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Nagano Entrance Mathematics study materials
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Nagano Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Compute: (-7) + 3 × (-2)

  1. -1
  2. -13 Correct answer.
  3. 8
  4. 17

Explanation

Order of operations: multiplication before addition. 3 × (-2) = -6, then (-7) + (-6) = -13. The answer is -13.

Question 2Numbers and Algebraic Expressions

Compute: 2/3 - 1/4

  1. 5/12 Correct answer.
  2. 11/12
  3. 1/12
  4. 7/12

Explanation

The LCD of 3 and 4 is 12. Convert: 2/3 = 8/12 and 1/4 = 3/12. Then 8/12 - 3/12 = 5/12.

Question 3Numbers and Algebraic Expressions

Compute: 0.3 × 0.04

  1. 0.012 Correct answer.
  2. 0.12
  3. 0.0012
  4. 1.2

Explanation

Multiply 3 × 4 = 12. The factors have 1 + 2 = 3 decimal places, so the product is 0.012.

Question 4Numbers and Algebraic Expressions

What is the prime factorization of 84?

  1. 2 × 3 × 7
  2. 2² × 3 × 7 Correct answer.
  3. 2² × 3 × 7²
  4. 2³ × 3 × 7

Explanation

84 = 4 × 21 = 2² × 3 × 7. Verify: 4 × 3 × 7 = 84.

Question 5Numbers and Algebraic Expressions

Find the GCD of 36 and 48.

  1. 144
  2. 6
  3. 4
  4. 12 Correct answer.

Explanation

36 = 2² × 3² and 48 = 2⁴ × 3. The GCD takes the minimum power of each prime: 2² × 3 = 12.

Question 6Numbers and Algebraic Expressions

Expand (x + 3)(x - 4)

  1. x² - x - 12 Correct answer.
  2. x² + x - 12
  3. x² - x + 12
  4. x² - 7x - 12

Explanation

Using FOIL: x·x = x², x·(-4) = -4x, 3·x = 3x, 3·(-4) = -12. Combine: x² - 4x + 3x - 12 = x² - x - 12.

Question 7Numbers and Algebraic Expressions

Factor x² - 8x + 15

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

Explanation

Find two numbers that multiply to 15 and add to -8. Those are -3 and -5, so x² - 8x + 15 = (x - 3)(x - 5).

Question 8Numbers and Algebraic Expressions

Compute: |-5| + |2 - 7|

  1. -10
  2. 7
  3. 10 Correct answer.
  4. 0

Explanation

|-5| = 5 and |2 - 7| = |-5| = 5. So 5 + 5 = 10.

Question 9Linear Equations and Functions

Solve the system: x + y = 7 and x - y = 1. What is (x, y)?

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

Explanation

Add the equations: 2x = 8, so x = 4. Substitute into x + y = 7: 4 + y = 7, so y = 3. The solution is (4, 3).

Question 10Linear Equations and Functions

Tickets for a school event cost 500 yen for adults and 300 yen for children. If 8 tickets were sold for a total of 3,000 yen, how many adult tickets were sold?

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

Explanation

Let a = adult tickets, c = child tickets. Then a + c = 8 and 500a + 300c = 3000. Substitute c = 8 - a: 500a + 300(8 - a) = 3000, so 200a + 2400 = 3000, giving 200a = 600 and a = 3.

Question 11Linear Equations and Functions

Solve: 3(x - 2) + 4 = 2x + 1

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

Explanation

Distribute: 3x - 6 + 4 = 2x + 1, so 3x - 2 = 2x + 1. Subtract 2x: x - 2 = 1. Add 2: x = 3.

Question 12Linear Equations and Functions

Solve x² - 5x + 6 = 0

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

Explanation

Factor: x² - 5x + 6 = (x - 2)(x - 3) = 0. Setting each factor to zero gives x = 2 or x = 3.

Question 13Linear Equations and Functions

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

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

Explanation

Using x = (4 ± √(16 - 4)) / 2 = (4 ± √12) / 2 = (4 ± 2√3) / 2 = 2 ± √3. The discriminant is b² - 4ac = 16 - 4 = 12.

Question 14Linear Equations and Functions

The length of a rectangle is 3 cm more than its width. The area is 28 cm². Find the width.

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

Explanation

Let w = width, so length = w + 3. Then w(w + 3) = 28, giving w² + 3w - 28 = 0. Factor: (w + 7)(w - 4) = 0. Since width must be positive, w = 4 cm.

