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Akita Entrance Mathematics study materials
Akita Entrance Mathematics study resources

Akita Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate: (−8) + (+5).

  1. 3
  2. −40
  3. −3 Correct answer.
  4. −13

Explanation

Adding a positive number to a negative number means finding the difference of their absolute values and keeping the sign of the larger absolute value. Since |−8| = 8 is greater than |+5| = 5, the answer is −(8 − 5) = −3.

Question 2Numbers and Algebraic Expressions

Calculate: (−6) × (−4).

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

Explanation

The product of two numbers with the same sign is positive. Multiply the absolute values: 6 × 4 = 24, and since both factors are negative, the result is +24.

Question 3Numbers and Algebraic Expressions

Calculate: (−18) ÷ (+3).

  1. 6
  2. −9
  3. −6 Correct answer.
  4. −15

Explanation

The quotient of two numbers with different signs is negative. Divide the absolute values: 18 ÷ 3 = 6, and because the signs differ, the answer is −6.

Question 4Numbers and Algebraic Expressions

Calculate: (−2)^3.

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

Explanation

(−2)^3 means (−2) × (−2) × (−2). The first pair gives +4, then 4 × (−2) = −8. An odd power of a negative number is negative.

Question 5Numbers and Algebraic Expressions

Simplify: 3x + 2x − 4x.

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

Explanation

Combine like terms by adding the coefficients: 3 + 2 − 4 = 1. So the expression simplifies to 1x, which is written simply as x.

Question 6Numbers and Algebraic Expressions

Find the value of 2a − 3b when a = 4 and b = −2.

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

Explanation

Substitute the values: 2(4) − 3(−2) = 8 + 6, because −3 × (−2) = +6. Therefore 8 + 6 = 14.

Question 7Numbers and Algebraic Expressions

Solve the equation: x + 7 = 3.

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

Explanation

Subtract 7 from both sides: x = 3 − 7 = −4. Check: −4 + 7 = 3, which is correct.

Question 8Numbers and Algebraic Expressions

Calculate: 12 ÷ (−3) + 5.

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

Explanation

Division comes before addition: 12 ÷ (−3) = −4. Then −4 + 5 = 1.

Question 9Numbers and Algebraic Expressions

Solve the equation: 3x − 5 = 16.

  1. 63
  2. 11/3
  3. 7 Correct answer.
  4. −7

Explanation

Add 5 to both sides: 3x = 21. Divide both sides by 3: x = 7. Check: 3(7) − 5 = 21 − 5 = 16.

Question 10Numbers and Algebraic Expressions

Solve the equation: 0.4x − 1 = 0.2x + 3.

  1. 15
  2. 10
  3. 20 Correct answer.
  4. 40

Explanation

Subtract 0.2x from both sides: 0.2x − 1 = 3. Add 1: 0.2x = 4. Multiplying both sides by 5 (or dividing by 0.2) gives x = 20.

Question 11Numbers and Algebraic Expressions

Solve the simultaneous equations: x + y = 10 and x − y = 4. What is the value of x?

  1. 6
  2. 14
  3. 3
  4. 7 Correct answer.

Explanation

Add the two equations to eliminate y: 2x = 14, so x = 7. Substituting back gives y = 3, and indeed 7 + 3 = 10 and 7 − 3 = 4.

Question 12Numbers and Algebraic Expressions

Expand: (x + 3)(x − 5).

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

Explanation

Use the pattern (x + a)(x + b) = x^2 + (a + b)x + ab. Here a = 3 and b = −5, so the x-coefficient is 3 + (−5) = −2 and the constant is 3 × (−5) = −15, giving x^2 − 2x − 15.

Question 13Numbers and Algebraic Expressions

Factor: x^2 − 7x + 12.

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

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). Expanding confirms it: x^2 − 4x − 3x + 12 = x^2 − 7x + 12.

Question 14Numbers and Algebraic Expressions

Rewrite √48 in the form a√b, where b is as small as possible.

