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Aomori Entrance Mathematics study materials
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Aomori Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate: (−7) + (+4).

  1. −11
  2. 3
  3. −28
  4. −3 Correct answer.

Explanation

Adding a positive number to a negative number means subtracting the smaller absolute value from the larger and keeping the sign of the larger absolute value. Since |−7| = 7 is greater than |+4| = 4, the result is −(7 − 4) = −3.

Question 2Numbers and Algebraic Expressions

Calculate: (−5) × (−6).

  1. −11
  2. 11
  3. 30 Correct answer.
  4. −30

Explanation

The product of two numbers with the same sign is positive. Multiply the absolute values: 5 × 6 = 30, and since both factors are negative, the result is +30.

Question 3Numbers and Algebraic Expressions

Calculate: (−24) ÷ (+6).

  1. −4 Correct answer.
  2. 4
  3. −3
  4. −18

Explanation

The quotient of two numbers with different signs is negative. Divide the absolute values: 24 ÷ 6 = 4, and because the signs differ, the answer is −4.

Question 4Numbers and Algebraic Expressions

Calculate: (−3)^2.

  1. −6
  2. 9 Correct answer.
  3. 6
  4. −9

Explanation

(−3)^2 means (−3) × (−3). The product of two negative numbers is positive, so the answer is 9. An even power of a negative number is always positive.

Question 5Numbers and Algebraic Expressions

Simplify: 5a − 3a + 2a.

  1. −4a
  2. 4a Correct answer.
  3. 10a
  4. 6a

Explanation

Combine like terms by adding the coefficients: 5 − 3 + 2 = 4. So the expression simplifies to 4a.

Question 6Numbers and Algebraic Expressions

Find the value of 2x − 5 when x = −3.

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

Explanation

Substitute x = −3 into the expression: 2 × (−3) − 5 = −6 − 5 = −11. Multiply first, then subtract.

Question 7Numbers and Algebraic Expressions

Solve the equation: x − 7 = −2.

  1. 5 Correct answer.
  2. −9
  3. −5
  4. 9

Explanation

Add 7 to both sides to isolate x: x = −2 + 7 = 5. Adding a positive number to a negative number gives the difference of the absolute values with the sign of the larger one.

Question 8Numbers and Algebraic Expressions

Calculate: (−2) × 3 + 8.

  1. 2 Correct answer.
  2. −14
  3. −2
  4. 14

Explanation

Multiplication comes before addition: (−2) × 3 = −6, then −6 + 8 = 2. The sum is positive because 8 has the larger absolute value.

Question 9Numbers and Algebraic Expressions

Solve the equation: 3x + 5 = 2x − 1.

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

Explanation

Subtract 2x from both sides to get x + 5 = −1, then subtract 5: x = −6. Check: 3(−6) + 5 = −13 and 2(−6) − 1 = −13.

Question 10Numbers and Algebraic Expressions

Calculate: (−3)^2 − 4 × (−2).

  1. 10
  2. 1
  3. 17 Correct answer.
  4. −17

Explanation

Evaluate the power and the product first: (−3)^2 = 9 and 4 × (−2) = −8. Then 9 − (−8) = 9 + 8 = 17.

Question 11Numbers and Algebraic Expressions

Simplify: (2x − 3) − (x − 5).

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

Explanation

Distribute the minus sign over the second parentheses: 2x − 3 − x + 5. Combine like terms: (2x − x) + (−3 + 5) = x + 2.

Question 12Numbers and Algebraic Expressions

Simplify: 12x^2y ÷ (−3xy).

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

Explanation

Divide the coefficients: 12 ÷ (−3) = −4. For the variables, x^2 ÷ x = x and y ÷ y = 1. The result is −4x.

Question 13Numbers and Algebraic Expressions

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

  1. x^2 + 8x + 8
  2. x^2 + 8x + 15 Correct answer.
  3. x^2 + 15x + 8
  4. x^2 + 2x + 15

Explanation

Multiply each term: x·x = x^2, x·5 = 5x, 3·x = 3x, and 3·5 = 15. Combining the middle terms gives 5x + 3x = 8x, so the answer is x^2 + 8x + 15.

Question 14Numbers and Algebraic Expressions

Factor: x^2 + 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 product is 12 and whose sum is 7. Those numbers are 3 and 4, so x^2 + 7x + 12 = (x + 3)(x + 4).

Question 15Numbers and Algebraic Expressions

Calculate: √12 × √3.

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

Explanation

Use the rule √a × √b = √(a × b): √12 × √3 = √36 = 6.

Question 16Numbers and Algebraic Expressions

Simplify: √50 − √18.

  1. 5√2
  2. 2√2 Correct answer.
  3. √32
  4. 8√2

Explanation

Rewrite each radical in simplest form: √50 = 5√2 and √18 = 3√2. Then 5√2 − 3√2 = 2√2.

Question 17Numbers and Algebraic Expressions

Solve the system of equations: x + y = 7 and x − y = 3. What is the value of x?

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

Explanation

Add the two equations to eliminate y: 2x = 10, so x = 5. Substituting back gives y = 2, which satisfies both equations.

