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Toyama Prefectural Public High School Entrance Examination - Mathematics study materials
Toyama Prefectural Public High School Entrance Examination - Mathematics study resources

Toyama Prefectural Public High School Entrance Examination - Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate 7 − 3 × 2 + 4.

  1. 5 Correct answer.
  2. 14
  3. 8
  4. 12

Explanation

Multiplication is performed before addition and subtraction, so 3 × 2 = 6. Then 7 − 6 + 4 = 5, working left to right.

Question 2Numbers and Algebraic Expressions

Calculate (−5) + (−8) − (−3).

  1. 0
  2. −16
  3. −10 Correct answer.
  4. −6

Explanation

Subtracting a negative is the same as adding a positive, so the expression becomes −5 − 8 + 3. Combining gives −13 + 3 = −10.

Question 3Numbers and Algebraic Expressions

Which expression is equal to 3(2a − 4) − 2(a − 5)?

  1. 8a − 2
  2. 4a − 2 Correct answer.
  3. 4a + 2
  4. 4a − 22

Explanation

Expanding gives 6a − 12 − 2a + 10. Collecting like terms gives 4a − 2.

Question 4Numbers and Algebraic Expressions

Factorise x² − 7x + 12.

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

Explanation

Two numbers whose product is 12 and whose sum is −7 are −3 and −4. Hence x² − 7x + 12 = (x − 3)(x − 4).

Question 5Numbers and Algebraic Expressions

Simplify (2√3 + √2)(√3 − √2).

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

Explanation

Expanding gives 2√3·√3 − 2√3·√2 + √2·√3 − √2·√2 = 6 − 2√6 + √6 − 2. This simplifies to 4 − √6.

Question 6Numbers and Algebraic Expressions

Write 0.00036 in standard form (scientific notation).

  1. 36 × 10⁻⁵
  2. 3.6 × 10⁻⁴ Correct answer.
  3. 3.6 × 10⁻³
  4. 3.6 × 10⁴

Explanation

Moving the decimal point 4 places to the right gives 3.6, so the exponent is −4. Hence 0.00036 = 3.6 × 10⁻⁴.

Question 7Numbers and Algebraic Expressions

Evaluate 2⁻³ × 2⁵ ÷ 2².

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

Explanation

Using the laws of indices, add the exponents for multiplication and subtract for division: −3 + 5 − 2 = 0. Since 2⁰ = 1, the value is 1.

Question 8Numbers and Algebraic Expressions

If a = −2 and b = 3, what is the value of a² − 2ab?

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

Explanation

Substituting gives (−2)² − 2(−2)(3) = 4 + 12 = 16. The product of two negatives in −2ab is positive.

Question 9Numbers and Algebraic Expressions

Solve the equation 5x − 3 = 2x + 12.

  1. x = 5 Correct answer.
  2. x = 15
  3. x = 3
  4. x = 9

Explanation

Subtract 2x from both sides to get 3x − 3 = 12, then add 3 to get 3x = 15. Dividing by 3 gives x = 5.

Question 10Numbers and Algebraic Expressions

Solve the simultaneous equations x + y = 7 and 2x − y = 2.

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

Explanation

Adding the two equations eliminates y: 3x = 9, so x = 3. Substituting into x + y = 7 gives y = 4.

Question 11Numbers and Algebraic Expressions

Expand (x + 3)².

  1. x² + 9
  2. x² + 6x + 9 Correct answer.
  3. x² + 3x + 9
  4. x² + 6x + 3

Explanation

(x + 3)² means (x + 3)(x + 3), which expands to x² + 3x + 3x + 9 = x² + 6x + 9.

Question 12Numbers and Algebraic Expressions

Which of the following is an irrational number?

  1. 22/7
  2. 0.25
  3. √16
  4. √7 Correct answer.

Explanation

A number is irrational if it cannot be written as a fraction of integers. Since 7 is not a perfect square, √7 is irrational.

Question 13Numbers and Algebraic Expressions

Simplify (6x²y) ÷ (2xy²) for x ≠ 0 and y ≠ 0.

