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Ishikawa Entrance Mathematics study materials
Ishikawa Entrance Mathematics study resources

Ishikawa Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Compute (−3) × 4 − (−2).

  1. −10 Correct answer.
  2. −14
  3. 10
  4. −16

Explanation

(−3) × 4 = −12, then −12 − (−2) = −12 + 2 = −10.

Question 2Numbers and Algebraic Expressions

What is the prime factorization of 84?

  1. 2² × 3 × 7 Correct answer.
  2. 2 × 3² × 7
  3. 2 × 3 × 7
  4. 2² × 3² × 7

Explanation

84 = 4 × 21 = 2² × 3 × 7. Check: 4 × 21 = 84.

Question 3Numbers and Algebraic Expressions

Find the greatest common divisor of 48 and 72.

  1. 8
  2. 12
  3. 6
  4. 24 Correct answer.

Explanation

48 = 2⁴ × 3 and 72 = 2³ × 3²; GCD = 2³ × 3 = 24. Check: 48 ÷ 24 = 2, 72 ÷ 24 = 3.

Question 4Numbers and Algebraic Expressions

Find the least common multiple of 12 and 18.

  1. 72
  2. 6
  3. 216
  4. 36 Correct answer.

Explanation

12 = 2² × 3, 18 = 2 × 3²; LCM = 2² × 3² = 36.

Question 5Numbers and Algebraic Expressions

Simplify 12a³b² ÷ 4ab.

  1. 3a²b²
  2. 3a²b Correct answer.
  3. 8a²b
  4. 3ab

Explanation

Divide the coefficients and subtract the repeated factors separately: 12 ÷ 4 = 3, a³ ÷ a = a², and b² ÷ b = b. The quotient is therefore 3a²b.

Question 6Numbers and Algebraic Expressions

A distance measured as 25,000 m is to be written to two significant figures. Which form is correct?

  1. 2.5 × 10³ m
  2. 2.5 × 10⁴ m Correct answer.
  3. 25 × 10³ m
  4. 2.50 × 10⁴ m

Explanation

In the significant-figure form the first factor is written as a number from 1 up to (but not including) 10, multiplied by a power of 10. Two significant figures give 2.5, and 25,000 = 2.5 × 10⁴, so the answer is 2.5 × 10⁴ m.

Question 7Numbers and Algebraic Expressions

Expand (x + 3)(x − 5).

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

Explanation

x·x = x², x·(−5) + 3·x = −2x, 3·(−5) = −15; sum = x² − 2x − 15.

Question 8Numbers and Algebraic Expressions

Factor x² − 7x + 12.

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

Explanation

Need two numbers whose product is 12 and sum −7: −3 and −4, so (x − 3)(x − 4).

Question 9Numbers and Algebraic Expressions

Compute √18 + √50 − √8.

  1. 4√2
  2. 5√2
  3. 6√2 Correct answer.
  4. 7√2

Explanation

√18 = 3√2, √50 = 5√2, √8 = 2√2; 3√2 + 5√2 − 2√2 = (3 + 5 − 2)√2 = 6√2.

Question 10Numbers and Algebraic Expressions

Compute 3√2 × 2√6.

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

Explanation

3 × 2 = 6 and √2 × √6 = √12 = 2√3, giving 6 × 2√3 = 12√3.

Question 11Numbers and Algebraic Expressions

Which expression is equivalent to (x − 2)² − (x + 1)(x − 1)?

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

Explanation

(x−2)² = x² − 4x + 4 and (x+1)(x−1) = x² − 1; difference = (x² − 4x + 4) − (x² − 1) = −4x + 5.

Question 12Numbers and Algebraic Expressions

Which is a rational number?

  1. π
  2. √3
  3. √2
  4. 0.333... (= 1/3) Correct answer.

Explanation

0.333... = 1/3 is a ratio of integers, hence rational. √2, π, and √3 are irrational.

Question 13Linear Equations and Functions

Solve 3x − 7 = 2x + 5.

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

Explanation

3x − 2x = 5 + 7, so x = 12. Check: 3(12)−7 = 29 = 2(12)+5 = 29.

Question 14Linear Equations and Functions

Solve 4(x − 2) = 2x + 6.

  1. x = 1
  2. x = 5
  3. x = 7 Correct answer.
  4. x = 14

Explanation

4x − 8 = 2x + 6 → 2x = 14 → x = 7. Check: 4(5) = 20 vs 2(7)+6 = 20.

Question 15Linear Equations and Functions

Solve the simultaneous equations x + y = 8, 2x − y = 7.

  1. x = 5, y = 3 Correct answer.
  2. x = 5, y = −3
  3. x = 1, y = 7
  4. x = 3, y = 5

Explanation

Adding the two equations eliminates y: 3x = 15, so x = 5; then y = 8 − 5 = 3. Check: 2(5)−3 = 7.

Question 16Linear Equations and Functions

Solve 2x + 3y = 1, 3x − y = 7.

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

Explanation

From the second equation y = 3x − 7; substitute: 2x + 3(3x − 7) = 1 → 11x − 21 = 1 → 11x = 22 → x = 2, y = −1. Check: 2(2)+3(−1) = 1 ✓, 3(2)−(−1) = 7 ✓.

