correct.

Question 15Numbers and Algebraic Expressions

Solve the simultaneous equations 3x + 2y = 12 and x - y = -1. Give the answer as (x, y).

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

Explanation

From the second equation, y = x + 1. Substitute into the first: 3x + 2(x + 1) = 12, so 5x + 2 = 12 and x = 2. Then y = 3. Check: 3(2) + 2(3) = 12.

Question 16Numbers and Algebraic Expressions

Three times a number minus 8 equals the number plus 22. Find the number.

  1. 11
  2. 30
  3. 15 Correct answer.
  4. 7

Explanation

Let the number be x. Then 3x - 8 = x + 22. Subtracting x and adding 8 gives 2x = 30, so x = 15. Check: 3(15) - 8 = 37 and 15 + 22 = 37.

Question 17Numbers and Algebraic Expressions

Express √48 in simplest radical form.

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

Explanation

Factor 48 = 16 × 3, where 16 is a perfect square. Then √48 = √16 × √3 = 4√3. Check: (4√3)² = 16 × 3 = 48.

Question 18Numbers and Algebraic Expressions

Calculate √12 × √6 and express the result in simplest radical form.

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

Explanation

Combine the radicals: √12 × √6 = √72. Factor 72 = 36 × 2, so √72 = 6√2. Check: (6√2)² = 36 × 2 = 72, and 12 × 6 = 72.

Question 19Numbers and Algebraic Expressions

Calculate (2 + √3)(2 - √3).

  1. -1
  2. 1 Correct answer.
  3. 4 + 2√3
  4. 7

Explanation

This is a difference of squares: (a + b)(a - b) = a² - b². So (2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1.

Question 20Numbers and Algebraic Expressions

Calculate √2(√8 + √2).

  1. 2√10
  2. 8
  3. 4
  4. 6 Correct answer.

Explanation

Distribute: √2 × √8 = √16 = 4, and √2 × √2 = 2. Add the results: 4 + 2 = 6.

Question 21Numbers and Algebraic Expressions

Calculate (√5 + 2)² - 4√5.

  1. 1 + 4√5
  2. 9 Correct answer.
  3. 13
  4. 5

Explanation

Expand the square: (√5 + 2)² = 5 + 4√5 + 4 = 9 + 4√5. Subtract 4√5: 9 + 4√5 - 4√5 = 9. The radical terms cancel exactly.

Question 22Numbers and Algebraic Expressions

Solve the equation x² = 49.

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

Explanation

Taking the square root of both sides gives x = ±√49 = ±7. A quadratic equation has two solutions because both 7² = 49 and (-7)² = 49.

Question 23Numbers and Algebraic Expressions

Solve the equation x² - 5x + 6 = 0.

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

Explanation

Factor the left side: x² - 5x + 6 = (x - 2)(x - 3) = 0. So x = 2 or x = 3. Check: 2² - 5(2) + 6 = 0 and 3² - 5(3) + 6 = 0.

Question 24Numbers and Algebraic Expressions

Solve the equation x² + 4x - 12 = 0.

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

Explanation

Factor: x² + 4x - 12 = (x + 6)(x - 2) = 0, since 6 and -2 sum to 4 and multiply to -12. The solutions are x = -6 and x = 2. Check: (-6)² + 4(-6) - 12 = 36 - 24 - 12 = 0.

Question 25Numbers and Algebraic Expressions

Solve the equation 2x² - 3x - 1 = 0 using the quadratic formula.

  1. x = (-3 ± √17)/4
  2. x = (3 ± √7)/4
  3. x = (3 ± √17)/2
  4. x = (3 ± √17)/4 Correct answer.

Explanation

Apply x = (-b ± √(b² - 4ac))/2a with a = 2, b = -3, c = -1. The discriminant is 9 - 4(2)(-1) = 9 + 8 = 17. So x = (3 ± √17)/4.

Question 26Numbers and Algebraic Expressions

A rectangle has a perimeter of 26 cm and an area of 40 cm². Find the lengths of its two sides.

  1. 4 cm and 9 cm
  2. 2 cm and 11 cm
  3. 5 cm and 8 cm Correct answer.
  4. 6 cm and 7 cm

Explanation

Let the sides be a and b. Then 2(a + b) = 26, so a + b = 13, and ab = 40. The numbers with sum 13 and product 40 are 5 and 8, since they are the roots of t² - 13t + 40 = (t - 5)(t - 8) = 0.

