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Chiba Entrance Mathematics study materials
Chiba Entrance Mathematics study resources

Chiba Entrance Mathematics practice questions

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Question 1Numbers and Algebraic Expressions

Calculate: (−9) + (+2).

  1. 7
  2. −11
  3. −7 Correct answer.
  4. −18

Explanation

Adding numbers with different signs means subtracting the smaller absolute value from the larger and keeping the sign of the larger. Since |−9| = 9 > |+2| = 2, the result is −(9 − 2) = −7.

Question 2Numbers and Algebraic Expressions

Calculate: (−8) × (−7).

  1. −56
  2. 15
  3. −15
  4. 56 Correct answer.

Explanation

The product of two numbers with the same sign is positive. Multiply the absolute values: 8 × 7 = 56, and since both factors are negative, the result is +56.

Question 3Numbers and Algebraic Expressions

Calculate: (−54) ÷ (+9).

  1. 6
  2. −6 Correct answer.
  3. −7
  4. −45

Explanation

The quotient of two numbers with different signs is negative. Divide the absolute values: 54 ÷ 9 = 6, and the result is −6.

Question 4Numbers and Algebraic Expressions

Calculate: −(−4)^2.

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

Explanation

Evaluate the power inside the parentheses first: (−4)^2 = (−4) × (−4) = 16. Then apply the leading minus sign: −16.

Question 5Numbers and Algebraic Expressions

Simplify: 7x − 2x − 4x + 3x.

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

Explanation

Combine like terms by adding the coefficients in order: 7 − 2 − 4 + 3 = 4. So the expression simplifies to 4x.

Question 6Numbers and Algebraic Expressions

Find the value of 3a + 7 when a = −4.

  1. −19
  2. 5
  3. −5 Correct answer.
  4. 19

Explanation

Substitute a = −4: 3 × (−4) + 7 = −12 + 7 = −5. Multiply first, then add, and note that −12 + 7 is negative because 12 has the larger absolute value.

Question 7Numbers and Algebraic Expressions

Solve the equation: x + 6 = −4.

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

Explanation

Subtract 6 from both sides: x = −4 − 6 = −10. Check: −10 + 6 = −4.

Question 8Numbers and Algebraic Expressions

Calculate: (−4) × 5 + 12.

  1. −32
  2. 32
  3. 8
  4. −8 Correct answer.

Explanation

Multiplication comes before addition: (−4) × 5 = −20, then −20 + 12 = −8. The result is negative because 20 has the larger absolute value.

Question 9Numbers and Algebraic Expressions

Solve the equation: 5x − 3 = 3x + 7.

  1. −5
  2. 2
  3. 10
  4. 5 Correct answer.

Explanation

Subtract 3x from both sides: 2x − 3 = 7. Add 3: 2x = 10, so x = 5. Check: 5(5) − 3 = 22 and 3(5) + 7 = 22.

Question 10Numbers and Algebraic Expressions

Calculate: (−2)^3 + 5 × (−3).

  1. 7
  2. −21
  3. −23 Correct answer.
  4. 23

Explanation

Evaluate the power and the product first: (−2)^3 = −8 (an odd power of a negative number is negative) and 5 × (−3) = −15. Then −8 + (−15) = −23.

Question 11Numbers and Algebraic Expressions

Simplify: (3a − 5) − (4a − 1).

  1. −a − 6
  2. a − 4
  3. −a − 4 Correct answer.
  4. 7a − 6

Explanation

Distribute the minus sign over the second parentheses: 3a − 5 − 4a + 1. Combine like terms: (3a − 4a) + (−5 + 1) = −a − 4.

Question 12Numbers and Algebraic Expressions

Simplify: 15ab^2 ÷ 5b.

  1. 3ab Correct answer.
  2. 10ab
  3. 3b
  4. 3ab^2

Explanation

Divide the coefficients: 15 ÷ 5 = 3. For the variables, a stays and b^2 ÷ b = b. The result is 3ab.

Question 13Numbers and Algebraic Expressions

Expand: (x + 2)(x + 7).

