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Kagawa Entrance Mathematics study materials
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Kagawa Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Evaluate (−3) + 5 − (−2).

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

Explanation

Work left to right: (−3) + 5 = 2, then 2 − (−2) = 2 + 2 = 4. Subtracting a negative number is the same as adding its opposite, so the result is 4.

Question 2Numbers and Algebraic Expressions

Compute (−4) × 3 ÷ (−2).

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

Explanation

First (−4) × 3 = −12. Then −12 ÷ (−2) = 6. A negative divided by a negative is positive, so the answer is 6.

Question 3Numbers and Algebraic Expressions

Simplify the expression 3x − 2(x + 4) − 7.

  1. x − 15 Correct answer.
  2. x + 1
  3. 5x − 15
  4. x − 1

Explanation

Distribute first: −2(x + 4) = −2x − 8. Then 3x − 2x − 8 − 7 = x − 15. Combining like terms gives x − 15.

Question 4Numbers and Algebraic Expressions

When x = −2 and y = 3, evaluate the expression x² − 2xy + y².

  1. 1
  2. 25 Correct answer.
  3. 5
  4. −5

Explanation

Substitute: x² = (−2)² = 4, 2xy = 2(−2)(3) = −12, y² = 9. So 4 − (−12) + 9 = 4 + 12 + 9 = 25. Equivalently, x² − 2xy + y² = (x − y)² = (−5)² = 25.

Question 5Numbers and Algebraic Expressions

Expand (x + 3)(x − 5).

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

Explanation

Use FOIL: x·x = x², x·(−5) = −5x, 3·x = 3x, 3·(−5) = −15. Combine the middle terms: −5x + 3x = −2x. So the result is x² − 2x − 15.

Question 6Numbers and Algebraic Expressions

Factor the expression x² − 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 that multiply to +12 and add to −7. Both must be negative: −3 and −4 give (−3)(−4) = 12 and −3 + (−4) = −7. So x² − 7x + 12 = (x − 3)(x − 4).

Question 7Numbers and Algebraic Expressions

Factor 2x² − 5x − 3.

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

Explanation

We seek a factorization (2x + a)(x + b) with a·b = −3 and 2b + a = −5. Try a = 1, b = −3: a·b = −3 and 2(−3) + 1 = −5. Both conditions hold, so 2x² − 5x − 3 = (2x + 1)(x − 3).

Question 8Numbers and Algebraic Expressions

Which of the following is equal to √72 − √32?

  1. √40
  2. √2
  3. 4√2
  4. 2√2 Correct answer.

Explanation

Simplify each radical: √72 = √(36·2) = 6√2, and √32 = √(16·2) = 4√2. Then 6√2 − 4√2 = 2√2. Like radicals combine by adding or subtracting their coefficients.

Question 9Numbers and Algebraic Expressions

Compute (2 + √3)(2 − √3).

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

Explanation

This is a difference of squares: (a + b)(a − b) = a² − b² with a = 2, b = √3. So 2² − (√3)² = 4 − 3 = 1. The radicals cancel, leaving the rational number 1.

Question 10Numbers and Algebraic Expressions

Rationalize and simplify 6 / √3.

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

Explanation

Multiply numerator and denominator by √3: 6√3 / (√3·√3) = 6√3 / 3 = 2√3. The denominator becomes the rational number 3, and 6/3 = 2.

Question 11Numbers and Algebraic Expressions

If a = 2 and b = −3, find the value of a²b + ab².

  1. 12
  2. 0
  3. −6
  4. 6 Correct answer.

Explanation

Compute each term: a²b = (2)²(−3) = 4(−3) = −12, and ab² = (2)(−3)² = 2(9) = 18. Adding: −12 + 18 = 6. Alternatively, factor as ab(a + b) = (2)(−3)(2 + (−3)) = (−6)(−1) = 6.

Question 12Numbers and Algebraic Expressions

Simplify 3√2 + √8 − √18.

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

Explanation

Simplify each radical: √8 = √(4·2) = 2√2, and √18 = √(9·2) = 3√2. Then 3√2 + 2√2 − 3√2 = 2√2. The first and last terms cancel, leaving 2√2.

Question 13Numbers and Algebraic Expressions

Which expression is the expanded form of (x − 2)²?

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

Explanation

Use the identity (a − b)² = a² − 2ab + b² with a = x, b = 2: x² − 2(x)(2) + 2² = x² − 4x + 4. The middle term is negative and the constant is positive.

Question 14Numbers and Algebraic Expressions

Find the value of (−2)³ × (−1)².

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

Explanation

Compute each power: (−2)³ = −8, and (−1)² = 1. Then −8 × 1 = −8. The cube of a negative is negative, and the square of a negative is positive.