Question 27Numbers and Algebraic Expressions

The sum of three consecutive integers is 72. Find the largest of the three integers.

  1. 23
  2. 26
  3. 24
  4. 25 Correct answer.

Explanation

Let the smallest integer be n. Then n + (n + 1) + (n + 2) = 72, so 3n + 3 = 72 and n = 23. The largest integer is n + 2 = 25.

Question 28Numbers and Algebraic Expressions

Calculate -3² + (-3)².

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

Explanation

Note the precedence: -3² = -(3²) = -9, while (-3)² = 9. Adding them gives -9 + 9 = 0. The two expressions differ because the exponent binds before the leading minus sign.

Question 29Numbers and Algebraic Expressions

Evaluate (a² - b)/(a + b) when a = 3 and b = -1.

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

Explanation

Substitute: a² - b = 9 - (-1) = 10, and a + b = 3 + (-1) = 2. Then 10 ÷ 2 = 5.

Question 30Numbers and Algebraic Expressions

Given x + y = 7 and xy = 10, find the value of x² + y².

  1. 29 Correct answer.
  2. 39
  3. 69
  4. 49

Explanation

Use the identity x² + y² = (x + y)² - 2xy. Substitute: 7² - 2(10) = 49 - 20 = 29.

Question 31Linear Equations and Functions

y is proportional to x, and y = 6 when x = 2. Find the value of y when x = 5.

  1. 15 Correct answer.
  2. 10
  3. 30
  4. 12

Explanation

For direct proportion, y = ax with a = 6 ÷ 2 = 3. So y = 3x, and when x = 5, y = 3 × 5 = 15.

Question 32Linear Equations and Functions

y is inversely proportional to x, and y = 4 when x = 3. Find the value of y when x = 6.

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

Explanation

For inverse proportion, y = a/x with a = xy = 3 × 4 = 12. So y = 12/x, and when x = 6, y = 12 ÷ 6 = 2.

Question 33Linear Equations and Functions

Find the slope of the line passing through the points (1, 3) and (3, 7).

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

Explanation

Slope = (change in y)/(change in x) = (7 - 3)/(3 - 1) = 4/2 = 2.

Question 34Linear Equations and Functions

Find the equation of the line with slope 3 that passes through the point (2, 1).

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

Explanation

Write y = 3x + b and substitute the point: 1 = 3(2) + b, so b = 1 - 6 = -5. The equation is y = 3x - 5. Check: at x = 2, y = 6 - 5 = 1.

Question 35Linear Equations and Functions

Find the y-coordinate of the point where the line 2x + 3y = 12 crosses the y-axis.

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

Explanation

On the y-axis, x = 0. Substituting gives 3y = 12, so y = 4. The line crosses the y-axis at (0, 4).

Question 36Linear Equations and Functions

Find the equation of the line passing through the points (-2, 5) and (4, -1).

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

Explanation

The slope is (-1 - 5)/(4 - (-2)) = -6/6 = -1. Using y = -x + b with the point (4, -1): -1 = -4 + b, so b = 3. The equation is y = -x + 3. Check with (-2, 5): -(-2) + 3 = 5.

Question 37Linear Equations and Functions

For the function y = 2x - 4, find the value of x when y = 6.

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

Explanation

Substitute y = 6: 6 = 2x - 4. Adding 4 gives 10 = 2x, so x = 5. Check: 2(5) - 4 = 6.

Question 38Quadratic Functions and Equations

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

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

Explanation

Substitute the point: 12 = a(2²) = 4a. Dividing by 4 gives a = 3. Check: y = 3x² at x = 2 gives 3 × 4 = 12.

Question 39Quadratic Functions and Equations

For the function y = 2x², find the rate of change (average rate of change) as x increases from 1 to 3.

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

Explanation

At x = 1, y = 2; at x = 3, y = 18. The rate of change is (18 - 2)/(3 - 1) = 16/2 = 8. For y = ax² this also equals a(p + q) = 2(1 + 3) = 8.

Question 40Quadratic Functions and Equations

For the function y = 3x², find the rate of change as x increases from -1 to 4.

  1. 15
  2. 3
  3. 12
  4. 9 Correct answer.

Explanation

At x = -1, y = 3; at x = 4, y = 48. The rate of change is (48 - 3)/(4 - (-1)) = 45/5 = 9. This matches the shortcut a(p + q) = 3(-1 + 4) = 9.