Question 15Numbers and Algebraic Expressions

Expand (2x + 3)(2x - 3)

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

Explanation

This is the difference-of-squares pattern (a + b)(a - b) = a² - b² with a = 2x and b = 3. So (2x)² - 3² = 4x² - 9.

Question 16Numbers and Algebraic Expressions

Factor x² + 6x + 9

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

Explanation

This is a perfect-square trinomial: x² + 6x + 9 = (x + 3)², since 2 × 3 = 6 and 3² = 9.

Question 17Numbers and Algebraic Expressions

Simplify: 3√2 + 5√2 - √8

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

Explanation

First simplify √8 = 2√2. Then combine like radicals: 3√2 + 5√2 - 2√2 = (3 + 5 - 2)√2 = 6√2.

Question 18Numbers and Algebraic Expressions

Compute √12 × √3

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

Explanation

Combine under one radical: √12 × √3 = √(12 × 3) = √36 = 6. Alternatively, √12 = 2√3, so 2√3 × √3 = 2 × 3 = 6.

Question 19Numbers and Algebraic Expressions

Which of the following is an irrational number?

  1. √2 Correct answer.
  2. 3.14
  3. 7/3
  4. 0.5

Explanation

√2 cannot be expressed as a ratio of two integers; it is irrational. The other choices (0.5 = 1/2, 7/3, and 3.14 = 314/100) are all rational numbers.

Question 20Linear Equations and Functions

Solve the inequality: 2(x - 1) < 3x + 2

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

Explanation

Expand: 2x - 2 < 3x + 2. Subtract 2x and add 2: -4 < x, which means x > -4.

Question 21Numbers and Algebraic Expressions

If a/b = 3/4 and b/c = 8/5, find a/c.

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

Explanation

Multiply: a/c = (a/b) × (b/c) = (3/4) × (8/5). Simplify by canceling: 3 × (8/4) / 5 = 3 × 2 / 5 = 6/5.

Question 22Linear Equations and Functions

A train travels at 60 km/h for 30 minutes, then at 80 km/h for 45 minutes. What is the total distance traveled?

  1. 90 km Correct answer.
  2. 120 km
  3. 100 km
  4. 70 km

Explanation

Convert times to hours: 30 min = 0.5 h, 45 min = 0.75 h. Distance = 60 × 0.5 + 80 × 0.75 = 30 + 60 = 90 km.

Question 23Linear Equations and Functions

Pipe A fills a tank in 6 hours and Pipe B fills it in 3 hours. How long does it take if both pipes work together?

  1. 3 hours
  2. 4.5 hours
  3. 2 hours Correct answer.
  4. 9 hours

Explanation

Rate A = 1/6 tank per hour, Rate B = 1/3 tank per hour. Combined rate = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2. Time = 1 ÷ (1/2) = 2 hours.

Question 24Linear Equations and Functions

A shop gives a 20% discount on an item marked 2,500 yen. What is the final price?

  1. 500 yen
  2. 2,000 yen Correct answer.
  3. 3,000 yen
  4. 2,300 yen

Explanation

A 20% discount means paying 80%: 2,500 × 0.80 = 2,000 yen. Alternatively, the discount is 2,500 × 0.20 = 500 yen, so the final price is 2,500 - 500 = 2,000 yen.

Question 25Linear Equations and Functions

If y is directly proportional to x and y = 15 when x = 5, find y when x = 8.

  1. 12
  2. 15
  3. 40
  4. 24 Correct answer.

Explanation

Direct proportion means y = kx. Find k: 15 = k × 5, so k = 3. Then y = 3 × 8 = 24.

Question 26Linear Equations and Functions

If y is inversely proportional to x and y = 6 when x = 2, find y when x = 3.

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

Explanation

Inverse proportion means y = k/x. Find k: 6 = k/2, so k = 12. Then y = 12/3 = 4.

Question 27Linear Equations and Functions

Find the slope of the line passing through (1, 2) and (4, 8).

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

Explanation

Slope = (y₂ - y₁)/(x₂ - x₁) = (8 - 2)/(4 - 1) = 6/3 = 2.

Question 28Linear Equations and Functions

What is the equation of a line with slope 3 and y-intercept -2?

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

Explanation

In slope-intercept form y = mx + b, the slope m = 3 and the y-intercept b = -2. So the equation is y = 3x - 2.

Question 29Linear Equations and Functions

For y = 2x + 5, what is y when x = -3?