Question 18Numbers and Algebraic Expressions

Solve the equation: x^2 = 49.

  1. x = 7 and x = −7 Correct answer.
  2. x = 7 only
  3. x = −7 only
  4. no solution

Explanation

Taking the square root of both sides gives x = ±√49 = ±7. Both 7 and −7 satisfy the equation because squaring either gives 49.

Question 19Numbers and Algebraic Expressions

Solve the equation: x^2 − 5x + 6 = 0.

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

Explanation

Factor the left side: find two numbers with product 6 and sum −5, which are −2 and −3. So (x − 2)(x − 3) = 0, giving x = 2 or x = 3.

Question 20Numbers and Algebraic Expressions

Simplify: 6/√3 + √27.

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

Explanation

Rationalize the first term: 6/√3 = 6√3/3 = 2√3. Simplify the second: √27 = 3√3. Then 2√3 + 3√3 = 5√3.

Question 21Numbers and Algebraic Expressions

Solve the equation: x^2 + 2x − 4 = 0.

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

Explanation

This quadratic does not factor with integers, so use the quadratic formula: x = (−2 ± √(4 + 16))/2 = (−2 ± √20)/2 = (−2 ± 2√5)/2 = −1 ± √5.

Question 22Numbers and Algebraic Expressions

Expand: (2x − 3)^2.

  1. 4x^2 − 6x + 9
  2. 4x^2 + 9
  3. 4x^2 − 12x + 9 Correct answer.
  4. 4x^2 − 12x − 9

Explanation

Use the formula (a − b)^2 = a^2 − 2ab + b^2 with a = 2x and b = 3: (2x)^2 = 4x^2, 2·(2x)·3 = 12x, and 3^2 = 9. The result is 4x^2 − 12x + 9.

Question 23Numbers and Algebraic Expressions

Factor completely: 3x^2 − 12.

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

Explanation

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

Question 24Numbers and Algebraic Expressions

Given x + y = 8 and xy = 12, find the value of x^2 + y^2.

  1. 52
  2. 40 Correct answer.
  3. 16
  4. 64

Explanation

Use the identity x^2 + y^2 = (x + y)^2 − 2xy. Substituting gives 8^2 − 2(12) = 64 − 24 = 40.

Question 25Numbers and Algebraic Expressions

Solve the equation: (x + 1)(x − 3) = 5.

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

Explanation

Expand the left side: x^2 − 2x − 3 = 5. Move 5 to the left: x^2 − 2x − 8 = 0. Factor: (x − 4)(x + 2) = 0, so x = 4 or x = −2.

Question 26Linear Equations and Functions

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

  1. 81
  2. 7
  3. 34
  4. 12 Correct answer.

Explanation

Direct proportion means y = ax, so substitute x = 4: y = 3 × 4 = 12.

Question 27Linear Equations and Functions

y is inversely proportional to x, and y = 12/x. Find the value of y when x = 3.

  1. 9
  2. 4 Correct answer.
  3. 36
  4. 15

Explanation

Inverse proportion means y = a/x with a = 12. Substitute x = 3: y = 12 ÷ 3 = 4.

Question 28Linear Equations and Functions

What is the slope of the line y = 2x + 5?

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

Explanation

A linear function written as y = ax + b has slope a and y-intercept b. Here a = 2, so the slope is 2.

Question 29Linear Equations and Functions

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

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

Explanation

In y = ax + b, the y-intercept is b, the value of y when x = 0. Here b = 3, so the line crosses the y-axis at (0, 3).

Question 30Quadratic Functions and Equations

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

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

Explanation

Substitute x = 2 and y = 12: 12 = a × 2^2 = 4a. Dividing both sides by 4 gives a = 3.

Question 31Linear Equations and Functions

Which of the following functions has a y-value that always increases as x increases?

  1. y = −x^2
  2. y = 2x Correct answer.
  3. y = −2x
  4. y = 12/x

Explanation

A linear function y = ax increases for all x when a is positive. Since 2 > 0, y = 2x always increases as x increases.

Question 32Linear Equations and Functions

Find the equation of the line with slope −2 that passes through the point (0, 1).

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

Explanation

A line through (0, 1) has y-intercept 1, so b = 1. With slope −2, the equation is y = −2x + 1.

Question 33Linear Equations and Functions

Find the equation of the line passing through the points (2, 5) and (4, 9).

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

Explanation

The slope is (9 − 5)/(4 − 2) = 4/2 = 2. Substituting (2, 5) into y = 2x + b gives 5 = 4 + b, so b = 1. The equation is y = 2x + 1.

Question 34Quadratic Functions and Equations

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

  1. 12 Correct answer.
  2. 9
  3. 24
  4. 6

Explanation

When x = 1, y = 3; when x = 3, y = 27. The rate of change is (27 − 3)/(3 − 1) = 24/2 = 12.

Question 35Linear Equations and Functions

Find the x-intercept of the line y = −2x + 6.

