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Miyagi Entrance Mathematics study materials
Miyagi Entrance Mathematics study resources

Miyagi Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Evaluate: -3 + 5 - 8

  1. 6
  2. -10
  3. -6 Correct answer.
  4. 0

Explanation

Work left to right: -3 + 5 = 2, then 2 - 8 = -6. The result is negative because the final subtraction exceeds the running total.

Question 2Numbers and Algebraic Expressions

Compute: (-2)^3 × (-3)

  1. -48
  2. 24 Correct answer.
  3. -24
  4. 48

Explanation

First (-2)^3 = -8. Then (-8) × (-3) = 24. A negative times a negative is positive, so the product is positive 24.

Question 3Numbers and Algebraic Expressions

Find the value of sqrt(144) - sqrt(64).

  1. 4 Correct answer.
  2. 8
  3. 20
  4. 80

Explanation

sqrt(144) = 12 and sqrt(64) = 8, so 12 - 8 = 4. Square roots are evaluated first, then subtracted.

Question 4Numbers and Algebraic Expressions

Simplify 2^3 × 2^2.

  1. 10
  2. 16
  3. 64
  4. 32 Correct answer.

Explanation

Using the law of exponents, 2^3 × 2^2 = 2^(3+2) = 2^5 = 32. Equivalently, 8 × 4 = 32.

Question 5Numbers and Algebraic Expressions

Evaluate -5 × (-2 + 4).

  1. -10 Correct answer.
  2. -3
  3. 10
  4. 40

Explanation

First simplify the parentheses: -2 + 4 = 2. Then -5 × 2 = -10. The single negative sign makes the product negative.

Question 6Numbers and Algebraic Expressions

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

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

Explanation

Using FOIL: x·x = x^2, x·(-2) = -2x, 3·x = 3x, 3·(-2) = -6. Combining -2x + 3x = x, so the result is x^2 + x - 6.

Question 7Numbers and Algebraic Expressions

Factor x^2 - 7x + 12 completely.

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

Explanation

We need 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 8Numbers and Algebraic Expressions

Evaluate a^2 - 2ab + b^2 when a = 3 and b = 1.

  1. 10
  2. 4 Correct answer.
  3. 2
  4. 16

Explanation

Substituting: 3^2 - 2(3)(1) + 1^2 = 9 - 6 + 1 = 4. Notice this equals (a - b)^2 = (3 - 1)^2 = 4.

Question 9Numbers and Algebraic Expressions

Simplify 3*sqrt(2) + sqrt(8).

  1. 5*sqrt(2) Correct answer.
  2. 5
  3. 3*sqrt(2)
  4. sqrt(10)

Explanation

First simplify sqrt(8) = sqrt(4·2) = 2*sqrt(2). Then 3*sqrt(2) + 2*sqrt(2) = 5*sqrt(2). Like radicals are combined just like like terms.

Question 10Numbers and Algebraic Expressions

Expand (2x - 3)^2.

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

Explanation

Using (a - b)^2 = a^2 - 2ab + b^2: (2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9. The middle term is twice the product, with a negative sign.

Question 11Numbers and Algebraic Expressions

Factor x^2 - 9.

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

Explanation

This is a difference of squares: a^2 - b^2 = (a - b)(a + b). With a = x and b = 3, x^2 - 9 = (x - 3)(x + 3).

Question 12Numbers and Algebraic Expressions

Simplify (x + 2)^2 - (x - 2)^2.

  1. 0
  2. 4x
  3. 8
  4. 8x Correct answer.

Explanation

Using the difference of squares a^2 - b^2 = (a + b)(a - b) with a = x + 2 and b = x - 2: the sum is 2x and the difference is 4, so the product is 2x × 4 = 8x. Alternatively, expanding gives (x^2 + 4x + 4) - (x^2 - 4x + 4) = 8x.

Question 13Numbers and Algebraic Expressions

Compute sqrt(50) × sqrt(8).

  1. 20 Correct answer.
  2. 200
  3. sqrt(58)
  4. 10*sqrt(2)

Explanation

Use the product property of radicals: sqrt(50) × sqrt(8) = sqrt(50 × 8) = sqrt(400) = 20. Alternatively, sqrt(50) = 5*sqrt(2) and sqrt(8) = 2*sqrt(2), so 5*sqrt(2) × 2*sqrt(2) = 10 × 2 = 20.