  1. 3x²/y
  2. 3/y
  3. 3xy
  4. 3x/y Correct answer.

Explanation

Divide the coefficients: 6 ÷ 2 = 3. For the variables, x² ÷ x = x and y ÷ y² = 1/y, giving 3x/y.

Question 14Numbers and Algebraic Expressions

The expression 2x + 3 is multiplied by 4 and then 5 is subtracted from the result. Which expression represents this?

  1. 8x + 12
  2. 8x + 7 Correct answer.
  3. 8x − 2
  4. 8x + 17

Explanation

Multiplying gives 4(2x + 3) = 8x + 12, then subtracting 5 gives 8x + 7.

Question 15Numbers and Algebraic Expressions

Solve the inequality 4 − 3x > 13.

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

Explanation

Subtract 4 to get −3x > 9, then divide by −3. Dividing by a negative number reverses the inequality, giving x < −3.

Question 16Numbers and Algebraic Expressions

What is the value of √50 − √18 + √8?

  1. 4√2
  2. 3√2 Correct answer.
  3. √40
  4. 5√2

Explanation

Simplifying each surd gives 5√2 − 3√2 + 2√2. Collecting like terms gives 4√2, so the expression equals 4√2.

Question 17Numbers and Algebraic Expressions

Make x the subject of the formula y = (3x − 1)/2.

  1. x = 2y + 1
  2. x = (y + 1)/6
  3. x = (2y + 1)/3 Correct answer.
  4. x = (2y − 1)/3

Explanation

Multiply both sides by 2 to get 2y = 3x − 1, then add 1 to get 2y + 1 = 3x. Dividing by 3 gives x = (2y + 1)/3.

Question 18Numbers and Algebraic Expressions

If x² = 49 and x < 0, what is the value of x³?

  1. −49
  2. 343
  3. −343 Correct answer.
  4. 49

Explanation

From x² = 49, x is either 7 or −7; the condition x < 0 selects x = −7. Then x³ = (−7)³ = −343.

Question 19Numbers and Algebraic Expressions

The sum of three consecutive integers is 72. What is the largest of the three integers?

  1. 24
  2. 23
  3. 26
  4. 25 Correct answer.

Explanation

Let the middle integer be n; then (n − 1) + n + (n + 1) = 3n = 72, so n = 24. The largest is n + 1 = 25.

Question 20Numbers and Algebraic Expressions

Which expression is equivalent to (a − b)² − (a + b)²?

  1. 2a² + 2b²
  2. −4ab Correct answer.
  3. 0
  4. −2ab

Explanation

Expanding gives (a² − 2ab + b²) − (a² + 2ab + b²) = −4ab. The squared terms cancel.

Question 21Numbers and Algebraic Expressions

Express 1/√5 with a rational denominator.

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

Explanation

Multiply numerator and denominator by √5: (1 × √5)/(√5 × √5) = √5/5. The denominator is now rational.

Question 22Numbers and Algebraic Expressions

A number is increased by 20% and the result is then decreased by 20%. Compared with the original number, the final value is:

  1. 4% greater
  2. 2% smaller
  3. 4% smaller Correct answer.
  4. the same

Explanation

Multiplying by 1.2 and then by 0.8 gives 0.96, which is 96% of the original. So the final value is 4% smaller.

Question 23Numbers and Algebraic Expressions

Simplify (x² − 9)/(x + 3) for x ≠ −3.

  1. x + 3
  2. x − 3 Correct answer.
  3. x² − 3
  4. x − 9

Explanation

The numerator factorises as (x + 3)(x − 3). Cancelling the common factor (x + 3) leaves x − 3.

Question 24Numbers and Algebraic Expressions

If 2^x = 8 and 3^y = 81, what is the value of x + y?

  1. 11
  2. 4
  3. 7 Correct answer.
  4. 5

Explanation

Since 8 = 2³, x = 3; since 81 = 3⁴, y = 4. Therefore x + y = 7.