  1. x^2 + 9x + 9
  2. x^2 + 14x + 9
  3. x^2 + 5x + 14
  4. x^2 + 9x + 14 Correct answer.

Explanation

Multiply each pair of terms: x·x = x^2, x·7 = 7x, 2·x = 2x, and 2·7 = 14. Combining the middle terms gives 7x + 2x = 9x, so the answer is x^2 + 9x + 14.

Question 14Numbers and Algebraic Expressions

Factor: x^2 + 2x − 15.

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

Explanation

Find two numbers whose product is −15 and whose sum is +2. Those numbers are +5 and −3, so x^2 + 2x − 15 = (x + 5)(x − 3). Check by expanding: x^2 − 3x + 5x − 15 = x^2 + 2x − 15.

Question 15Numbers and Algebraic Expressions

Calculate: √8 × √6.

  1. 2√6
  2. 4√3 Correct answer.
  3. √14
  4. 48

Explanation

Use the rule √a × √b = √(a × b): √8 × √6 = √48. Since 48 = 16 × 3, we get √48 = 4√3.

Question 16Numbers and Algebraic Expressions

Simplify: √27 + √12.

  1. 3√5
  2. √39
  3. 6√3
  4. 5√3 Correct answer.

Explanation

Rewrite each radical in simplest form: √27 = 3√3 and √12 = 2√3. Then 3√3 + 2√3 = 5√3. Like radicals combine like like terms.

Question 17Numbers and Algebraic Expressions

Solve the system of equations: 2x + y = 8 and x − y = 1. What is the value of y?

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

Explanation

Add the two equations to eliminate y: 3x = 9, so x = 3. Substitute into x − y = 1: 3 − y = 1, so y = 2. Both equations check: 2(3) + 2 = 8.

Question 18Numbers and Algebraic Expressions

Solve the equation: x^2 = 81.

  1. x = 9 and x = −9 Correct answer.
  2. x = −9 only
  3. no solution
  4. x = 9 only

Explanation

Taking the square root of both sides gives x = ±√81 = ±9. Both 9 and −9 satisfy the equation because squaring either gives 81.

Question 19Numbers and Algebraic Expressions

Solve the equation: x^2 − 8x + 15 = 0.

  1. x = −3, −5
  2. x = 1, 15
  3. x = 3, 5 Correct answer.
  4. x = 2, 6

Explanation

Factor the left side: find two numbers with product 15 and sum −8, which are −3 and −5. So (x − 3)(x − 5) = 0, giving x = 3 or x = 5.

Question 20Numbers and Algebraic Expressions

Simplify: 10/√5 + √20.

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

Explanation

Rationalize the first term: 10/√5 = 10√5/5 = 2√5. Simplify the second: √20 = 2√5. Then 2√5 + 2√5 = 4√5.

Question 21Numbers and Algebraic Expressions

Solve the equation: x^2 + 6x + 1 = 0.

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

Explanation

This quadratic does not factor with integers, so use the quadratic formula: x = (−6 ± √(36 − 4))/2 = (−6 ± √32)/2 = (−6 ± 4√2)/2 = −3 ± 2√2.

Question 22Numbers and Algebraic Expressions

Expand: (3x + 2)^2.

  1. 9x^2 + 4
  2. 9x^2 + 6x + 4
  3. 3x^2 + 12x + 4
  4. 9x^2 + 12x + 4 Correct answer.

Explanation

Use the formula (a + b)^2 = a^2 + 2ab + b^2 with a = 3x and b = 2: (3x)^2 = 9x^2, 2·(3x)·2 = 12x, and 2^2 = 4. The result is 9x^2 + 12x + 4.

Question 23Numbers and Algebraic Expressions

Factor completely: 2x^2 − 18.

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

Explanation

First factor out the common factor 2: 2(x^2 − 9). Then use the difference of squares: x^2 − 9 = (x + 3)(x − 3). The complete factorization is 2(x + 3)(x − 3).

Question 24Numbers and Algebraic Expressions

Given x + y = 6 and xy = 4, find the value of x^2 + y^2.

  1. 36
  2. 20
  3. 32
  4. 28 Correct answer.