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

Explanation

Factor out the largest perfect square: 48 = 16 × 3, so √48 = √16 × √3 = 4√3.

Question 15Numbers and Algebraic Expressions

Calculate: √12 × √3.

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

Explanation

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

Question 16Numbers and Algebraic Expressions

Solve the equation: x^2 = 49.

  1. x = −7 only
  2. x = 7 only
  3. x = 7 and x = −7 Correct answer.
  4. x = 24.5

Explanation

Taking the square root of both sides gives x = ±√49 = ±7, because both 7^2 and (−7)^2 equal 49. A quadratic equation of this form always has two solutions.

Question 17Numbers and Algebraic Expressions

Solve the equation: (x − 2)(x + 5) = 0.

  1. x = 2 and x = 5
  2. x = 2 and x = −5 Correct answer.
  3. x = −2 and x = −5
  4. x = −2 and x = 5

Explanation

A product is zero only when at least one factor is zero. So x − 2 = 0 gives x = 2, and x + 5 = 0 gives x = −5. The solutions are x = 2 and x = −5.

Question 18Numbers and Algebraic Expressions

Expand: (2x − 3)^2.

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

Explanation

Use (a − b)^2 = a^2 − 2ab + b^2 with a = 2x and b = 3: (2x)^2 − 2(2x)(3) + 3^2 = 4x^2 − 12x + 9.

Question 19Numbers and Algebraic Expressions

Factor completely: 3x^2 − 12.

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

Explanation

First factor out the common factor 3: 3x^2 − 12 = 3(x^2 − 4). Then use the difference of squares: x^2 − 4 = (x + 2)(x − 2), giving 3(x + 2)(x − 2).

Question 20Numbers and Algebraic Expressions

Calculate: (−3)^2 − 4 × (−2).

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

Explanation

Evaluate powers and products first: (−3)^2 = 9 and 4 × (−2) = −8. Then 9 − (−8) = 9 + 8 = 17.

Question 21Numbers and Algebraic Expressions

Solve the quadratic equation: x^2 + 4x + 1 = 0.

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

Explanation

Apply the quadratic formula with a = 1, b = 4, c = 1: x = (−4 ± √(16 − 4))/2 = (−4 ± √12)/2 = (−4 ± 2√3)/2 = −2 ± √3.

Question 22Numbers and Algebraic Expressions

Solve the quadratic equation: x^2 − 3x − 10 = 0.

  1. x = −5 and x = −2
  2. x = −5 and x = 2
  3. x = 5 and x = 2
  4. x = 5 and x = −2 Correct answer.

Explanation

Factor by finding two numbers with product −10 and sum −3: they are −5 and +2, so (x − 5)(x + 2) = 0. Setting each factor to zero gives x = 5 and x = −2.

Question 23Numbers and Algebraic Expressions

Calculate: 6/√3 − √12.

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

Explanation

Rationalize the first term: 6/√3 = 6√3/3 = 2√3. Simplify the second: √12 = 2√3. Therefore 2√3 − 2√3 = 0.

Question 24Numbers and Algebraic Expressions

Solve the simultaneous equations: x/2 + y/3 = 7 and x − y = 4.

  1. x = 10, y = 6 Correct answer.
  2. x = 12, y = 8
  3. x = 6, y = 2
  4. x = 8, y = 4

Explanation

Clear fractions by multiplying the first equation by 6: 3x + 2y = 42. From the second equation, x = y + 4. Substitute: 3(y + 4) + 2y = 42, so 5y + 12 = 42, y = 6, and then x = 10. Check: 10/2 + 6/3 = 5 + 2 = 7.

Question 25Numbers and Algebraic Expressions

Given x + y = 7 and xy = 10, find the value of x^2 + y^2.

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

Explanation

Use the identity x^2 + y^2 = (x + y)^2 − 2xy. Substitute: 7^2 − 2(10) = 49 − 20 = 29.