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

Explanation

Substitute x = -3: y = 2(-3) + 5 = -6 + 5 = -1.

Question 30Quadratic Functions and Equations

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

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

Explanation

The 変化の割合 is the change in y divided by the change in x. Here y(1) = 2 and y(3) = 18, so the rate is (18 − 2) ÷ (3 − 1) = 16 ÷ 2 = 8. A useful shortcut for y = ax² is a(x₁ + x₂) = 2(1 + 3) = 8. Unlike a straight line, this rate changes depending on the interval chosen.

Question 31Linear Equations and Functions

A 一次関数 (linear function) passes through the points (-2, 5) and (2, -3). What is its equation?

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

Explanation

The slope is the change in y over the change in x: (-3 - 5) ÷ (2 - (-2)) = -8 ÷ 4 = -2. Substituting (-2, 5) into y = -2x + b gives 5 = 4 + b, so b = 1. The equation is y = -2x + 1, and checking x = 2 gives -4 + 1 = -3 as required.

Question 32Quadratic Functions and Equations

Which point lies on the graph of y = 3x²?

  1. (3, 9)
  2. (2, 6)
  3. (1, 1)
  4. (2, 12) Correct answer.

Explanation

Substitute x = 2: y = 3(2²) = 3(4) = 12. So (2, 12) lies on the graph.

Question 33Quadratic Functions and Equations

Find the intersection points of y = x and y = x².

  1. (-1, -1) and (1, 1)
  2. No intersection
  3. (0, 1) and (1, 0)
  4. (0, 0) and (1, 1) Correct answer.

Explanation

Set x = x², so x² - x = 0, giving x(x - 1) = 0. Thus x = 0 or x = 1. The corresponding points are (0, 0) and (1, 1).

Question 34Linear Equations and Functions

A linear function has slope 2. When x = 3, y = 7. Find the y-intercept.

  1. 1 Correct answer.
  2. -1
  3. 7
  4. 13

Explanation

Use y = mx + b: 7 = 2(3) + b, so 7 = 6 + b, giving b = 1.

Question 35Linear Equations and Functions

Two lines are parallel. One has equation y = 2x + 1. The other passes through (0, 4). What is its equation?

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

Explanation

Parallel lines have the same slope, so the slope is 2. The line passes through (0, 4), which is the y-intercept, so b = 4. The equation is y = 2x + 4.

Question 36Linear Equations and Functions

A water tank is being filled at a constant rate. After 2 minutes it contains 30 liters, and after 5 minutes it contains 60 liters. How many liters were in the tank initially?

  1. 0 liters
  2. 15 liters
  3. 30 liters
  4. 10 liters Correct answer.

Explanation

Rate = (60 - 30)/(5 - 2) = 30/3 = 10 L/min. At t = 2: 30 = initial + 10 × 2, so initial = 30 - 20 = 10 liters.

Question 37Linear Equations and Functions

A spring is 12 cm long with no load, and it stretches 0.4 cm for every 10 g hung on it. What is its total length when a 150 g weight is attached?

  1. 18 cm Correct answer.
  2. 24 cm
  3. 72 cm
  4. 16 cm

Explanation

The extension is proportional to the mass. A 150 g load is 15 lots of 10 g, so the spring stretches 15 × 0.4 = 6 cm. Added to the 12 cm natural length, the total is 18 cm. As a 一次関数 this is y = 0.04x + 12, where x is the mass in grams.

Question 38Linear Equations and Functions

The graph of the 反比例 (inverse proportion) y = a/x passes through the point (4, 3). What is y when x = 6?

  1. 2 Correct answer.
  2. 4.5
  3. 8
  4. 18

Explanation

In 反比例 the product xy stays constant and equals a. From the point (4, 3), a = 4 × 3 = 12, so the relation is y = 12/x. When x = 6, y = 12 ÷ 6 = 2.

Question 39Quadratic Functions and Equations

A ball rolls down a slope so that the distance y metres it has travelled x seconds after starting is given by y = 3x². How many seconds after starting has it travelled 48 m?

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

Explanation

Substitute y = 48 into y = 3x²: 48 = 3x², so x² = 16 and x = 4 (time cannot be negative). Note the signature of a y = ax² relation - doubling the time from 2 s to 4 s multiplies the distance by four, from 12 m to 48 m.

Question 40Quadratic Functions and Equations

If y is proportional to x² and y = 12 when x = 2, find y when x = 3.