Question 41Linear Equations and Functions

Find the coordinates of the intersection point of the lines y = x + 2 and y = 2x - 1.

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

Explanation

Set the expressions equal: x + 2 = 2x - 1, so x = 3. Then y = 3 + 2 = 5. The intersection is (3, 5). Check in the second line: 2(3) - 1 = 5.

Question 42Quadratic Functions and Equations

Find all intersection points of the parabola y = 2x² and the line y = x + 3.

  1. (-1, 2) and (3/2, 9/2) Correct answer.
  2. (1, 2) and (-3/2, 9/2)
  3. (-1, 2) and (3/2, 3/2)
  4. (1, 2) and (3, 6)

Explanation

Set 2x² = x + 3, so 2x² - x - 3 = 0. Using the quadratic formula: x = (1 ± √(1 + 24))/4 = (1 ± 5)/4, giving x = 3/2 or x = -1. At x = -1, y = 2(-1)² = 2; at x = 3/2, y = 2(9/4) = 9/2. The points are (-1, 2) and (3/2, 9/2).

Question 43Linear Equations and Functions

For the function y = -3x + 6, find the value of y when x = -2.

  1. -12
  2. 0
  3. 6
  4. 12 Correct answer.

Explanation

Substitute x = -2: y = -3(-2) + 6 = 6 + 6 = 12. The product of two negatives is positive.

Question 44Linear Equations and Functions

A job takes 12 workers exactly 10 days to complete. Assuming each worker works at the same rate, how many days would 8 workers take?

  1. 15 Correct answer.
  2. 20
  3. 7
  4. 12

Explanation

The total work is 12 × 10 = 120 worker-days. With 8 workers, the time is 120 ÷ 8 = 15 days. Workers and days are inversely proportional.

Question 45Linear Equations and Functions

A spring stretches 3 cm for every 50 g of weight hung from it. How far does it stretch under a 175 g weight?

  1. 12 cm
  2. 9 cm
  3. 7 cm
  4. 10.5 cm Correct answer.

Explanation

The stretch is proportional to the weight: 3 cm per 50 g means 3/50 cm per gram. For 175 g: 175 × 3/50 = 525/50 = 10.5 cm.

Question 46Linear Equations and Functions

A tank holds 50 L of water and drains at a constant rate of 4 L per minute. After how many minutes will 14 L remain in the tank?

  1. 9 minutes Correct answer.
  2. 12.5 minutes
  3. 16 minutes
  4. 8 minutes

Explanation

Let x be the minutes elapsed. Then 50 - 4x = 14, so 4x = 36 and x = 9. Check: after 9 minutes, 50 - 36 = 14 L remain.

Question 47Quadratic Functions and Equations

The graph of y = ax² passes through the point (-3, 18). Which of the following points also lies on this graph?

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

Explanation

Substitute the given point into y = ax²: 18 = a(-3)² = 9a, so a = 2 and the graph is y = 2x². At x = 2 this gives y = 2 × 4 = 8, so (2, 8) lies on the graph.

Question 48Linear Equations and Functions

A linear function has a rate of change of -2, and y = 7 when x = 1. Find its equation.

  1. y = -2x - 9
  2. y = 2x + 9
  3. y = -2x + 5
  4. y = -2x + 9 Correct answer.

Explanation

A linear function with rate of change -2 has the form y = -2x + b. Substitute (1, 7): 7 = -2 + b, so b = 9. The equation is y = -2x + 9. Check: at x = 1, y = -2 + 9 = 7.

Question 49Quadratic Functions and Equations

For the function y = 5x², find the value of y when x = -2.

  1. 20 Correct answer.
  2. -20
  3. 100
  4. 10

Explanation

Substitute: y = 5(-2)² = 5 × 4 = 20. The square of -2 is positive 4.

Question 50Linear Equations and Functions

A candle is 20 cm long and burns completely in 25 minutes at a constant rate. After how many minutes is the candle 8 cm long?

  1. 10 minutes
  2. 15 minutes Correct answer.
  3. 12.5 minutes
  4. 20 minutes

Explanation

The burn rate is 20 ÷ 25 = 0.8 cm per minute, so the length is y = 20 - 0.8x. Set 8 = 20 - 0.8x: 0.8x = 12, so x = 15. Check: 20 - 0.8(15) = 20 - 12 = 8.