Question 14Linear Equations and Functions

Solve 2x + 5 = 11.

  1. x = 4
  2. x = 3 Correct answer.
  3. x = 8
  4. x = -3

Explanation

Subtract 5 from both sides: 2x = 6. Then divide by 2: x = 3. Check: 2(3) + 5 = 11.

Question 15Linear Equations and Functions

Solve 3(x - 2) = 9.

  1. x = -1
  2. x = 9
  3. x = 3
  4. x = 5 Correct answer.

Explanation

First distribute: 3x - 6 = 9. Then add 6: 3x = 15. Divide by 3: x = 5. Alternatively, divide both sides by 3 first: x - 2 = 3, so x = 5.

Question 16Linear Equations and Functions

Solve the simultaneous equations: x + y = 7, x - y = 1.

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

Explanation

Add the two equations: (x + y) + (x - y) = 7 + 1, so 2x = 8 and x = 4. Substitute back: 4 + y = 7, so y = 3.

Question 17Linear Equations and Functions

Solve x^2 - 5x + 6 = 0.

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

Explanation

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

Question 18Linear Equations and Functions

Solve the inequality 2x + 3 <= 11.

  1. x <= 7
  2. x >= 4
  3. x <= 4.5
  4. x <= 4 Correct answer.

Explanation

Subtract 3 from both sides: 2x <= 8. Then divide by 2 (positive, so the inequality direction is preserved): x <= 4.

Question 19Linear Equations and Functions

Pens cost 200 yen each and notebooks cost 300 yen each. You buy 8 items totaling 1900 yen. How many of each did you buy?

  1. 3 pens, 5 notebooks
  2. 4 pens, 4 notebooks
  3. 6 pens, 2 notebooks
  4. 5 pens, 3 notebooks Correct answer.

Explanation

Let p = pens, n = notebooks. Then p + n = 8 and 200p + 300n = 1900. Substituting p = 8 - n: 200(8 - n) + 300n = 1900, so 1600 + 100n = 1900, giving n = 3 and p = 5. Check: 5 + 3 = 8 and 1000 + 900 = 1900.

Question 20Linear Equations and Functions

Solve the simultaneous equations: 2x + y = 8, x + y = 5.

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

Explanation

Subtract the second equation from the first: (2x + y) - (x + y) = 8 - 5, so x = 3. Substitute into x + y = 5: 3 + y = 5, so y = 2.

Question 21Linear Equations and Functions

Solve x^2 - 4x + 1 = 0 using the quadratic formula.

  1. -2 + sqrt(3) or -2 - sqrt(3)
  2. 2 + sqrt(3) or 2 - sqrt(3) Correct answer.
  3. 2 + sqrt(12) or 2 - sqrt(12)
  4. 4 + sqrt(3) or 4 - sqrt(3)

Explanation

Using x = (-b ± sqrt(b^2 - 4ac)) / (2a) with a = 1, b = -4, c = 1: the discriminant is 16 - 4 = 12, so x = (4 ± sqrt(12)) / 2 = (4 ± 2*sqrt(3)) / 2 = 2 ± sqrt(3).

Question 22Linear Equations and Functions

Solve the inequality -2x < 6.

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

Explanation

Divide both sides by -2. When dividing an inequality by a negative number, the direction reverses: x > -3. This sign reversal is the key step that is often missed.

Question 23Linear Equations and Functions

A car travels at 60 km/h for some time and at 80 km/h for 2 hours. The total distance is 220 km. How long did it travel at 60 km/h?

  1. 1.5 hours
  2. 0.5 hours
  3. 2 hours
  4. 1 hour Correct answer.

Explanation

The 80 km/h portion covers 80 × 2 = 160 km. The remaining distance is 220 - 160 = 60 km. At 60 km/h, the time is 60 / 60 = 1 hour.

Question 24Linear Equations and Functions

Solve x^2 + 6x + 9 = 0.

  1. x = 3
  2. x = -3 Correct answer.
  3. x = 3 or x = -3
  4. x = 0

Explanation

Recognize the perfect square: x^2 + 6x + 9 = (x + 3)^2 = 0. Taking the square root gives x + 3 = 0, so x = -3 (a double root).

Question 25Linear Equations and Functions

Given y = 2x - 1 and x + y = 5, find x and y.

