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Aichi Entrance Mathematics study materials
Aichi Entrance Mathematics study resources

Aichi Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate: (−7) + (+4) − (−3).

  1. −8
  2. −6
  3. −14
  4. 0 Correct answer.

Explanation

Removing parentheses gives −7 + 4 + 3. Working left to right: −7 + 4 = −3, then −3 + 3 = 0. Subtracting a negative number is the same as adding its opposite.

Question 2Numbers and Algebraic Expressions

Calculate: 12 ÷ (−3) × 2.

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

Explanation

Division and multiplication have equal priority, so work left to right: 12 ÷ (−3) = −4, then −4 × 2 = −8.

Question 3Numbers and Algebraic Expressions

Simplify: 3(2x − 4) − (x − 5).

  1. 5x + 7
  2. 5x − 7 Correct answer.
  3. 5x + 1
  4. 5x − 17

Explanation

Distribute the 3: 3(2x − 4) = 6x − 12. Removing the second parentheses flips both signs: −(x − 5) = −x + 5. Combining like terms gives 6x − x − 12 + 5 = 5x − 7.

Question 4Numbers and Algebraic Expressions

Calculate: (1/2 + 1/3) ÷ (5/6).

  1. 1 Correct answer.
  2. 5/6
  3. 25/36
  4. 5

Explanation

First add inside the parentheses: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. Dividing by 5/6 is the same as multiplying by 6/5: (5/6) × (6/5) = 1.

Question 5Numbers and Algebraic Expressions

Solve the equation 3x + 7 = 22.

  1. x = 3
  2. x = 5 Correct answer.
  3. x = 29/3
  4. x = 7

Explanation

Transpose the constant term: 3x = 22 − 7 = 15. Dividing both sides by 3 gives x = 5. Check: 3 × 5 + 7 = 22.

Question 6Numbers and Algebraic Expressions

Solve the simultaneous equations: 2x + 3y = 12 and x − y = 1.

  1. x = 0, y = 4
  2. x = 2, y = 3
  3. x = 6, y = 0
  4. x = 3, y = 2 Correct answer.

Explanation

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

Question 7Numbers and Algebraic Expressions

Factorize x² − 7x + 12.

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

Explanation

Look for two numbers whose sum is −7 and whose product is 12: −3 and −4. Therefore x² − 7x + 12 = (x − 3)(x − 4).

Question 8Numbers and Algebraic Expressions

Factorize x² − 9.

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

Explanation

This is a difference of squares: x² − 9 = x² − 3² = (x + 3)(x − 3).

Question 9Numbers and Algebraic Expressions

Calculate √12 × √3.

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

Explanation

√12 × √3 = √36 = 6. Alternatively, √12 = 2√3, so 2√3 × √3 = 2 × 3 = 6.

Question 10Numbers and Algebraic Expressions

Calculate √50 − √8 + √18.

  1. 4√2
  2. 6√2 Correct answer.
  3. √60
  4. 10√2

Explanation

Simplify each term first: √50 = 5√2, √8 = 2√2, √18 = 3√2. Then 5√2 − 2√2 + 3√2 = 6√2.

Question 11Numbers and Algebraic Expressions

Solve x² − 5x + 6 = 0.

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

Explanation

Factor using two numbers with sum −5 and product 6: (x − 2)(x − 3) = 0. So x = 2 or x = 3.

Question 12Numbers and Algebraic Expressions

Solve x² + 2x − 4 = 0 using the quadratic formula.

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

Explanation

With a = 1, b = 2, c = −4: x = (−2 ± √(4 + 16))/2 = (−2 ± √20)/2 = (−2 ± 2√5)/2 = −1 ± √5.

Question 13Numbers and Algebraic Expressions

Find the value of x² − 2xy when x = −2 and y = 3.

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

Explanation

Substitute: x² = (−2)² = 4 and −2xy = −2 × (−2) × 3 = 12. So the value is 4 + 12 = 16.

Question 14Numbers and Algebraic Expressions

Expand (x + 3)(x − 5).