Question 15Numbers and Algebraic Expressions

Expand and simplify 2(x + 3) − 3(x − 1) + x.

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

Explanation

Distribute: 2x + 6 − 3x + 3 + x. Combine like terms: (2x − 3x + x) + (6 + 3) = 0x + 9 = 9. The x-terms cancel completely, leaving the constant 9.

Question 16Numbers and Algebraic Expressions

If √x = 5, find the value of x − 6.

  1. 19 Correct answer.
  2. 25
  3. −1
  4. 1

Explanation

From √x = 5, square both sides: x = 25. Then x − 6 = 25 − 6 = 19. The square root undoes the squaring for non-negative x.

Question 17Numbers and Algebraic Expressions

Factor the expression 4x² − 9.

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

Explanation

This is a difference of squares: a² − b² = (a − b)(a + b) with a = 2x, b = 3. So 4x² − 9 = (2x)² − 3² = (2x − 3)(2x + 3). The two factors are conjugates.

Question 18Numbers and Algebraic Expressions

The expression (x + 1)(x + 2)(x + 3) is expanded. What is the coefficient of x²?

  1. 11
  2. 6 Correct answer.
  3. 1
  4. 5

Explanation

First multiply (x + 1)(x + 2) = x² + 3x + 2. Then multiply by (x + 3): (x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6 = x³ + 6x² + 11x + 6. The coefficient of x² is 6.

Question 19Linear Equations and Functions

Solve for x: 3x − 7 = 2x + 5.

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

Explanation

Subtract 2x from both sides: x − 7 = 5. Add 7: x = 12. Check: 3(12) − 7 = 29 and 2(12) + 5 = 29, confirming the solution.

Question 20Linear Equations and Functions

Solve the equation 2(x − 3) = 4 − x.

  1. x = 10/3 Correct answer.
  2. x = 2
  3. x = −2
  4. x = 10

Explanation

Distribute: 2x − 6 = 4 − x. Add x: 3x − 6 = 4. Add 6: 3x = 10. Divide by 3: x = 10/3. Checking: 2(10/3 − 3) = 2(1/3) = 2/3 and 4 − 10/3 = 2/3.

Question 21Linear Equations and Functions

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

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

Explanation

Add the two equations: (x + y) + (x − y) = 7 + 1, so 2x = 8, x = 4. Substitute into x + y = 7: 4 + y = 7, y = 3. The solution is x = 4, y = 3.

Question 22Linear Equations and Functions

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

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

Explanation

From x − y = 1, x = y + 1. Substitute into 2(y + 1) + 3y = 12: 2y + 2 + 3y = 12, so 5y = 10, y = 2. Then x = 2 + 1 = 3. The solution is x = 3, y = 2.

Question 23Linear Equations and Functions

Solve the simultaneous equations: 3x + 2y = 16, 2x + y = 9.

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

Explanation

From 2x + y = 9, y = 9 − 2x. Substitute: 3x + 2(9 − 2x) = 16, so 3x + 18 − 4x = 16, −x = −2, x = 2. Then y = 9 − 2(2) = 5. The solution is x = 2, y = 5.

Question 24Linear Equations and Functions

Solve x² − 5x + 6 = 0.

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

Explanation

Factor: x² − 5x + 6 = (x − 2)(x − 3) = 0. By the zero-product property, x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3. Both are positive since the sum is +5 and the product is +6.

Question 25Linear Equations and Functions

Solve x² − 6x − 7 = 0.

  1. x = 7 or x = 1
  2. x = −7 or x = 1
  3. x = 7 or x = −1 Correct answer.
  4. x = −7 or x = −1

Explanation

Factor: we need two numbers multiplying to −7 and adding to −6. Those are −7 and +1: (x − 7)(x + 1) = 0, so x = 7 or x = −1. The product is negative, so the roots have opposite signs.

Question 26Linear Equations and Functions

Using the quadratic formula, solve x² − 4x − 1 = 0. Which gives the solutions?

  1. x = −2 ± √5
  2. x = 2 ± √5 Correct answer.
  3. x = 2 ± √3
  4. x = 4 ± √20

Explanation

Here a = 1, b = −4, c = −1. The discriminant is b² − 4ac = 16 + 4 = 20, so √(discriminant) = √20 = 2√5. Then x = (4 ± 2√5)/2 = 2 ± √5. The two roots are 2 + √5 and 2 − √5.

Question 27Linear Equations and Functions

Solve the inequality 2x − 5 < 7 and express the solution as a range.

