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Osaka Public High School Entrance Examination - Mathematics study materials
Osaka Public High School Entrance Examination - Mathematics study resources

Osaka Public High School Entrance Examination - Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate: (−3) × 4 − (−2).

  1. 10
  2. −6
  3. −10 Correct answer.
  4. −14

Explanation

(−3) × 4 = −12, and −12 − (−2) = −12 + 2 = −10. Subtracting a negative is adding the positive.

Question 2Numbers and Algebraic Expressions

Evaluate: −5 + 8 − 3.

  1. 0 Correct answer.
  2. 2
  3. 16
  4. −6

Explanation

Working left to right: −5 + 8 = 3, then 3 − 3 = 0.

Question 3Numbers and Algebraic Expressions

Compute 3/4 + 1/6 and give the answer in lowest terms.

  1. 2/5
  2. 4/24
  3. 11/12 Correct answer.
  4. 4/10

Explanation

Common denominator 12: 3/4 = 9/12 and 1/6 = 2/12, so 9/12 + 2/12 = 11/12, already in lowest terms.

Question 4Numbers and Algebraic Expressions

Simplify: 2/3 ÷ 4/5.

  1. 3/4
  2. 5/6 Correct answer.
  3. 8/15
  4. 8/9

Explanation

Division by a fraction equals multiplication by its reciprocal: 2/3 × 5/4 = 10/12 = 5/6.

Question 5Numbers and Algebraic Expressions

Simplify the square root: √72.

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

Explanation

√72 = √(36 × 2) = √36 × √2 = 6√2, since 36 is the largest perfect-square factor of 72.

Question 6Numbers and Algebraic Expressions

Simplify: √27 + √12.

  1. 3√39
  2. 5√3 Correct answer.
  3. 3√3
  4. √39

Explanation

√27 = 3√3 and √12 = 2√3, so √27 + √12 = 3√3 + 2√3 = 5√3. Like surds add.

Question 7Numbers and Algebraic Expressions

Solve for x: 3x − 7 = 2x + 5.

  1. x = 1/5
  2. x = −12
  3. x = 2
  4. x = 12 Correct answer.

Explanation

Subtract 2x: x − 7 = 5. Add 7: x = 12. Checking: 3(12)−7 = 29 and 2(12)+5 = 29.

Question 8Numbers and Algebraic Expressions

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

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

Explanation

Adding the two equations eliminates y: 2x = 14, so x = 7. Then 7 + y = 10 gives y = 3.

Question 9Numbers and Algebraic Expressions

Expand and simplify: (x + 3)(x − 5).

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

Explanation

Using FOIL: x·x = x², x·(−5) = −5x, 3·x = 3x, 3·(−5) = −15. Combine −5x + 3x = −2x, so x² − 2x − 15.

Question 10Numbers and Algebraic Expressions

Factorise: x² − 9.

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

Explanation

x² − 9 = x² − 3² is a difference of squares, factoring as (x − 3)(x + 3).

Question 11Numbers and Algebraic Expressions

Solve: 2(x − 4) = 3x + 1.

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

Explanation

Expand: 2x − 8 = 3x + 1. Subtract 2x: −8 = x + 1. Subtract 1: x = −9. Check: 2(−13) = −26 vs 3(−9)+1 = −26.

Question 12Numbers and Algebraic Expressions

If a number n satisfies 3n + 5 = 2(n + 4), find n.

  1. n = 7
  2. n = 1
  3. n = −3
  4. n = 3 Correct answer.

Explanation

Expand the right side: 3n + 5 = 2n + 8. Subtract 2n: n + 5 = 8. Subtract 5: n = 3. Check: 3(3)+5 = 14 = 2(7).

Question 13Numbers and Algebraic Expressions

Solve the simultaneous equations: 2x + 3y = 16, x − y = 1.

  1. x = 7, y = 2
  2. x = 4, y = 1
  3. x = 5, y = 4
  4. x = 19/5, y = 14/5 Correct answer.

Explanation

From x − y = 1, x = y + 1. Substitute: 2(y+1) + 3y = 16 ⇒ 5y + 2 = 16 ⇒ 5y = 14 ⇒ y = 14/5, and x = 19/5.

Question 14Numbers and Algebraic Expressions

The sum of two numbers is 25 and their difference is 7. Find the larger number.

  1. 18
  2. 16 Correct answer.
  3. 9
  4. 14

Explanation

Let the numbers be a (larger) and b. a + b = 25, a − b = 7. Adding: 2a = 32, so a = 16. Then b = 9.

Question 15Numbers and Algebraic Expressions

Solve for x: x² − 5x + 6 = 0.

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

Explanation

Factorise: (x − 2)(x − 3) = 0, so x = 2 or x = 3. Both satisfy the equation.

Question 16Numbers and Algebraic Expressions

Solve: x² = 49.

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

Explanation

Taking the square root of both sides gives x = 7 or x = −7, written x = ±7. Both 7² and (−7)² equal 49.

Question 17Numbers and Algebraic Expressions

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

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

Explanation

Let width = w; length = w + 3. Perimeter 2(w + w + 3) = 26 ⇒ 4w + 6 = 26 ⇒ 4w = 20 ⇒ w = 5. Length = 8 cm.