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

Explanation

Substitute y = 2x - 1 into x + y = 5: x + (2x - 1) = 5, so 3x = 6 and x = 2. Then y = 2(2) - 1 = 3. Check: 2 + 3 = 5.

Question 26Linear Equations and Functions

You have 1200 yen. You want to buy at least 3 pens at 200 yen each, and as many notebooks as possible at 150 yen each. What is the maximum number of notebooks you can buy?

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

Explanation

First spend on pens: 3 × 200 = 600 yen. Remaining: 1200 - 600 = 600 yen. Each notebook costs 150 yen, so 600 / 150 = 4 notebooks. The constraint is 'at least 3 pens', so buying exactly 3 pens maximizes the notebook budget.

Question 27Linear Equations and Functions

Solve (x - 1)(x + 3) = 0.

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

Explanation

By the zero product property, if (x - 1)(x + 3) = 0 then x - 1 = 0 or x + 3 = 0. So x = 1 or x = -3.

Question 28Linear Equations and Functions

Solve x^2 - 2x - 8 = 0.

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

Explanation

Factor: find two numbers whose product is -8 and sum is -2. Those are -4 and 2, so x^2 - 2x - 8 = (x - 4)(x + 2) = 0, giving x = 4 or x = -2.

Question 29Linear Equations and Functions

Solve the simultaneous equations: 3x + 2y = 16, x - y = 2.

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

Explanation

From the second equation: x = y + 2. Substitute into the first: 3(y + 2) + 2y = 16, so 5y + 6 = 16, giving y = 2. Then x = 2 + 2 = 4.

Question 30Linear Equations and Functions

If y is directly proportional to x, and y = 12 when x = 4, find the constant of proportionality k.

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

Explanation

Direct proportion means y = kx. Substituting y = 12 and x = 4: 12 = k × 4, so k = 3. This means y = 3x for all values.

Question 31Linear Equations and Functions

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

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

Explanation

First find k: y = k/x, so 6 = k/2, giving k = 12. Then with y = 3: 3 = 12/x, so x = 4. As x increases, y decreases, which is the hallmark of inverse proportion.

Question 32Linear Equations and Functions

For the linear function y = 2x + 3, find y when x = -1.

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

Explanation

Substitute x = -1: y = 2(-1) + 3 = -2 + 3 = 1. The negative input produces a smaller output, but the y-intercept of 3 keeps it positive.

Question 33Linear Equations and Functions

Find the equation of the line passing through (1, 2) and (3, 6).

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

Explanation

Slope m = (6 - 2)/(3 - 1) = 4/2 = 2. Using point-slope with (1, 2): y - 2 = 2(x - 1), so y = 2x. Check with (3, 6): 2(3) = 6.

Question 34Quadratic Functions and Equations

For the quadratic function y = ax^2, if y = 8 when x = 2, find the value of a.

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

Explanation

Substitute x = 2 and y = 8: 8 = a(2^2) = 4a, so a = 2. The function is y = 2x^2, a parabola opening upward with vertex at the origin.

Question 35Quadratic Functions and Equations

For y = -2x^2, find y when x = 3.

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

Explanation

Substitute x = 3: y = -2(3^2) = -2(9) = -18. The negative coefficient means the parabola opens downward, so all outputs for nonzero x are negative.

Question 36Quadratic Functions and Equations

For y = x^2, find the change in y as x goes from 1 to 3.

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

Explanation

When x = 1, y = 1. When x = 3, y = 9. The change is 9 - 1 = 8. Note this is not proportional: tripling x from 1 to 3 multiplies y by 9, not by 3.

Question 37Quadratic Functions and Equations

Find the coordinates of the vertex of y = x^2 - 4x.

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

Explanation

Complete the square: y = x^2 - 4x = (x - 2)^2 - 4. The vertex is at (2, -4), and the axis of symmetry is the line x = 2. The parabola opens upward since the coefficient of x^2 is positive.

Question 38Linear Equations and Functions

Find the intersection point of y = 2x and y = -x + 3.

  1. (2, 1)
  2. (1, 2) Correct answer.
  3. (3, 0)
  4. (0, 3)

Explanation

Set the two expressions equal: 2x = -x + 3, so 3x = 3 and x = 1. Then y = 2(1) = 2. The intersection point is (1, 2), which satisfies both equations.