Question 25Numbers and Algebraic Expressions

The quadratic equation x² + kx + 9 = 0 has exactly one real solution. Which value of k is possible?

  1. −6 Correct answer.
  2. 9
  3. 0
  4. 3

Explanation

A quadratic has exactly one real solution when the discriminant b² − 4ac equals zero. Here k² − 36 = 0, so k = 6 or k = −6.

Question 26Numbers and Algebraic Expressions

What is the value of 3 − 2 × (−4)?

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

Explanation

Multiplication is performed before subtraction, so 2 × (−4) = −8. Then 3 − (−8) = 3 + 8 = 11.

Question 27Numbers and Algebraic Expressions

What is the value of (−2)³ + (−3)²?

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

Explanation

A negative number raised to an odd power is negative, so (−2)³ = −8. A negative number raised to an even power is positive, so (−3)² = 9, and −8 + 9 = 1.

Question 28Numbers and Algebraic Expressions

Which expression shows 0.00000052 written using a power of 10?

  1. 52 × 10⁻⁹
  2. 5.2 × 10⁷
  3. 5.2 × 10⁻⁶
  4. 5.2 × 10⁻⁷ Correct answer.

Explanation

Move the decimal point 7 places to the right to get 5.2, so the multiplier is 10⁻⁷. The exponent is negative because the original number is less than 1.

Question 29Numbers and Algebraic Expressions

What is the value of 5 − 2(3 − 7)?

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

Explanation

Evaluate the bracket first: 3 − 7 = −4. Then 5 − 2(−4) = 5 + 8 = 13.

Question 30Numbers and Algebraic Expressions

Simplify 3(2a − 5) − 2(a − 4).

  1. 4a − 23
  2. 8a − 7
  3. 4a − 7 Correct answer.
  4. 4a + 7

Explanation

Expanding gives 6a − 15 − 2a + 8. Collecting like terms gives 4a − 7.

Question 31Numbers and Algebraic Expressions

Expand and simplify (2x − 1)(x + 4).

  1. 2x² − 7x − 4
  2. 2x² + 9x − 4
  3. 2x² + 7x + 4
  4. 2x² + 7x − 4 Correct answer.

Explanation

Each term of the first bracket multiplies each term of the second: 2x² + 8x − x − 4. Collecting gives 2x² + 7x − 4.

Question 32Numbers and Algebraic Expressions

Factorise 6x² − 9x completely.

  1. x(6x − 9)
  2. 3x(2x − 9)
  3. 3(2x² − 3x)
  4. 3x(2x − 3) Correct answer.

Explanation

The highest common factor of 6x² and 9x is 3x. Taking it out gives 3x(2x − 3).

Question 33Numbers and Algebraic Expressions

Simplify (3x²y³) × (2x³y) ÷ (6xy²), where x and y are nonzero.

  1. x⁴y² Correct answer.
  2. 6x⁴y²
  3. x⁵y²
  4. x⁴y³

Explanation

The product of the first two factors is 6x⁵y⁴. Dividing by 6xy² gives x⁴y² because the coefficients cancel and the exponents subtract.

Question 34Numbers and Algebraic Expressions

Solve the inequality 7 − 2x ≥ 1.

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

Explanation

Subtract 7 from both sides to get −2x ≥ −6. Dividing by −2 reverses the inequality sign, giving x ≤ 3.

Question 35Numbers and Algebraic Expressions

A shop raises a price by 30%, then later reduces the new price by 30%. The final price compared with the original price is

  1. 30% lower
  2. 9% higher
  3. the same
  4. 9% lower Correct answer.

Explanation

Multiplying by 1.30 and then by 0.70 gives 0.91 of the original. So the final price is 9% lower.

Question 36Numbers and Algebraic Expressions

If 3^(x+1) = 81, what is the value of x?

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

Explanation

Since 81 = 3⁴, the exponents must be equal: x + 1 = 4. Hence x = 3.

Question 37Numbers and Algebraic Expressions

What is the value of (√7 + √3)(√7 − √3)?