Question 17Linear Equations and Functions

Solve x² − 5x + 6 = 0.

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

Explanation

Factor: (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Question 18Linear Equations and Functions

Solve x² − 7x + 10 = 0.

  1. x = −2 or x = −5
  2. x = 1 or x = 10
  3. x = 2 or x = 5 Correct answer.
  4. x = −1 or x = −10

Explanation

Two numbers with product 10 and sum 7 are 2 and 5: (x − 2)(x − 5) = 0, so x = 2 or x = 5.

Question 19Linear Equations and Functions

Solve x² = 49 (over the reals).

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

Explanation

Taking the square root of both sides of x² = 49 gives x = ±√49 = ±7, so the equation has the two real roots 7 and −7. A quadratic of the form x² = k with k > 0 always has two roots of opposite sign.

Question 20Linear Equations and Functions

Using the quadratic formula, solve x² − 4x − 5 = 0.

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

Explanation

a=1, b=−4, c=−5; D = 16 + 20 = 36, √D = 6; x = (4 ± 6)/2 → x = 5 or x = −1. (Factor check: (x−5)(x+1) = x²−4x−5 ✓.)

Question 21Linear Equations and Functions

How many real solutions does x² + 2x + 5 = 0 have?

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

Explanation

Discriminant D = b² − 4ac = 4 − 20 = −16 < 0, so there are 0 real solutions.

Question 22Linear Equations and Functions

Solve −3x + 6 ≤ 12 and note the key step.

  1. x ≤ −2
  2. x ≥ 2
  3. x ≤ 6
  4. x ≥ −2 (inequality flips when dividing by −3) Correct answer.

Explanation

−3x ≤ 6 → dividing by −3 flips the sign → x ≥ −2. The flip is the key step.

Question 23Linear Equations and Functions

Solve the system 3x + 2y = 16, x − y = 2.

  1. x = 4, y = 2 Correct answer.
  2. x = 4, y = 6
  3. x = 6, y = 4
  4. x = 2, y = 0

Explanation

From x − y = 2, x = y + 2; substitute: 3(y+2) + 2y = 16 → 5y + 6 = 16 → 5y = 10 → y = 2, x = 4. Check: 3(4)+2(2) = 16 ✓.

Question 24Linear Equations and Functions

A number is tripled and 4 is subtracted to give 17. What is the number?

  1. 13
  2. 7 Correct answer.
  3. 5
  4. 21

Explanation

Let the number be x: 3x − 4 = 17 → 3x = 21 → x = 7. Check: 3(7)−4 = 17.

Question 25Linear Equations and Functions

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

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

Explanation

Let the numbers be a (larger) and b: a + b = 20, a − b = 4. Adding: 2a = 24 → a = 12. (Then b = 8.)

Question 26Linear Equations and Functions

A father is 4 times as old as his son; in 20 years he will be twice as old. How old is the son now?

  1. 10 Correct answer.
  2. 5
  3. 20
  4. 8

Explanation

Let son = s, father = 4s. In 20 years: 4s + 20 = 2(s + 20) → 4s + 20 = 2s + 40 → 2s = 20 → s = 10. Check: father 40, in 20 years 60 vs son 30 = 2× ✓.

Question 27Linear Equations and Functions

Pencils cost ¥80 each and erasers ¥50 each. If 10 items cost ¥680, how many pencils were bought?

  1. 6 Correct answer.
  2. 8
  3. 5
  4. 4

Explanation

Let p = pencils, e = erasers; p + e = 10 and 80p + 50e = 680. From e = 10 − p: 80p + 50(10−p) = 680 → 30p + 500 = 680 → 30p = 180 → p = 6. Check: 80(6)+50(4) = 680 ✓.

Question 28Linear Equations and Functions

Solve x² + 6x + 9 = 0.

  1. x = 3
  2. x = −3 (a repeated root) Correct answer.
  3. no real root
  4. x = 3 or x = −3

Explanation

x² + 6x + 9 = (x + 3)² = 0, so x = −3 is a double (repeated) root. Discriminant = 0 confirms one distinct root.

Question 29Linear Equations and Functions

Solve the simultaneous equations 5x + 2y = 19, 3x − 2y = 5.

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

Explanation

The y-terms are +2y and −2y, so adding the two equations eliminates y: 8x = 24, giving x = 3. Substituting into the first equation gives 15 + 2y = 19, so 2y = 4 and y = 2. Check in the second equation: 3(3) − 2(2) = 5 ✓.

Question 30Linear Equations and Functions

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

  1. 20
  2. 12
  3. 24 Correct answer.
  4. 40

Explanation

y = kx; 15 = 5k → k = 3; so y = 3·8 = 24.

Question 31Linear Equations and Functions

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

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

Explanation

y = k/x; 6 = k/4 → k = 24; so y = 24/8 = 3.

Question 32Linear Equations and Functions

A linear function passes through (0, −3) and has slope 2. What is its equation?

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

Explanation

Slope-intercept form y = mx + b with m = 2 and y-intercept b = −3 gives y = 2x − 3.