Question 39Linear Equations and Functions

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

  1. 24 Correct answer.
  2. 8
  3. 3
  4. 15

Explanation

Find k: y = kx, so 15 = 5k, giving k = 3. Then y = 3(8) = 24. Direct proportion means y scales linearly with x.

Question 40Linear Equations and Functions

For y = 3x - 2, find x when y = 7.

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

Explanation

Set y = 7: 7 = 3x - 2, so 3x = 9 and x = 3. Check: 3(3) - 2 = 7.

Question 41Linear Equations and Functions

If y is inversely proportional to x (y = k/x), what happens to y when x is doubled?

  1. y is doubled
  2. y is unchanged
  3. y is tripled
  4. y is halved Correct answer.

Explanation

If y = k/x and x becomes 2x, then y' = k/(2x) = (1/2)(k/x) = y/2. In inverse proportion, when one variable doubles, the other is halved, keeping the product xy = k constant.

Question 42Linear Equations and Functions

A line has slope -2 and passes through (1, 5). Find its equation.

  1. y = 2x + 7
  2. y = -2x + 7 Correct answer.
  3. y = -2x - 7
  4. y = -2x + 5

Explanation

Using y = mx + b: 5 = -2(1) + b, so b = 7. The equation is y = -2x + 7. Check: when x = 1, y = -2 + 7 = 5.

Question 43Quadratic Functions and Equations

For y = 3x^2, find the change in y as x goes from 2 to 4.

  1. 48
  2. 24
  3. 12
  4. 36 Correct answer.

Explanation

When x = 2, y = 3(4) = 12. When x = 4, y = 3(16) = 48. The change is 48 - 12 = 36. The rate of change of a quadratic function is not constant; it accelerates as x moves away from the vertex.

Question 44Linear Equations and Functions

Find the slope of the line passing through (-1, 3) and (2, -3).

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

Explanation

Slope m = (y2 - y1)/(x2 - x1) = (-3 - 3)/(2 - (-1)) = -6/3 = -2. The negative slope means the line descends as x increases.

Question 45Quadratic Functions and Equations

The quadratic function y = ax^2 passes through (3, 18). Find y when x = 2.

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

Explanation

First find a: 18 = a(9), so a = 2 and y = 2x^2. Then y(2) = 2(4) = 8. Because y scales with x^2, reducing x from 3 to 2 cuts y from 18 to 8, not by the same factor as x.

Question 46Linear Equations and Functions

What is the relationship between the lines y = 3x + 1 and y = 3x - 2?

  1. The same line
  2. Perpendicular
  3. Intersecting at one point
  4. Parallel Correct answer.

Explanation

Both lines have slope 3 but different y-intercepts (1 and -2). Lines with the same slope and different intercepts are parallel and never intersect. The y-intercept difference means they are distinct, not the same line.

Question 47Quadratic Functions and Equations

Find the minimum value of y = x^2 - 6x + 5.

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

Explanation

Complete the square: y = (x - 3)^2 - 4. Since (x - 3)^2 is always non-negative, the minimum occurs when (x - 3)^2 = 0, giving y = -4 at x = 3. The parabola opens upward, so this is a minimum.

Question 48Quadratic Functions and Equations

For y = x^2 - 5x + 6, find x when y = 0.

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

Explanation

Factor: x^2 - 5x + 6 = (x - 2)(x - 3) = 0. By the zero product property, x = 2 or x = 3. These are the x-intercepts of the parabola.

Question 49Linear Equations and Functions

A water tank is filled at a constant rate of 50 liters per minute. How long does it take to fill 300 liters?

  1. 5 minutes
  2. 150 minutes
  3. 50 minutes
  4. 6 minutes Correct answer.

Explanation

Using the linear relationship t = V/r: t = 300/50 = 6 minutes. The volume is directly proportional to time, so the graph is a straight line through the origin with slope 50.

Question 50Geometry: Plane Figures and Angles

Two angles of a triangle measure 50 degrees and 60 degrees. Find the third angle.

  1. 60 degrees
  2. 110 degrees
  3. 50 degrees
  4. 70 degrees Correct answer.

Explanation

The sum of angles in a triangle is 180 degrees. The third angle is 180 - 50 - 60 = 70 degrees. This is the triangle angle sum theorem.