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

Explanation

Expand term by term: x² − 5x + 3x − 15 = x² − 2x − 15.

Question 15Numbers and Algebraic Expressions

Factorize 2x² + 4x completely.

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

Explanation

The greatest common factor of 2x² and 4x is 2x. Factoring it out: 2x² + 4x = 2x(x + 2).

Question 16Numbers and Algebraic Expressions

Calculate (√5 + 2)(√5 − 2).

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

Explanation

Use the difference of squares pattern: (√5 + 2)(√5 − 2) = (√5)² − 2² = 5 − 4 = 1.

Question 17Numbers and Algebraic Expressions

Solve (x + 2)/3 = 4.

  1. x = 2
  2. x = 14
  3. x = 6
  4. x = 10 Correct answer.

Explanation

Multiply both sides by 3: x + 2 = 12. Then x = 12 − 2 = 10. Check: (10 + 2)/3 = 4.

Question 18Numbers and Algebraic Expressions

Solve the simultaneous equations: 3x + 2y = 7 and 5x − 2y = 1.

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

Explanation

Adding the two equations eliminates y: 8x = 8, so x = 1. Substituting into the first equation: 3 + 2y = 7, so y = 2.

Question 19Numbers and Algebraic Expressions

Find the value of ab² − a²b when a = 3 and b = −4.

  1. 12
  2. −84
  3. −12
  4. 84 Correct answer.

Explanation

ab² = 3 × (−4)² = 3 × 16 = 48, and a²b = 9 × (−4) = −36. So ab² − a²b = 48 − (−36) = 84.

Question 20Numbers and Algebraic Expressions

Calculate √75 ÷ √3 + √2 × √8.

  1. 41
  2. 7
  3. 9 Correct answer.
  4. 5 + 2√2

Explanation

√75 ÷ √3 = √25 = 5, and √2 × √8 = √16 = 4. So the value is 5 + 4 = 9.

Question 21Numbers and Algebraic Expressions

Factorize x² − 4x + 4.

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

Explanation

This fits the perfect-square pattern a² − 2ab + b² = (a − b)² with a = x and b = 2: x² − 4x + 4 = (x − 2)².

Question 22Numbers and Algebraic Expressions

Solve x² − 6x + 2 = 0.

  1. x = −3 ± √7
  2. x = 3 ± √7 Correct answer.
  3. x = 3 ± √11
  4. x = 6 ± √7

Explanation

By the quadratic formula: x = (6 ± √(36 − 8))/2 = (6 ± √28)/2 = (6 ± 2√7)/2 = 3 ± √7.

Question 23Numbers and Algebraic Expressions

Calculate (−3/4) ÷ (9/8).

  1. −27/32
  2. −2/3 Correct answer.
  3. 2/3
  4. −3/2

Explanation

Dividing by 9/8 is the same as multiplying by 8/9: (−3/4) × (8/9) = −24/36 = −2/3.

Question 24Numbers and Algebraic Expressions

Solve the simultaneous equations: x + y = 10 and x − y = 4.

  1. x = 7, y = 3 Correct answer.
  2. x = 6, y = 4
  3. x = 8, y = 2
  4. x = 3, y = 7

Explanation

Adding the equations gives 2x = 14, so x = 7. Then y = 10 − 7 = 3. Check: 7 − 3 = 4.

Question 25Linear Equations and Functions

y is inversely proportional to x, and y = 8 when x = 3. Find the value of y when x = 4.

  1. 12
  2. 96
  3. 6 Correct answer.
  4. 32/3

Explanation

For inverse proportion the product xy is constant: a = 3 × 8 = 24, so y = 24/x. When x = 4, y = 24/4 = 6.

Question 26Linear Equations and Functions

Find the slope of the line passing through the points (2, 1) and (6, 9).

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

Explanation

Slope = (change in y)/(change in x) = (9 − 1)/(6 − 2) = 8/4 = 2.

Question 27Linear Equations and Functions

A line has slope −2 and y-intercept 3. Write its equation.