  1. 2√21
  2. 4 Correct answer.
  3. 10
  4. 7

Explanation

This is a difference of squares: (√7)² − (√3)² = 7 − 3 = 4.

Question 38Numbers and Algebraic Expressions

Simplify (x² − 4x)/(x² − 16) for x ≠ 4 and x ≠ −4.

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

Explanation

Factorise: the numerator is x(x − 4) and the denominator is (x − 4)(x + 4). Cancelling the common factor x − 4 leaves x/(x + 4).

Question 39Numbers and Algebraic Expressions

For the equation x² + kx + 16 = 0 to have exactly one real solution, which value of k is possible?

  1. 16
  2. 8 Correct answer.
  3. 4
  4. 0

Explanation

Exactly one real solution requires the discriminant k² − 64 to be zero. So k² = 64, giving k = 8 or k = −8.

Question 40Numbers and Algebraic Expressions

If x + y = 5 and xy = 3, what is the value of x² + y²?

  1. 16
  2. 22
  3. 25
  4. 19 Correct answer.

Explanation

Using (x + y)² = x² + 2xy + y², we get x² + y² = 25 − 2 × 3 = 19.

Question 41Equations and Inequalities

Solve the equation 3x - 7 = 11.

  1. x = 18
  2. x = 4
  3. x = 3
  4. x = 6 Correct answer.

Explanation

Add 7 to both sides to get 3x = 18, then divide both sides by 3 to obtain x = 6.

Question 42Equations and Inequalities

Solve the inequality 2x + 5 > 13.

  1. x > 4 Correct answer.
  2. x > 9
  3. x > 8
  4. x < 4

Explanation

Subtract 5 from both sides to get 2x > 8, then divide by the positive number 2, which keeps the inequality direction, giving x > 4.

Question 43Equations and Inequalities

What is the solution of the equation 5(x - 2) = 3x + 4?

  1. x = 1
  2. x = 3
  3. x = 7 Correct answer.
  4. x = -7

Explanation

Expand the left side to get 5x - 10 = 3x + 4. Subtracting 3x and adding 10 to both sides gives 2x = 14, so x = 7.

Question 44Equations and Inequalities

Solve the inequality -3x ≤ 12.

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

Explanation

Dividing both sides by the negative number -3 reverses the inequality sign, giving x ≥ -4.

Question 45Equations and Inequalities

Which value of x satisfies the equation x² - 5x + 6 = 0?

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

Explanation

Factor the quadratic as (x - 2)(x - 3) = 0, so x = 2 or x = 3. Only x = 3 appears among the options.

Question 46Equations and Inequalities

Solve the system of equations: x + y = 7 and x - y = 3.

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

Explanation

Adding the two equations eliminates y and gives 2x = 10, so x = 5. Substituting into x + y = 7 gives y = 2.

Question 47Equations and Inequalities

Solve the equation 4x - 3 = 2x + 9.

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

Explanation

Subtract 2x from both sides to get 2x - 3 = 9, then add 3 to get 2x = 12, so x = 6.

Question 48Equations and Inequalities

How many solutions does the equation 2(x + 3) = 2x + 6 have?

  1. No solution
  2. Infinitely many solutions Correct answer.
  3. Exactly one solution
  4. Exactly two solutions

Explanation

Expanding the left side gives 2x + 6 = 2x + 6, which is true for every value of x, so there are infinitely many solutions.

Question 49Equations and Inequalities

Solve the inequality 3(x - 1) < 2x + 4.

  1. x > 7
  2. x < 5
  3. x < 7 Correct answer.
  4. x < 1

Explanation

Expanding gives 3x - 3 < 2x + 4. Subtracting 2x and adding 3 to both sides gives x < 7.

Question 50Equations and Inequalities

The equation x² + kx + 9 = 0 has exactly one real solution. What is a possible value of the positive constant k?

  1. k = 3
  2. k = 6 Correct answer.
  3. k = 18
  4. k = 9

Explanation

A quadratic has exactly one real solution when its discriminant b² - 4ac equals zero. Here k² - 36 = 0, so k = 6 or k = -6; the positive value is 6.