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Gifu Entrance Mathematics study materials
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Gifu Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate the value of (-8) + (+5) - (-3).

  1. 0 Correct answer.
  2. -6
  3. -16
  4. 6

Explanation

Subtracting a negative number becomes addition, so -(-3) = +3. The expression is -8 + 5 + 3. Computing left to right: -8 + 5 = -3, then -3 + 3 = 0.

Question 2Numbers and Algebraic Expressions

Calculate the value of (-3)² + (-2) × 4 - 6 ÷ (-2).

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

Explanation

Follow the order of operations: exponents and multiplication/division before addition/subtraction. (-3)² = 9, (-2) × 4 = -8, and 6 ÷ (-2) = -3, so the expression is 9 + (-8) - (-3) = 9 - 8 + 3 = 4.

Question 3Numbers and Algebraic Expressions

Calculate 3/4 - 5/6.

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

Explanation

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

Question 4Numbers and Algebraic Expressions

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

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

Explanation

Substitute the values: 5(-2) - 3(4) = -10 - 12. Both terms are negative, so the total is -22.

Question 5Numbers and Algebraic Expressions

Calculate (5/6) ÷ (-2/3).

  1. 5/4
  2. -5/9
  3. -4/5
  4. -5/4 Correct answer.

Explanation

Dividing by a fraction means multiplying by its reciprocal: (5/6) ÷ (-2/3) = (5/6) × (-3/2). Multiplying gives -15/12, which simplifies to -5/4.

Question 6Numbers and Algebraic Expressions

Simplify the expression 3(2a - 5) - 4(a - 3).

  1. 2a - 3 Correct answer.
  2. 2a + 3
  3. 2a - 27
  4. -2a - 3

Explanation

Distribute each factor: 3(2a - 5) = 6a - 15 and -4(a - 3) = -4a + 12. Combining like terms gives 6a - 4a - 15 + 12 = 2a - 3.

Question 7Numbers and Algebraic Expressions

Calculate √12 × √3.

  1. 6 Correct answer.
  2. √15
  3. 36
  4. 2√3

Explanation

Use the rule √a × √b = √(ab): √12 × √3 = √36. Since 36 = 6², the value is 6.

Question 8Numbers and Algebraic Expressions

Expand the expression (2x + 3)(x - 4).

  1. 2x² - 5x - 12 Correct answer.
  2. 2x² - 5x + 12
  3. 2x² - 11x - 12
  4. 2x² + 5x - 12

Explanation

Multiply each term: 2x·x = 2x², 2x·(-4) = -8x, 3·x = 3x, and 3·(-4) = -12. Combining the middle terms gives -8x + 3x = -5x, so the result is 2x² - 5x - 12.

Question 9Numbers and Algebraic Expressions

Factor the expression x² + 7x + 12.

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

Explanation

Look for two numbers whose product is 12 and whose sum is 7. Those numbers are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).

Question 10Numbers and Algebraic Expressions

Factor the expression x² - 9.

  1. (x - 3)²
  2. (x - 9)(x + 1)
  3. (x + 3)(x - 3) Correct answer.
  4. (x + 3)²

Explanation

This is a difference of squares: x² - 9 = x² - 3². Using a² - b² = (a + b)(a - b), the factorization is (x + 3)(x - 3).

Question 11Numbers and Algebraic Expressions

Calculate (x + 5)² - (x - 5)².

  1. 20x Correct answer.
  2. 2x² + 50
  3. 10x
  4. 50

Explanation

Expand each square: (x + 5)² = x² + 10x + 25 and (x - 5)² = x² - 10x + 25. Subtracting cancels x² and 25, leaving 10x - (-10x) = 20x.

Question 12Numbers and Algebraic Expressions

Calculate √48 - √27 + √12.

  1. 9√3
  2. 5√3
  3. 3√3 Correct answer.
  4. √33

Explanation

Simplify each radical: √48 = 4√3, √27 = 3√3, and √12 = 2√3. Then 4√3 - 3√3 + 2√3 = (4 - 3 + 2)√3 = 3√3.

Question 13Numbers and Algebraic Expressions

Calculate (√5 + 2)(√5 - 2).

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

Explanation

Use the difference-of-squares pattern (a + b)(a - b) = a² - b². Here (√5)² - 2² = 5 - 4 = 1.

Question 14Numbers and Algebraic Expressions

Simplify (3a²b)² × 2ab ÷ 6a³b².