  1. 36
  2. 27 Correct answer.
  3. 18
  4. 9

Explanation

y = kx², so 12 = k(4), giving k = 3. Then y = 3(3²) = 3(9) = 27.

Question 41Linear Equations and Functions

A car travels at 40 km/h for 6 hours. How long would the same journey take at 60 km/h?

  1. 4 hours Correct answer.
  2. 9 hours
  3. 10 hours
  4. 6 hours

Explanation

Distance = 40 × 6 = 240 km. At 60 km/h: time = 240/60 = 4 hours. Since speed and time are inversely proportional, faster speed means less time.

Question 42Quadratic Functions and Equations

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

  1. 3 Correct answer.
  2. 12
  3. 48
  4. 6

Explanation

Substitute x = 2, y = 12: 12 = a(2²) = 4a, so a = 12/4 = 3.

Question 43Quadratic Functions and Equations

Find the intersection points of y = 2x and y = x².

  1. No intersection
  2. (0, 0) and (2, 4) Correct answer.
  3. (0, 2) and (2, 0)
  4. (1, 2) only

Explanation

Set 2x = x², so x² - 2x = 0, giving x(x - 2) = 0. Thus x = 0 or x = 2. The points are (0, 0) and (2, 4).

Question 44Linear Equations and Functions

The line y = -(3/2)x + 6 crosses the y-axis at A and the x-axis at B. Taking O as the origin, what is the area of triangle OAB?

  1. 6
  2. 10
  3. 12 Correct answer.
  4. 24

Explanation

Setting x = 0 gives A(0, 6); setting y = 0 gives (3/2)x = 6, so B(4, 0). Triangle OAB has a right angle at the origin, with legs OA = 6 and OB = 4 along the axes. Its area is (1/2) × 4 × 6 = 12.

Question 45Quadratic Functions and Equations

For the function y = x² with the domain (xの変域) -2 ≤ x ≤ 3, what is the range (yの変域)?

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

Explanation

Squaring never gives a negative result, and x = 0 lies inside the domain, so the smallest value is y = 0. The largest comes from whichever endpoint is furthest from zero - here x = 3, giving y = 9. The range is therefore 0 ≤ y ≤ 9.

Question 46Linear Equations and Functions

For the linear function y = ax + b, when x = 1, y = 5 and when x = 3, y = 11. Find a and b.

  1. a = 3, b = 2 Correct answer.
  2. a = 3, b = 5
  3. a = 2, b = 3
  4. a = 6, b = 2

Explanation

Slope a = (11 - 5)/(3 - 1) = 6/2 = 3. Substitute into y = 3x + b: 5 = 3(1) + b, so b = 2.

Question 47Linear Equations and Functions

A tank holding 60 L of water drains at a constant rate and is completely empty after 24 minutes. Writing y for the litres remaining after x minutes, which equation is correct?

  1. y = 60 - 2.5x Correct answer.
  2. y = 60 - 24x
  3. y = 60 + 2.5x
  4. y = 2.5x

Explanation

The tank starts full, so the value at x = 0 is 60. Draining 60 L in 24 minutes is a constant rate of 60 ÷ 24 = 2.5 L per minute, and the amount remaining decreases, giving y = 60 - 2.5x. Checking at x = 24: y = 60 - 60 = 0, as required.

Question 48Geometry: Plane Figures and Angles

In a triangle, two angles are 55° and 75°. What is the third angle?

  1. 80°
  2. 50° Correct answer.
  3. 130°
  4. 25°

Explanation

The angles of a triangle sum to 180°. The third angle is 180° - 55° - 75° = 50°.

Question 49Geometry: Congruence, Similarity and Measurement

Two similar triangles have sides in ratio 2:5. If the smaller triangle has area 8 cm², what is the area of the larger?

  1. 32 cm²
  2. 20 cm²
  3. 200 cm²
  4. 50 cm² Correct answer.

Explanation

For similar figures, the area ratio is the square of the side ratio: (5/2)² = 25/4. Larger area = 8 × 25/4 = 50 cm².

Question 50Geometry: Congruence, Similarity and Measurement

Which condition guarantees that two triangles are congruent?

  1. SSA (Side-Side-Angle)
  2. SSS (Side-Side-Side) Correct answer.
  3. All angles equal
  4. AAA (Angle-Angle-Angle)

Explanation

SSS guarantees congruence: if all three corresponding sides are equal, the triangles are congruent. SSA is ambiguous (the 'ambiguous case'), and AAA only guarantees similarity.