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

Explanation

The x-intercept occurs where y = 0: 0 = −2x + 6, so 2x = 6 and x = 3. The line crosses the x-axis at (3, 0).

Question 36Linear Equations and Functions

y is inversely proportional to x. When x = 4, y = 6. Find the value of y when x = 8.

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

Explanation

For inverse proportion, xy is constant: 4 × 6 = 24. When x = 8, y = 24 ÷ 8 = 3. Doubling x halves y.

Question 37Linear Equations and Functions

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

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

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).

Question 38Quadratic Functions and Equations

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

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

Explanation

Because the interval includes x = 0, the minimum value of y is 0^2 = 0. The maximum occurs at the endpoint farthest from 0: 2^2 = 4. So the range is 0 ≤ y ≤ 4.

Question 39Linear Equations and Functions

A water tank holds 60 liters. Water drains out at a constant rate of 5 liters per minute. How many liters remain in the tank after 8 minutes?

  1. 100 L
  2. 20 L Correct answer.
  3. 40 L
  4. 12 L

Explanation

The amount drained in 8 minutes is 5 × 8 = 40 L. Subtracting from the initial amount: 60 − 40 = 20 L remain. This is the linear function y = 60 − 5x evaluated at x = 8.

Question 40Quadratic Functions and Equations

For the function y = (1/2)x^2, find the rate of change as x increases from 2 to 6.

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

Explanation

When x = 2, y = 2; when x = 6, y = 18. The rate of change is (18 − 2)/(6 − 2) = 16/4 = 4.

Question 41Quadratic Functions and Equations

The parabola y = x^2 and the line y = x + 2 intersect at two points. What is the x-coordinate of the intersection point with the larger x-coordinate?

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

Explanation

Set x^2 = x + 2, giving x^2 − x − 2 = 0. Factor: (x − 2)(x + 1) = 0, so x = 2 or x = −1. The larger x-coordinate is 2.

Question 42Quadratic Functions and Equations

A ball's height is modeled by y = −x^2 + 4x, where x is the time in seconds after launch and y is the height in meters. What is the maximum height of the ball?

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

Explanation

Complete the square: y = −(x^2 − 4x) = −(x − 2)^2 + 4. Since the parabola opens downward, the vertex (2, 4) is the maximum point, so the maximum height is 4 m.

Question 43Linear Equations and Functions

A rectangle has a perimeter of 24 cm. If one side has length x cm and the adjacent side has length y cm, which equation expresses y as a function of x?

  1. y = 24 − 2x
  2. y = 24 − x
  3. y = 12 − x Correct answer.
  4. y = x − 12

Explanation

The perimeter satisfies 2x + 2y = 24. Dividing by 2 gives x + y = 12, so y = 12 − x.

Question 44Linear Equations and Functions

Find the area of the triangle formed by the line y = −x + 4, the x-axis, and the y-axis.

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

Explanation

The line meets the axes at (4, 0) and (0, 4), so the triangle has base 4 and height 4. Its area is (1/2) × 4 × 4 = 8.

Question 45Quadratic Functions and Equations

Point A lies on the parabola y = x^2 and has x-coordinate 3. Find the slope of the line through the origin O and point A.

  1. 3 Correct answer.
  2. 27
  3. 1/3
  4. 9

Explanation

At x = 3, y = 3^2 = 9, so A = (3, 9). The slope of OA is rise over run: 9/3 = 3.

Question 46Geometry: Plane Figures and Angles

Two straight lines intersect, forming an angle of 65°. What is the measure of the vertical (opposite) angle?

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

Explanation

Vertical angles formed by two intersecting lines are always equal. Therefore the angle opposite the 65° angle is also 65°.

Question 47Geometry: Plane Figures and Angles

In a triangle, two of the interior angles measure 50° and 60°. Find the measure of the third angle.

  1. 80°
  2. 60°
  3. 110°
  4. 70° Correct answer.

Explanation

The interior angles of a triangle sum to 180°. The third angle is 180° − 50° − 60° = 70°.

Question 48Geometry: Plane Figures and Angles

What is the sum of the interior angles of a pentagon?

  1. 900°
  2. 360°
  3. 540° Correct answer.
  4. 720°

Explanation

The sum of the interior angles of an n-sided polygon is (n − 2) × 180°. For a pentagon, n = 5, so the sum is 3 × 180° = 540°.

Question 49Geometry: Plane Figures and Angles

What is the measure of one exterior angle of a regular hexagon?

  1. 120°
  2. 90°
  3. 72°
  4. 60° Correct answer.

Explanation

The exterior angles of any polygon sum to 360°. A regular hexagon has 6 equal exterior angles, so each one is 360° ÷ 6 = 60°.

Question 50Geometry: Plane Figures and Angles

A right triangle has legs of length 3 cm and 4 cm. Find the length of the hypotenuse.

  1. 5 cm Correct answer.
  2. √7 cm
  3. 7 cm
  4. 25 cm

Explanation

By the Pythagorean theorem, the hypotenuse c satisfies c^2 = 3^2 + 4^2 = 9 + 16 = 25, so c = 5 cm.