  1. 3a²b Correct answer.
  2. 3ab²
  3. 3a³b
  4. a²b

Explanation

First square: (3a²b)² = 9a⁴b². Multiply by 2ab to get 18a⁵b³. Dividing by 6a³b² gives (18/6)a⁵⁻³b³⁻² = 3a²b.

Question 15Numbers and Algebraic Expressions

Find the value of x² - 4x + 4 when x = 2 - √3.

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

Explanation

Notice x² - 4x + 4 = (x - 2)². Substituting x = 2 - √3 gives (2 - √3 - 2)² = (-√3)² = 3. Factoring first avoids messy radical expansion.

Question 16Numbers and Algebraic Expressions

Rationalize the denominator and simplify: 6/(√3 - 1).

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

Explanation

Multiply numerator and denominator by the conjugate √3 + 1: 6(√3 + 1)/((√3 - 1)(√3 + 1)) = 6(√3 + 1)/(3 - 1) = 6(√3 + 1)/2. Dividing 6 by 2 gives 3(√3 + 1) = 3√3 + 3.

Question 17Numbers and Algebraic Expressions

Given x + y = 7 and xy = 4, find the value of x² + y².

  1. 41 Correct answer.
  2. 57
  3. 33
  4. 49

Explanation

Use the identity x² + y² = (x + y)² - 2xy. Substituting gives 7² - 2(4) = 49 - 8 = 41.

Question 18Numbers and Algebraic Expressions

Calculate 0.3².

  1. 0.09 Correct answer.
  2. 0.06
  3. 0.9
  4. 0.6

Explanation

0.3² means 0.3 × 0.3. Since 3 × 3 = 9 and there are two decimal places in total, the product is 0.09.

Question 19Numbers and Algebraic Expressions

Express 252 as a product of prime factors.

  1. 2 × 3 × 6 × 7
  2. 2² × 3² × 7 Correct answer.
  3. 2² × 3³ × 7
  4. 2³ × 3² × 7

Explanation

Divide repeatedly by primes: 252 = 2 × 126 = 2 × 2 × 63 = 2² × 3 × 21 = 2² × 3 × 3 × 7. So 252 = 2² × 3² × 7.

Question 20Numbers and Algebraic Expressions

How many integers n satisfy √10 < n < √50?

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

Explanation

Since √10 ≈ 3.16 and √50 ≈ 7.07, the integers strictly between them are 4, 5, 6, and 7. That is 4 integers.

Question 21Linear Equations and Functions

Solve the equation 3x - 7 = 11.

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

Explanation

Add 7 to both sides: 3x = 18. Dividing both sides by 3 gives x = 6. Checking: 3(6) - 7 = 18 - 7 = 11.

Question 22Linear Equations and Functions

Solve the equation 5(x - 2) = 3x + 4.

  1. -3
  2. 3
  3. 14
  4. 7 Correct answer.

Explanation

Distribute: 5x - 10 = 3x + 4. Move the x-terms left and constants right: 5x - 3x = 4 + 10, so 2x = 14 and x = 7. Checking: 5(7 - 2) = 25 and 3(7) + 4 = 25.

Question 23Linear Equations and Functions

Solve the inequality 2x - 5 < 7.

  1. x < 12
  2. x < 1
  3. x > 6
  4. x < 6 Correct answer.

Explanation

Add 5 to both sides: 2x < 12. Dividing by the positive number 2 keeps the inequality direction: x < 6.

Question 24Linear Equations and Functions

Solve the simultaneous equations: 2x + y = 10 and x - y = 2.

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

Explanation

Add the two equations to eliminate y: 3x = 12, so x = 4. Substitute into x - y = 2: 4 - y = 2, giving y = 2. Checking the first equation: 2(4) + 2 = 10.

Question 25Linear Equations and Functions

Solve the quadratic equation x² - 5x + 6 = 0.

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

Explanation

Factor by finding two numbers with product 6 and sum -5: those are -2 and -3. So (x - 2)(x - 3) = 0, giving x = 2 or x = 3.

Question 26Linear Equations and Functions

Solve the quadratic equation x² = 7x.

  1. x = 0, x = -7
  2. x = 0, x = 7 Correct answer.
  3. x = 7
  4. x = √7, x = -√7

Explanation

Move all terms to one side: x² - 7x = 0, then factor: x(x - 7) = 0. The product is zero when x = 0 or x - 7 = 0, so x = 0 or x = 7.