Question 18Numbers and Algebraic Expressions

A shop sells pens at ¥120 each. If you buy 3 pens the third is half price. What is the total for 3 pens?

  1. ¥360
  2. ¥240
  3. ¥420
  4. ¥300 Correct answer.

Explanation

Two pens at ¥120 = ¥240, plus the third at half price ¥60, total ¥240 + ¥60 = ¥300.

Question 19Numbers and Algebraic Expressions

Simplify: 2√3 × √6.

  1. 6√3
  2. 2√18
  3. 2√9
  4. 6√2 Correct answer.

Explanation

√3 × √6 = √18 = √(9×2) = 3√2, so 2√3 × √6 = 2 × 3√2 = 6√2.

Question 20Numbers and Algebraic Expressions

Solve using the quadratic formula: x² − 4x + 1 = 0.

  1. x = 2 ± √3 Correct answer.
  2. x = 2 ± √5
  3. x = ±1
  4. x = 4 ± √12

Explanation

a=1, b=−4, c=1. Discriminant = 16 − 4 = 12. x = (4 ± √12)/2 = (4 ± 2√3)/2 = 2 ± √3.

Question 21Numbers and Algebraic Expressions

Rationalise the denominator and simplify: 1/√2.

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

Explanation

Multiply numerator and denominator by √2: 1/√2 × √2/√2 = √2/2. The denominator becomes 2.

Question 22Numbers and Algebraic Expressions

Which inequality is equivalent to −2x > 6?

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

Explanation

Dividing both sides by −2 reverses the inequality: x < −3. Dividing or multiplying by a negative flips the sign.

Question 23Numbers and Algebraic Expressions

Expand: (2x + 1)².

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

Explanation

(2x + 1)² = (2x)² + 2(2x)(1) + 1² = 4x² + 4x + 1, using (a + b)² = a² + 2ab + b².

Question 24Numbers and Algebraic Expressions

Factorise completely: 2x² − 8.

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

Explanation

First take out 2: 2(x² − 4). Then factor the difference of squares: 2(x − 2)(x + 2).

Question 25Numbers and Algebraic Expressions

If a = 2 and b = −3, evaluate a² − 2ab + b².

  1. 1
  2. −5
  3. 25 Correct answer.
  4. 13

Explanation

a² = 4, 2ab = 2(2)(−3) = −12, b² = 9. So 4 − (−12) + 9 = 4 + 12 + 9 = 25. Equivalently (a − b)² = (5)² = 25.

Question 26Linear Equations and Functions

Which of the following is a direct proportion?

  1. y = x + 5
  2. y = 5/x
  3. y = 5x Correct answer.
  4. y = x²

Explanation

Direct proportion has the form y = kx (a constant times x). y = 5x fits; its graph is a line through the origin.

Question 27Linear Equations and Functions

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

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

Explanation

Inverse proportion: y = k/x. From y = 6 at x = 2: k = 6 × 2 = 12. At x = 3: y = 12/3 = 4.

Question 28Linear Equations and Functions

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

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

Explanation

In y = mx + b form, the coefficient of x is the slope m. Here m = −2 and the y-intercept is 5.

Question 29Linear Equations and Functions

What is the y-intercept of the line passing through (0, 3) with slope 2?

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

Explanation

The point (0, 3) lies on the y-axis, so the y-intercept is 3. The line is y = 2x + 3.

Question 30Linear Equations and Functions

The line y = ax + b passes through (1, 5) and (3, 11). Find a.

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

Explanation

Slope a = (11 − 5)/(3 − 1) = 6/2 = 3. So a = 3; then b = 5 − 3(1) = 2.

Question 31Linear Equations and Functions

Find the equation of the line through (2, 7) with slope 4.

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

Explanation

y = 4x + b. Substitute (2, 7): 7 = 8 + b, so b = −1. The line is y = 4x − 1.

Question 32Linear Equations and Functions

Two lines are parallel. If one is y = 3x − 4, which could be the other?

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

Explanation

Parallel lines share the same slope 3 but have different intercepts. y = 3x + 2 is parallel (and distinct). y = 3x − 4 is the identical line, not a different parallel line.

Question 33Quadratic Functions and Equations

For y = x², what is the value of y when x = −4?

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

Explanation

y = (−4)² = 16. The square of a negative is positive; the parabola y = x² is symmetric about the y-axis.

Question 34Quadratic Functions and Equations

What are the coordinates of the vertex of y = x²?

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

Explanation

y = x² has its minimum at the origin (0, 0), where the axis of symmetry x = 0 meets the parabola.

Question 35Quadratic Functions and Equations

For the parabola y = x², what is the axis of symmetry?

  1. x = 1
  2. x = 0 (the y-axis) Correct answer.
  3. y = 0 (the x-axis)
  4. y = x

Explanation

The parabola y = x² is symmetric about the y-axis, i.e. the line x = 0, since (−a)² = a² for every a.

Question 36Quadratic Functions and Equations

How many points do the graphs of y = x² and y = 4 have in common?