Question 26Linear Equations and Functions

y is directly proportional to x, and y = 6 when x = 2. What is the constant of proportionality?

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

Explanation

Direct proportion means y = ax. Substitute the given values: 6 = 2a, so a = 3. The constant of proportionality is 3.

Question 27Linear Equations and Functions

Which of the following equations represents y being inversely proportional to x?

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

Explanation

Inverse proportion has the form y = a/x (equivalently xy = a), where a is a nonzero constant. Only y = 12/x has this form.

Question 28Linear Equations and Functions

For the linear function y = 2x + 3, find the value of y when x = 4.

  1. 14
  2. 24
  3. 11 Correct answer.
  4. 9

Explanation

Substitute x = 4: y = 2(4) + 3 = 8 + 3 = 11.

Question 29Linear Equations and Functions

What is the slope (rate of change) of the linear function y = −3x + 5?

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

Explanation

In the form y = ax + b, the slope is a, the coefficient of x. Here a = −3, so the slope is −3; the graph falls 3 units for every 1 unit it moves right.

Question 30Quadratic Functions and Equations

For the function y = ax^2, y = 12 when x = 2. Find the value of a.

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

Explanation

Substitute: 12 = a(2)^2 = 4a. Dividing both sides by 4 gives a = 3.

Question 31Quadratic Functions and Equations

For the function y = x^2, find the value of y when x = −3.

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

Explanation

Substitute x = −3: y = (−3)^2 = (−3) × (−3) = 9. Squaring a negative number always gives a positive result.

Question 32Linear Equations and Functions

Find the rate of change of the function y = 3x + 2 as x increases from 1 to 4.

  1. 2
  2. 9
  3. 11
  4. 3 Correct answer.

Explanation

The rate of change is (change in y)/(change in x). When x = 1, y = 5; when x = 4, y = 14. So (14 − 5)/(4 − 1) = 9/3 = 3. For any linear function the rate of change equals the slope.

Question 33Linear Equations and Functions

Find the equation of the line with slope −2 that passes through the point (0, 5).

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

Explanation

A line with slope a and y-intercept b is y = ax + b. The point (0, 5) lies on the y-axis, so the y-intercept is 5. With slope −2, the equation is y = −2x + 5.

Question 34Linear Equations and Functions

Find the equation of the line passing through the points (2, 3) and (4, 7).

  1. y = 2x + 3
  2. y = 2x − 1 Correct answer.
  3. y = 4x − 5
  4. y = (1/2)x + 2

Explanation

The slope is (7 − 3)/(4 − 2) = 4/2 = 2. Substituting (2, 3) into y = 2x + b gives 3 = 4 + b, so b = −1. The equation is y = 2x − 1, and indeed 2(4) − 1 = 7.

Question 35Quadratic Functions and Equations

Find the rate of change of the function y = 2x^2 as x increases from 1 to 3.

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

Explanation

When x = 1, y = 2; when x = 3, y = 18. The rate of change is (18 − 2)/(3 − 1) = 16/2 = 8. Unlike a linear function, this rate depends on the interval chosen.

Question 36Linear Equations and Functions

For which function does y decrease as x increases within the range x > 0?

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

Explanation

For x > 0, the parabola y = −2x^2 opens downward, so as x grows, x^2 grows and −2x^2 becomes more negative: y decreases. The other three functions all increase over x > 0.

Question 37Linear Equations and Functions

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

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

Explanation

Set the expressions equal: 2x + 1 = −x + 7, so 3x = 6 and x = 2. Then y = 2(2) + 1 = 5. Both lines pass through (2, 5).

Question 38Linear Equations and Functions

A linear function's graph passes through (1, 4) and (3, 10). Find its equation.

  1. y = 2x + 2
  2. y = 6x − 2
  3. y = 3x + 1 Correct answer.
  4. y = 3x + 10

Explanation

The slope is (10 − 4)/(3 − 1) = 6/2 = 3. Substitute (1, 4) into y = 3x + b: 4 = 3 + b, so b = 1. The equation is y = 3x + 1; check: 3(3) + 1 = 10.