Explanation

Use the identity x^2 + y^2 = (x + y)^2 − 2xy. Substituting gives 6^2 − 2(4) = 36 − 8 = 28.

Question 25Numbers and Algebraic Expressions

Solve the equation: (x − 2)(x + 4) = 7.

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

Explanation

Expand the left side: x^2 + 2x − 8 = 7. Move 7 to the left: x^2 + 2x − 15 = 0. Factor: (x + 5)(x − 3) = 0, so x = −5 or x = 3.

Question 26Numbers and Algebraic Expressions

Expand: (x + 5)(x − 5).

  1. x^2 − 25 Correct answer.
  2. x^2 − 10x − 25
  3. x^2 + 10x + 25
  4. x^2 + 25

Explanation

Use the difference-of-squares formula (a + b)(a − b) = a^2 − b^2. Here the middle terms cancel: 5x − 5x = 0, leaving x^2 − 25.

Question 27Numbers and Algebraic Expressions

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

  1. 5x + 11
  2. 5x + 7
  3. 5x + 9
  4. 7x + 11 Correct answer.

Explanation

Expand each part: (x + 3)^2 = x^2 + 6x + 9 and (x − 2)(x + 1) = x^2 − x − 2. Subtracting the second product flips every one of its terms: x^2 + 6x + 9 − x^2 + x + 2. The x^2 terms cancel and the middle terms add, 6x + x = 7x, leaving 7x + 11.

Question 28Numbers and Algebraic Expressions

Factor: x^2 − 6x + 9.

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

Explanation

Find two numbers with product 9 and sum −6: both are −3. So x^2 − 6x + 9 = (x − 3)(x − 3) = (x − 3)^2, a perfect-square trinomial.

Question 29Numbers and Algebraic Expressions

Calculate: √2(√8 + √18).

  1. 10 Correct answer.
  2. 4 + 3√2
  3. √52
  4. 10√2

Explanation

Simplify inside the parentheses first: √8 = 2√2 and √18 = 3√2, so the sum is 5√2. Then √2 × 5√2 = 5 × 2 = 10.

Question 30Numbers and Algebraic Expressions

Solve the equation: 2x^2 − 5x + 2 = 0.

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

Explanation

Factor by grouping or inspection: (2x − 1)(x − 2) = 2x^2 − 4x − x + 2 = 2x^2 − 5x + 2. Setting each factor to 0 gives x = 1/2 or x = 2. Check with the quadratic formula: x = (5 ± √(25 − 16))/4 = (5 ± 3)/4.

Question 31Linear Equations and Functions

y is directly proportional to x, and y = 5x. Find the value of y when x = 6.

  1. 30 Correct answer.
  2. 56
  3. 125
  4. 11

Explanation

Direct proportion means y = ax, so substitute x = 6: y = 5 × 6 = 30.

Question 32Linear Equations and Functions

y is inversely proportional to x, and y = 18/x. Find the value of y when x = 2.

  1. 9 Correct answer.
  2. 20
  3. 36
  4. 16

Explanation

Inverse proportion means y = a/x with a = 18. Substitute x = 2: y = 18 ÷ 2 = 9.

Question 33Linear Equations and Functions

What is the slope of the line y = −3x + 7?

  1. −3 Correct answer.
  2. 7
  3. 3
  4. −10

Explanation

A linear function written as y = ax + b has slope a and y-intercept b. Here a = −3, so the slope is −3.

Question 34Linear Equations and Functions

What is the y-intercept of the line y = 4x − 9?

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

Explanation

In y = ax + b, the y-intercept is b, the value of y when x = 0. Here b = −9, so the line crosses the y-axis at (0, −9).

Question 35Quadratic Functions and Equations

The graph of y = ax^2 passes through the point (3, 18). Find the value of a.

  1. 2 Correct answer.
  2. 9
  3. 6
  4. 54

Explanation

Substitute x = 3 and y = 18: 18 = a × 3^2 = 9a. Dividing both sides by 9 gives a = 2.

Question 36Linear Equations and Functions

Which of the following functions has a y-value that always decreases as x increases?