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

Explanation

Set x² = 4: x = 2 or x = −2. The line y = 4 cuts the parabola at (2, 4) and (−2, 4), two points.

Question 37Quadratic Functions and Equations

For y = x², as x increases from 1 to 3, how does y change?

  1. y decreases from 1 to 9
  2. y stays 1
  3. y increases from 1 to 9 Correct answer.
  4. y increases from 1 to 3

Explanation

At x = 1, y = 1; at x = 3, y = 9. On x > 0 the parabola is increasing, so y rises from 1 to 9.

Question 38Linear Equations and Functions

The graph of y = ax + b is shifted down by 3. The new function is y = 2x − 5. What was the original b?

  1. 2
  2. −5
  3. −2 Correct answer.
  4. −8

Explanation

Shifting down by 3 subtracts 3 from the intercept, so b − 3 = −5, giving b = −2.

Question 39Linear Equations and Functions

A quantity is directly proportional to x with constant 4, so y = 4x. When y = 20, what is x?

  1. 16
  2. 5 Correct answer.
  3. 80
  4. 4

Explanation

From y = 4x, set 4x = 20, so x = 5. Direct proportion gives a straight line through the origin.

Question 40Quadratic Functions and Equations

Find the x-intercepts of y = x² − 4.

  1. no x-intercepts
  2. x = 2 and x = −2 Correct answer.
  3. x = 0
  4. x = 4 and x = −4

Explanation

Set y = 0: x² − 4 = 0 ⇒ x² = 4 ⇒ x = ±2. The parabola crosses the x-axis at (2, 0) and (−2, 0).

Question 41Quadratic Functions and Equations

A function is given as y = 2x². Compared with y = x², how is the graph changed?

  1. Shifted up 2
  2. Narrower (steeper) Correct answer.
  3. Wider (shallower)
  4. Shifted right 2

Explanation

Multiplying x² by 2 (a > 1) stretches the parabola vertically, making it narrower and steeper than y = x².

Question 42Linear Equations and Functions

Given the points (0, 1), (1, 3), (2, 5) lie on one line. Find the function.

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

Explanation

Slope = (3 − 1)/(1 − 0) = 2, and the y-intercept is 1 (from (0, 1)), so y = 2x + 1. Check x = 2: y = 5. ✓

Question 43Linear Equations and Functions

A car uses fuel at a rate proportional to distance: 4 L per 50 km. How much for 200 km?

  1. 50 L
  2. 8 L
  3. 32 L
  4. 16 L Correct answer.

Explanation

Rate = 4/50 L/km. For 200 km: (4/50) × 200 = 16 L. Equivalently 200 is 4×50, so 4 × 4 = 16.

Question 44Linear Equations and Functions

Which graph represents an inverse proportion y = k/x (k > 0)?

  1. A straight line through the origin
  2. A parabola opening upward
  3. A curve in the 1st and 3rd quadrants, never touching the axes Correct answer.
  4. A V-shape with vertex at the origin

Explanation

y = k/x (k > 0) is a rectangular hyperbola lying in the 1st and 3rd quadrants, asymptotic to both axes.

Question 45Linear Equations and Functions

The line y = mx + b has m > 0 and b < 0. Which quadrant does it NOT pass through?

  1. Quadrant II Correct answer.
  2. Quadrant IV
  3. Quadrant III
  4. Quadrant I

Explanation

Positive slope rising, with a negative y-intercept: the line crosses below the origin on the y-axis and rises to the right, so it passes through I, III, IV but not II.

Question 46Quadratic Functions and Equations

For y = ax², the point (2, 8) lies on the curve. Find a.

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

Explanation

Substitute: 8 = a(2)² = 4a, so a = 2. The function is y = 2x².

Question 47Quadratic Functions and Equations

Two functions f(x) = x² and g(x) = 2x are equal at which x values?

  1. x = 0 only
  2. x = 2 only
  3. x = 1 and x = 2
  4. x = 0 and x = 2 Correct answer.

Explanation

Set x² = 2x: x² − 2x = 0 ⇒ x(x − 2) = 0, so x = 0 or x = 2. At both, x² = 2x.

Question 48Linear Equations and Functions

If y is proportional to x and y = 15 when x = 5, what is the constant of proportionality?

  1. 3 Correct answer.
  2. 5
  3. 75
  4. 1/3

Explanation

y = kx, so k = y/x = 15/5 = 3. The relation is y = 3x.

Question 49Quadratic Functions and Equations

The parabola y = −x² opens in which direction?

  1. To the right
  2. Downward (maximum at vertex) Correct answer.
  3. Upward (minimum at vertex)
  4. To the left

Explanation

A negative leading coefficient (a = −1 < 0) makes the parabola open downward, with a maximum at its vertex (0, 0).

Question 50Linear Equations and Functions

A point moves so that y = 6/x. As x doubles from 1 to 2, how does y change?

  1. y becomes 4
  2. y doubles from 6 to 12
  3. y stays 6
  4. y halves from 6 to 3 Correct answer.

Explanation

Inverse proportion: at x = 1, y = 6; at x = 2, y = 3. Doubling x halves y, the hallmark of inverse proportion.