  1. x ≤ 6
  2. x < 1
  3. x < 6 Correct answer.
  4. x > 6

Explanation

Add 5 to both sides: 2x < 12. Divide by 2 (positive, so the inequality direction is unchanged): x < 6. The inequality is strict (<), so 6 is not included.

Question 28Linear Equations and Functions

Solve the inequality −3x + 1 ≤ 10.

  1. x ≥ −3 Correct answer.
  2. x ≥ 3
  3. x ≤ −3
  4. x ≤ 3

Explanation

Subtract 1: −3x ≤ 9. Divide by −3 and flip the inequality (dividing by a negative reverses the sign): x ≥ −3. The flip is the key step many students miss.

Question 29Linear Equations and Functions

A number plus 5 equals three times the number minus 7. Find the number.

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

Explanation

Let the number be n. The equation is n + 5 = 3n − 7. Subtract n: 5 = 2n − 7. Add 7: 12 = 2n. Divide by 2: n = 6. Check: 6 + 5 = 11 and 3(6) − 7 = 11.

Question 30Linear Equations and Functions

The sum of two numbers is 20 and their difference is 4. Find the larger number.

  1. 12 Correct answer.
  2. 10
  3. 8
  4. 16

Explanation

Let the numbers be a and b with a > b. Then a + b = 20 and a − b = 4. Add the equations: 2a = 24, so a = 12. The larger number is 12 (and the smaller is 8).

Question 31Linear Equations and Functions

A shop sells pencils at 80 yen each and erasers at 120 yen each. A student buys 10 items in total for 920 yen. How many erasers did the student buy?

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

Explanation

Let p = pencils, e = erasers. Then p + e = 10 and 80p + 120e = 920. From the first equation p = 10 − e. Substitute: 80(10 − e) + 120e = 920, so 800 − 80e + 120e = 920, 40e = 120, e = 3. The student bought 3 erasers (and 7 pencils).

Question 32Linear Equations and Functions

The length of a rectangle is 3 cm more than its width. The perimeter is 26 cm. Find the width.

  1. 10 cm
  2. 5 cm Correct answer.
  3. 13 cm
  4. 8 cm

Explanation

Let the width be w. Then the length is w + 3. The perimeter is 2(w + w + 3) = 26, so 2(2w + 3) = 26, 4w + 6 = 26, 4w = 20, w = 5. The width is 5 cm (and the length is 8 cm).

Question 33Linear Equations and Functions

Solve x² + 2x − 8 = 0 by factoring.

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

Explanation

We need two numbers multiplying to −8 and adding to +2. Those are +4 and −2: (x + 4)(x − 2) = 0, so x = −4 or x = 2. The product is negative, so the roots have opposite signs.

Question 34Linear Equations and Functions

A father is now 4 times as old as his son. In 6 years, he will be 3 times as old as his son. How old is the son now?

  1. 6 Correct answer.
  2. 18
  3. 24
  4. 12

Explanation

Let the son's current age be s. Then the father is 4s now. In 6 years: 4s + 6 = 3(s + 6). Expand: 4s + 6 = 3s + 18. Subtract 3s: s + 6 = 18. Subtract 6: s = 12. Verify: father now 48, in 6 years 54; son now 12, in 6 years 18; 54 = 3(18) = 54. So the son is 12 now.

Question 35Linear Equations and Functions

Find all integer values of x that satisfy the inequality 2x − 1 < 5 and x + 3 ≥ 2 simultaneously.

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

Explanation

From 2x − 1 < 5: 2x < 6, x < 3. From x + 3 ≥ 2: x ≥ −1. Combining: −1 ≤ x < 3. The integers satisfying both are −1, 0, 1, 2. Note that 3 is excluded because the first inequality is strict.

Question 36Linear Equations and Functions

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

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

Explanation

Direct proportion means y = kx. Find k: 15 = k(5), so k = 3. Then when x = 8: y = 3(8) = 24. The ratio y/x = 3 is constant for all valid x.

Question 37Linear Equations and Functions

If y is inversely proportional to x, and y = 6 when x = 4, find y when x = 8.

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

Explanation

Inverse proportion means y = k/x. Find k: 6 = k/4, so k = 24. Then when x = 8: y = 24/8 = 3. As x doubles from 4 to 8, y halves from 6 to 3.

Question 38Linear Equations and Functions

A line passes through the points (1, 3) and (2, 7). What is the slope of this line?

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

Explanation

Slope = (y₂ − y₁)/(x₂ − x₁) = (7 − 3)/(2 − 1) = 4/1 = 4. The slope is the rate of change of y with respect to x, which is 4 here.

Question 39Linear Equations and Functions

A linear function y = ax + b has slope a = −2 and passes through the point (3, 4). Find the y-intercept b.