Question 27Linear Equations and Functions

Solve the quadratic equation 2x² - 4x - 6 = 0.

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

Explanation

Divide every term by 2 first: x² - 2x - 3 = 0. Then factor: (x - 3)(x + 1) = 0, giving x = 3 or x = -1.

Question 28Linear Equations and Functions

Three times a certain number plus 5 equals 26. Find the number.

  1. 11/3
  2. 31/3
  3. 21
  4. 7 Correct answer.

Explanation

Let the number be x. Then 3x + 5 = 26, so 3x = 21 and x = 7. Checking: 3(7) + 5 = 21 + 5 = 26.

Question 29Linear Equations and Functions

Buying 2 pencils and 3 erasers costs 340 yen, and buying 1 pencil and 4 erasers costs 320 yen. What is the price of one pencil?

  1. 20 yen
  2. 40 yen
  3. 60 yen
  4. 80 yen Correct answer.

Explanation

Let p be the pencil price and e the eraser price: 2p + 3e = 340 and p + 4e = 320. Double the second equation: 2p + 8e = 640, then subtract the first: 5e = 300, so e = 60. Then p = 320 - 4(60) = 80 yen.

Question 30Linear Equations and Functions

Solve the quadratic equation 3x² - 5x - 1 = 0 using the quadratic formula.

  1. x = (5 ± √37)/3
  2. x = (-5 ± √37)/6
  3. x = (5 ± √37)/6 Correct answer.
  4. x = (5 ± √13)/6

Explanation

With a = 3, b = -5, c = -1, the formula gives x = (5 ± √(25 - 4·3·(-1)))/(2·3) = (5 ± √(25 + 12))/6 = (5 ± √37)/6.

Question 31Linear Equations and Functions

Solve the system of inequalities: 3x - 2 ≥ 4x - 7 and 2x + 1 < 9.

  1. 4 ≤ x ≤ 5
  2. x < 4 Correct answer.
  3. x ≤ 5
  4. x ≤ 4

Explanation

From the first inequality: -x ≥ -5, and dividing by -1 flips the sign to x ≤ 5. From the second: 2x < 8, so x < 4. Both must hold, so the tighter condition x < 4 is the answer.

Question 32Linear Equations and Functions

Solve the equation x/3 + 1 = 5.

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

Explanation

Subtract 1 from both sides: x/3 = 4. Multiplying both sides by 3 gives x = 12. Checking: 12/3 + 1 = 4 + 1 = 5.

Question 33Linear Equations and Functions

A brother leaves home walking at 60 m/min. Ten minutes later, his sister leaves home and runs at 180 m/min along the same route. How many minutes after the sister starts does she catch up to her brother?

  1. 15 minutes
  2. 5 minutes Correct answer.
  3. 10/3 minutes
  4. 10 minutes

Explanation

Let t be the sister's running time. The brother has walked t + 10 minutes, so 180t = 60(t + 10). Then 120t = 600, giving t = 5 minutes.

Question 34Linear Equations and Functions

Solve the equation (x - 3)² = 16.

  1. x = 7, x = -1 Correct answer.
  2. x = 7
  3. x = 19, x = -13
  4. x = -7, x = 1

Explanation

Take square roots of both sides: x - 3 = ±4. So x - 3 = 4 gives x = 7, and x - 3 = -4 gives x = -1.

Question 35Linear Equations and Functions

A two-digit number has digits summing to 11. Reversing its digits produces a number 27 greater than the original. Find the original number.

  1. 74
  2. 47 Correct answer.
  3. 65
  4. 56

Explanation

Let the tens digit be x and ones digit y: x + y = 11 and 10y + x = 10x + y + 27. The second equation simplifies to 9y - 9x = 27, so y - x = 3. Solving the system gives x = 4, y = 7, so the number is 47.

Question 36Linear Equations and Functions

y is inversely proportional to x, and y = 24/x. What is the value of y when x = 6?

  1. 18
  2. 6
  3. 4 Correct answer.
  4. 144

Explanation

Substitute x = 6 into y = 24/x: y = 24/6 = 4. In inverse proportion, doubling x halves y, so a larger x gives a smaller y.

Question 37Linear Equations and Functions

What is the slope of the line y = 2x + 3?

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

Explanation

In the slope-intercept form y = mx + b, the slope is m, the coefficient of x. Here m = 2, so each increase of 1 in x increases y by 2.