Question 33Linear Equations and Functions

Find the y-intercept of y = 3x − 7.

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

Explanation

In y = mx + b, the y-intercept is b = −7 (the point (0, −7)).

Question 34Linear Equations and Functions

Find the x-intercept of y = 2x − 6.

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

Explanation

Set y = 0: 2x − 6 = 0 → 2x = 6 → x = 3.

Question 35Linear Equations and Functions

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

  1. (3, 4)
  2. (6, 13)
  3. (2, 5) Correct answer.
  4. (2, 3)

Explanation

Set equal: 2x + 1 = −x + 7 → 3x = 6 → x = 2; y = 2(2)+1 = 5. Check: −2+7 = 5 ✓.

Question 36Linear Equations and Functions

For y = 3x, when x increases by 1, y increases by:

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

Explanation

The rate of change of a linear function y = mx + b is the slope m = 3, so a unit increase in x raises y by 3.

Question 37Quadratic Functions and Equations

The graph of y = ax² is a parabola. What is its axis of symmetry?

  1. the line y = x
  2. no axis of symmetry
  3. the y-axis (x = 0) Correct answer.
  4. the x-axis (y = 0)

Explanation

For y = ax² the vertex is at the origin and the parabola opens up or down; reflecting across the y-axis leaves it unchanged, so the axis of symmetry is x = 0 (the y-axis).

Question 38Quadratic Functions and Equations

How does the graph of y = −x² compare to y = x²?

  1. It is shifted up by 1
  2. It is shifted right
  3. It is wider
  4. It is reflected across the x-axis (opens downward) Correct answer.

Explanation

The negative leading coefficient flips the parabola: y = x² opens upward, y = −x² opens downward (reflection across the x-axis).

Question 39Quadratic Functions and Equations

For y = x², between x = 1 and x = 2 the rate of change (average) is:

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

Explanation

Average rate of change = (y(2) − y(1))/(2 − 1) = (4 − 1)/1 = 3.

Question 40Quadratic Functions and Equations

Which is the vertex of y = −2x²?

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

Explanation

y = ax² (no linear or constant term) has its vertex at the origin (0, 0). Here a = −2 so it opens downward but the vertex is still (0, 0).

Question 41Quadratic Functions and Equations

Find the coordinates where y = x² meets y = 4.

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

Explanation

Set x² = 4 → x = ±2, with y = 4 in both cases, giving (−2, 4) and (2, 4).

Question 42Linear Equations and Functions

A point (3, k) lies on y = 2x − 5. Find k.

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

Explanation

Substitute x = 3: k = 2(3) − 5 = 6 − 5 = 1.

Question 43Linear Equations and Functions

Which function is inversely proportional?

  1. y = 6x
  2. y = 6/x Correct answer.
  3. y = x²
  4. y = x + 6

Explanation

An inverse proportion has the form y = k/x; y = 6/x fits, with the product xy = 6 constant.

Question 44Linear Equations and Functions

A car uses fuel at a constant rate; distance d km on f liters gives d = 12f. How many liters for 60 km?

  1. 5 Correct answer.
  2. 7
  3. 12
  4. 720

Explanation

60 = 12f → f = 60/12 = 5 liters. (This is a direct proportion with rate 12 km/L.)

Question 45Linear Equations and Functions

The inverse proportion y = k/x has its graph as a:

  1. circle
  2. parabola
  3. straight line through the origin
  4. hyperbola (two branches) Correct answer.

Explanation

y = k/x graphs as a hyperbola with two branches in opposite quadrants (e.g. for k > 0, quadrants I and III), approaching both axes asymptotically.

Question 46Quadratic Functions and Equations

For y = x², by how much does y increase as x goes from 2 to 3?

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

Explanation

y(2) = 4, y(3) = 9, so the increase is 9 − 4 = 5.

Question 47Geometry: Plane Figures and Angles

In a triangle, two interior angles are 50° and 70°. What is the third angle?

  1. 120°
  2. 70°
  3. 50°
  4. 60° Correct answer.

Explanation

The angles of a triangle sum to 180°, so the third is 180 − 50 − 70 = 60°.

Question 48Geometry: Plane Figures and Angles

In a right triangle the legs are 6 and 8. What is the hypotenuse?

  1. 14
  2. 12
  3. 10 Correct answer.
  4. 100

Explanation

By the Pythagorean theorem, c² = 6² + 8² = 36 + 64 = 100, so c = 10 (a 6-8-10 Pythagorean triple).

Question 49Geometry: Plane Figures and Angles

A right triangle has hypotenuse 13 and one leg 5. What is the other leg?

  1. 12 Correct answer.
  2. √194
  3. 18
  4. 8

Explanation

a² + 5² = 13² → a² = 169 − 25 = 144 → a = 12 (a 5-12-13 triple).

Question 50Geometry: Plane Figures and Angles

Two triangles satisfy SAS with the included angle equal. They are:

  1. similar but not congruent
  2. only equal in area
  3. congruent Correct answer.
  4. neither

Explanation

By the SAS congruence condition, two sides and the included angle being equal makes the triangles congruent (identical in size and shape).