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

Explanation

In y = ax + b, a is the slope and b is the y-intercept, so the equation is y = −2x + 3.

Question 28Quadratic Functions and Equations

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

  1. 6
  2. 24
  3. 3 Correct answer.
  4. 36

Explanation

Substitute x = 2, y = 12: 12 = a × 2² = 4a, so a = 3.

Question 29Quadratic 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. 5
  4. 8 Correct answer.

Explanation

When x = 1, y = 2; when x = 3, y = 18. Rate of change = (18 − 2)/(3 − 1) = 16/2 = 8.

Question 30Quadratic Functions and Equations

Points A(−1, 2) and B(3, 18) both lie on the graph of y = 2x². Find the slope of the line AB.

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

Explanation

Slope = (18 − 2)/(3 − (−1)) = 16/4 = 4. This is the same computation as the rate of change of the function between x = −1 and x = 3.

Question 31Linear Equations and Functions

Find all intersection points of the graphs y = 12/x and y = 3x.

  1. (2, 6) only
  2. (4, 3) and (−4, −3)
  3. (2, 6) and (−2, −6) Correct answer.
  4. (6, 2) and (−6, −2)

Explanation

Set 12/x = 3x. Multiplying both sides by x: 3x² = 12, so x² = 4 and x = ±2. Substituting back gives the points (2, 6) and (−2, −6).

Question 32Linear Equations and Functions

Find the equation of the line passing through the points (1, 5) and (3, 9).

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

Explanation

Slope = (9 − 5)/(3 − 1) = 4/2 = 2. Using the point (1, 5): 5 = 2 × 1 + b, so b = 3. The equation is y = 2x + 3.

Question 33Quadratic Functions and Equations

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

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

Explanation

Because the domain includes x = 0, the minimum value is y = 0. The maximum occurs at the endpoint farthest from 0: x = 2 gives y = 4 (while x = −1 gives only y = 1). So the range is 0 ≤ y ≤ 4.

Question 34Linear Equations and Functions

The line y = 3x + b passes through the point (2, 1). Find the value of b.

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

Explanation

Substitute x = 2, y = 1: 1 = 3 × 2 + b, so b = 1 − 6 = −5.

Question 35Linear Equations and Functions

What is the y-intercept of the line y = 2x − 3?

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

Explanation

The y-intercept is the constant term b in y = ax + b, the y-value where the line crosses the y-axis (x = 0). Here b = −3.

Question 36Linear Equations and Functions

y is proportional to x. When x = 5, y = −15. Write the equation relating x and y.

  1. y = 3x
  2. y = −15x
  3. y = −3x Correct answer.
  4. y = −(1/3)x

Explanation

Direct proportion has the form y = ax. Substituting: −15 = 5a, so a = −3 and the equation is y = −3x.

Question 37Quadratic Functions and Equations

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

  1. 15
  2. 9 Correct answer.
  3. 5
  4. 45

Explanation

At x = −1, y = 3; at x = 4, y = 48. Rate of change = (48 − 3)/(4 − (−1)) = 45/5 = 9.

Question 38Quadratic Functions and Equations

Find the intersection points of the parabola y = x² and the line y = x + 6.

  1. (3, 9) only
  2. (3, 9) and (−2, 4) Correct answer.
  3. (6, 36) and (−1, 1)
  4. (2, 4) and (−3, 9)

Explanation

Set x² = x + 6, so x² − x − 6 = 0, factoring as (x − 3)(x + 2) = 0. Thus x = 3 or x = −2, giving the points (3, 9) and (−2, 4).

Question 39Linear Equations and Functions

The graph of y = a/x passes through the point (−2, 5). Find the value of y when x = 5.

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

Explanation

Substituting the point: 5 = a/(−2), so a = −10 and the equation is y = −10/x. When x = 5, y = −10/5 = −2.

Question 40Linear Equations and Functions

A table shows x = 1, 2, 3 with corresponding values y = 4, 8, 12. Which equation expresses y as a function of x?