Question 38Linear Equations and Functions

What is the y-intercept of the line y = -x + 5?

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

Explanation

The y-intercept is the value of y when x = 0. Substituting x = 0 gives y = -0 + 5 = 5, so the line crosses the y-axis at (0, 5).

Question 39Quadratic Functions and Equations

For the function y = 2x², what is the value of y when x = 3?

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

Explanation

Substitute x = 3: y = 2 × 3² = 2 × 9 = 18. The square applies to x before multiplying by the coefficient 2.

Question 40Linear Equations and Functions

Find the equation of the line with slope 3 that passes through the point (2, 7).

  1. y = 3x + 1 Correct answer.
  2. y = 7x + 3
  3. y = 3x + 13
  4. y = 3x + 7

Explanation

Start with y = 3x + b and substitute the point: 7 = 3(2) + b, so b = 7 - 6 = 1. The equation is y = 3x + 1.

Question 41Linear Equations and Functions

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

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

Explanation

The slope is (8 - 2)/(3 - 1) = 6/2 = 3. Substituting (1, 2) into y = 3x + b gives 2 = 3 + b, so b = -1. The equation is y = 3x - 1.

Question 42Linear Equations and Functions

Find the coordinates of the intersection point of the lines y = 2x + 1 and y = -x + 7.

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

Explanation

Set the expressions equal: 2x + 1 = -x + 7, so 3x = 6 and x = 2. Substituting back: y = 2(2) + 1 = 5. The intersection is (2, 5), and it satisfies the second line too: -2 + 7 = 5.

Question 43Linear Equations and Functions

y is inversely proportional to x, and the graph passes through (4, 6). What is y when x = 8?

  1. 2
  2. 12
  3. 3 Correct answer.
  4. 48

Explanation

For inverse proportion, xy = a. From (4, 6), a = 4 × 6 = 24, so y = 24/x. When x = 8, y = 24/8 = 3.

Question 44Quadratic Functions and Equations

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

  1. 8
  2. 24
  3. 6
  4. 12 Correct answer.

Explanation

At x = 1, y = 3; at x = 3, y = 27. The rate of change is (27 - 3)/(3 - 1) = 24/2 = 12.

Question 45Quadratic Functions and Equations

For the function y = -2x² with domain -2 ≤ x ≤ 1, find the range of y.

  1. 0 ≤ y ≤ 8
  2. -8 ≤ y ≤ 0 Correct answer.
  3. -8 ≤ y ≤ -2
  4. -2 ≤ y ≤ 0

Explanation

Since the parabola opens downward with vertex at the origin, the maximum value in the domain is y = 0 at x = 0. The minimum occurs at the endpoint farthest from 0: at x = -2, y = -2(4) = -8. So the range is -8 ≤ y ≤ 0.

Question 46Quadratic Functions and Equations

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

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

Explanation

Substitute the point: 12 = a × 2² = 4a. Dividing by 4 gives a = 3, so the function is y = 3x².

Question 47Linear Equations and Functions

y is proportional to x, and the graph passes through (3, -12). Express y as a function of x.

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

Explanation

Direct proportion has the form y = ax. Substituting (3, -12): -12 = 3a, so a = -4 and y = -4x.

Question 48Linear Equations and Functions

Taro leaves home at 8:00 and walks at 60 m/min to a library 1800 m away. At what time does he arrive?

  1. 8:36
  2. 8:30 Correct answer.
  3. 8:45
  4. 8:20

Explanation

Time = distance ÷ speed = 1800 ÷ 60 = 30 minutes. Leaving at 8:00, he arrives at 8:30.

Question 49Quadratic Functions and Equations

For the function y = (1/2)x², find the rate of change as x increases from -4 to 2.

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

Explanation

At x = -4, y = (1/2)(16) = 8; at x = 2, y = (1/2)(4) = 2. The rate of change is (2 - 8)/(2 - (-4)) = -6/6 = -1.

Question 50Linear Equations and Functions

The line y = 2x - 6 intersects the x-axis and the y-axis at two points. Find the area of the triangle formed by these two points and the origin.

  1. 6
  2. 18
  3. 9 Correct answer.
  4. 12

Explanation

The x-intercept comes from 0 = 2x - 6, so x = 3, giving (3, 0). The y-intercept is (0, -6). The triangle has base 3 and height 6, so its area is (1/2) × 3 × 6 = 9.