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

Explanation

Substitute the point into y = −2x + b: 4 = −2(3) + b, so 4 = −6 + b, and b = 10. The y-intercept is 10, so the function is y = −2x + 10.

Question 40Linear Equations and Functions

The line y = 3x − 5 intersects the x-axis at which point?

  1. (0, 5)
  2. (5/3, 0) Correct answer.
  3. (0, −5)
  4. (−5/3, 0)

Explanation

On the x-axis, y = 0. Set 0 = 3x − 5, so 3x = 5, x = 5/3. The x-intercept is (5/3, 0). The y-intercept (0, −5) is a different point.

Question 41Linear Equations and Functions

Find the intersection point of the lines y = 2x + 1 and y = −x + 4.

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

Explanation

Set the equations equal: 2x + 1 = −x + 4. Add x: 3x + 1 = 4. Subtract 1: 3x = 3, x = 1. Then y = 2(1) + 1 = 3. The intersection is (1, 3). Verify: −1 + 4 = 3. ✓

Question 42Quadratic Functions and Equations

For the quadratic function y = x² − 4x + 3, find the coordinates of the vertex.

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

Explanation

The vertex x-coordinate is x = −b/(2a) = −(−4)/(2·1) = 2. Substitute: y = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1. The vertex is (2, −1), the minimum point of the parabola.

Question 43Quadratic Functions and Equations

For the quadratic function y = x² − 4x + 3, what is the equation of the axis of symmetry?

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

Explanation

The axis of symmetry is the vertical line x = −b/(2a) = −(−4)/(2·1) = 2. This is the x-coordinate of the vertex, and the parabola is symmetric about this line.

Question 44Quadratic Functions and Equations

For the function y = −2x², if x = 3, what is y?

  1. −12
  2. 12
  3. 18
  4. −18 Correct answer.

Explanation

Substitute x = 3: y = −2(3)² = −2(9) = −18. The negative coefficient means the parabola opens downward, and the square ensures y is always non-positive.

Question 45Quadratic Functions and Equations

The parabola y = x² − 6x + 5 intersects the x-axis at two points. What are they?

  1. (−1, 0) and (−5, 0)
  2. (1, 0) and (5, 0) Correct answer.
  3. (0, 5) and (0, 1)
  4. (1, 0) and (−5, 0)

Explanation

Set y = 0: x² − 6x + 5 = 0. Factor: (x − 1)(x − 5) = 0, so x = 1 or x = 5. The x-intercepts are (1, 0) and (5, 0). Both roots are positive since their sum is +6 and product is +5.

Question 46Linear Equations and Functions

A car travels at a constant speed. The distance d (km) after t hours is given by d = 60t. How far does the car travel in 2.5 hours?

  1. 240 km
  2. 60 km
  3. 120 km
  4. 150 km Correct answer.

Explanation

Substitute t = 2.5: d = 60(2.5) = 150. The car travels 150 km in 2.5 hours at 60 km/h. This is a direct proportion with constant of proportionality 60.

Question 47Linear Equations and Functions

The graph of y = 3x − 6 is shifted up by 2 units. What is the equation of the new line?

  1. y = 5x − 6
  2. y = 3x − 2
  3. y = 3x − 8
  4. y = 3x − 4 Correct answer.

Explanation

Shifting a graph up by 2 units means adding 2 to the y-value: y = (3x − 6) + 2 = 3x − 4. The slope stays the same; only the y-intercept changes from −6 to −4.

Question 48Quadratic Functions and Equations

For the function y = x², as x changes from 1 to 3, what is the average rate of change of y?

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

Explanation

At x = 1, y = 1. At x = 3, y = 9. The average rate of change is (9 − 1)/(3 − 1) = 8/2 = 4. This is the slope of the secant line connecting (1, 1) and (3, 9).

Question 49Linear Equations and Functions

The function y = k/x passes through the point (2, 5). Find the value of k.

  1. 10 Correct answer.
  2. 2/5
  3. 7
  4. 5/2

Explanation

Substitute the point: 5 = k/2, so k = 10. The constant of inverse proportionality is the product x·y, which is 2 × 5 = 10 for every point on the curve.

Question 50Linear Equations and Functions

A ball is thrown upward with height h (meters) after t seconds given by h = −5t² + 20t. At what time does the ball reach its maximum height?

  1. 1 second
  2. 0 seconds
  3. 4 seconds
  4. 2 seconds Correct answer.

Explanation

The maximum occurs at the vertex of the parabola h = −5t² + 20t. The t-coordinate is t = −b/(2a) = −20/(2·(−5)) = −20/(−10) = 2. The ball reaches maximum height at t = 2 seconds.