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

Explanation

Each y-value is 4 times the x-value, and the ratio y/x is constantly 4, so this is direct proportion: y = 4x.

Question 41Quadratic 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
  3. −6
  4. −1 Correct answer.

Explanation

At x = −4, y = 8; at x = 2, y = 2. Rate of change = (2 − 8)/(2 − (−4)) = −6/6 = −1.

Question 42Linear Equations and Functions

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

  1. (6, 13)
  2. (3, 7)
  3. (5, 2)
  4. (2, 5) Correct answer.

Explanation

Set 2x + 1 = −x + 7, so 3x = 6 and x = 2. Then y = 2 × 2 + 1 = 5. The intersection is (2, 5).

Question 43Geometry: Plane Figures and Angles

Two parallel lines ℓ and m are cut by a transversal. An angle of 65° is formed at line ℓ. What is the measure of the corresponding angle at line m?

  1. 90°
  2. 25°
  3. 65° Correct answer.
  4. 115°

Explanation

When parallel lines are cut by a transversal, corresponding angles are equal, so the corresponding angle is also 65°.

Question 44Geometry: Plane Figures and Angles

In a triangle, two interior angles measure 45° and 65°. Find the measure of the third interior angle.

  1. 80°
  2. 110°
  3. 70° Correct answer.
  4. 135°

Explanation

The interior angles of a triangle sum to 180°, so the third angle is 180° − 45° − 65° = 70°.

Question 45Geometry: Plane Figures and Angles

In triangle ABC, side BC is extended beyond C to form an exterior angle. The two non-adjacent interior angles are 50° and 70°. What is the measure of this exterior angle?

  1. 60°
  2. 130°
  3. 110°
  4. 120° Correct answer.

Explanation

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles: 50° + 70° = 120°.

Question 46Geometry: Congruence, Similarity and Measurement

Which of the following conditions does NOT guarantee that two triangles are congruent?

  1. Three angles respectively equal (AAA) Correct answer.
  2. Two sides and the included angle respectively equal (SAS)
  3. Three sides respectively equal (SSS)
  4. Two angles and the included side respectively equal (ASA)

Explanation

Equal angles only guarantee the same shape (similarity), not the same size; two triangles can share all three angles while one is an enlargement of the other. SSS, SAS, and ASA are the standard congruence conditions.

Question 47Geometry: Plane Figures and Angles

In a circle with center O, the central angle ∠AOB subtending arc AB is 80°. Point P is on the major arc AB. Find the inscribed angle ∠APB.

  1. 80°
  2. 160°
  3. 20°
  4. 40° Correct answer.

Explanation

By the inscribed angle theorem, an inscribed angle is half the central angle subtending the same arc: 80° ÷ 2 = 40°.

Question 48Geometry: Plane Figures and Angles

Point P lies on a circle, and AB is a diameter of the circle. What is the measure of the inscribed angle ∠APB?

  1. 180°
  2. 45°
  3. 90° Correct answer.
  4. 60°

Explanation

The central angle subtended by a diameter is 180°, and the inscribed angle is half of that: 90°. An angle inscribed in a semicircle is always a right angle.

Question 49Geometry: Plane Figures and Angles

In a circle with center O, points A and B form a central angle ∠AOB = 130° on the minor arc AB. Point P lies on the minor arc AB. Find the inscribed angle ∠APB.

  1. 115° Correct answer.
  2. 130°
  3. 65°
  4. 50°

Explanation

Since P is on the minor arc, ∠APB subtends the major arc AB, whose central angle is 360° − 130° = 230°. The inscribed angle is half of that: 230° ÷ 2 = 115°.

Question 50Geometry: Plane Figures and Angles

In a right triangle, the two legs are 6 cm and 8 cm. Find the length of the hypotenuse.

  1. 14 cm
  2. 12 cm
  3. 2√7 cm
  4. 10 cm Correct answer.

Explanation

By the Pythagorean theorem: hypotenuse² = 6² + 8² = 36 + 64 = 100, so the hypotenuse is √100 = 10 cm.