Question 39Quadratic Functions and Equations

Which of the following points lies on the graph of y = −x^2?

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

Explanation

Substitute x = −2: y = −(−2)^2 = −4, so the point (−2, −4) is on the graph. The parabola y = −x^2 opens downward and is symmetric about the y-axis.

Question 40Linear Equations and Functions

y is directly proportional to x, and y = −8 when x = 2. Find the value of y when x = −3.

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

Explanation

Find the constant: −8 = 2a, so a = −4 and y = −4x. When x = −3, y = −4 × (−3) = 12.

Question 41Quadratic Functions and Equations

For the function y = ax^2, the rate of change as x increases from −2 to 4 is 6. Find the value of a.

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

Explanation

For y = ax^2, the rate of change from x = p to x = q equals a(p + q). Here a(−2 + 4) = 2a = 6, so a = 3. Equivalently, (16a − 4a)/(4 − (−2)) = 12a/6 = 2a.

Question 42Linear Equations and Functions

The lines y = x + 2 and y = −2x + 8, together with the x-axis, form a triangle. Find the area of this triangle.

  1. 16
  2. 8
  3. 12 Correct answer.
  4. 24

Explanation

The lines meet the x-axis where y = 0: at (−2, 0) and (4, 0), giving a base of length 6. They intersect each other where x + 2 = −2x + 8, so x = 2 and y = 4; the height is 4. Area = (1/2) × 6 × 4 = 12.

Question 43Quadratic Functions and Equations

Find the rate of change of the function y = (1/2)x^2 as x increases from −4 to 2.

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

Explanation

When x = −4, y = (1/2)(16) = 8; when x = 2, y = (1/2)(4) = 2. The rate of change is (2 − 8)/(2 − (−4)) = −6/6 = −1.

Question 44Quadratic Functions and Equations

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

  1. (0, 0) and (3, 9) Correct answer.
  2. (1, 3) and (0, 0)
  3. (3, 3) and (0, 0)
  4. (0, 0) and (−3, 9)

Explanation

Set x^2 = 3x, so x^2 − 3x = 0 and x(x − 3) = 0, giving x = 0 or x = 3. The corresponding y-values are 0 and 9, so the intersection points are (0, 0) and (3, 9).

Question 45Geometry: Plane Figures and Angles

Two angles form a straight line together. If one angle measures 65°, what is the measure of the other angle?

  1. 65°
  2. 115° Correct answer.
  3. 25°
  4. 295°

Explanation

Angles on a straight line sum to 180°. So the other angle is 180° − 65° = 115°.

Question 46Geometry: Plane Figures and Angles

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

  1. 60°
  2. 250°
  3. 70° Correct answer.
  4. 80°

Explanation

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

Question 47Geometry: Plane Figures and Angles

Find the measure of one interior angle of a regular pentagon.

  1. 108° Correct answer.
  2. 100°
  3. 120°
  4. 72°

Explanation

The interior angles of an n-sided polygon sum to 180° × (n − 2). For a pentagon: 180° × 3 = 540°, and each angle of a regular pentagon is 540° ÷ 5 = 108°.

Question 48Geometry: Plane Figures and Angles

Find the sum of the interior angles of a hexagon.

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

Explanation

The formula is 180° × (n − 2). For n = 6: 180° × 4 = 720°. This also matches splitting the hexagon into 4 triangles from one vertex.

Question 49Geometry: Plane Figures and Angles

Find the measure of one exterior angle of a regular octagon.

  1. 40°
  2. 45° Correct answer.
  3. 135°
  4. 60°

Explanation

The exterior angles of any polygon sum to 360°. A regular octagon has 8 equal exterior angles, so each is 360° ÷ 8 = 45°.

Question 50Geometry: Plane Figures and Angles

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

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

Explanation

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