  1. y = 30/x
  2. y = −6x Correct answer.
  3. y = x^2
  4. y = 6x

Explanation

A linear function y = ax decreases for all x when a is negative. Since −6 < 0, y = −6x always decreases as x increases.

Question 37Linear Equations and Functions

Find the equation of the line with slope 3 that passes through the point (0, −4).

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

Explanation

A line through (0, −4) has y-intercept −4, so b = −4. With slope 3, the equation is y = 3x − 4.

Question 38Linear Equations and Functions

Find the equation of the line passing through the points (1, 3) and (3, 11).

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

Explanation

The slope is (11 − 3)/(3 − 1) = 8/2 = 4. Substituting (1, 3) into y = 4x + b gives 3 = 4 + b, so b = −1. The equation is y = 4x − 1. Check with (3, 11): 4(3) − 1 = 11.

Question 39Quadratic Functions and Equations

For the function y = 2x^2, find the rate of change (average rate of change) as x increases from 2 to 5.

  1. 7
  2. 42
  3. 14 Correct answer.
  4. 21

Explanation

When x = 2, y = 8; when x = 5, y = 50. The rate of change is (50 − 8)/(5 − 2) = 42/3 = 14.

Question 40Linear Equations and Functions

Find the x-intercept of the line y = 3x − 12.

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

Explanation

The x-intercept occurs where y = 0: 0 = 3x − 12, so 3x = 12 and x = 4. The line crosses the x-axis at (4, 0).

Question 41Linear Equations and Functions

y is inversely proportional to x. When x = 3, y = 10. Find the value of y when x = 5.

  1. 2
  2. 50
  3. 15
  4. 6 Correct answer.

Explanation

For inverse proportion, xy is constant: 3 × 10 = 30. When x = 5, y = 30 ÷ 5 = 6.

Question 42Linear Equations and Functions

Find the coordinates of the intersection point of the lines y = 2x − 1 and y = −x + 8.

  1. (4, 7)
  2. (3, 5) Correct answer.
  3. (3, 9)
  4. (5, 3)

Explanation

Set the expressions equal: 2x − 1 = −x + 8, so 3x = 9 and x = 3. Substituting back: y = 2(3) − 1 = 5. The intersection is (3, 5), and it also satisfies the second line: −3 + 8 = 5.

Question 43Quadratic Functions and Equations

For the function y = x^2 with domain −2 ≤ x ≤ 3, find the range of y.

  1. 4 ≤ y ≤ 9
  2. 0 ≤ y ≤ 3
  3. 0 ≤ y ≤ 9 Correct answer.
  4. −2 ≤ y ≤ 9

Explanation

Because the interval includes x = 0, the minimum value of y is 0^2 = 0. The maximum occurs at the endpoint farthest from 0: 3^2 = 9. So the range is 0 ≤ y ≤ 9.

Question 44Linear Equations and Functions

A water tank holds 80 liters. Water drains out at a constant rate of 4 liters per minute. How many liters remain in the tank after 12 minutes?

  1. 128 L
  2. 20 L
  3. 32 L Correct answer.
  4. 48 L

Explanation

The amount drained in 12 minutes is 4 × 12 = 48 L. Subtracting from the initial amount: 80 − 48 = 32 L remain. This is the linear function y = 80 − 4x evaluated at x = 12.

Question 45Quadratic Functions and Equations

For the function y = (1/4)x^2, find the rate of change as x increases from 2 to 6.

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

Explanation

When x = 2, y = 1; when x = 6, y = 9. The rate of change is (9 − 1)/(6 − 2) = 8/4 = 2.

Question 46Quadratic Functions and Equations

The parabola y = 2x^2 and the line y = 4x intersect at two points. What is the x-coordinate of the intersection point with the larger x-coordinate?

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

Explanation

Set 2x^2 = 4x, giving 2x^2 − 4x = 0. Factor: 2x(x − 2) = 0, so x = 0 or x = 2. The larger x-coordinate is 2.

Question 47Quadratic Functions and Equations

A ball's height is modeled by y = −x^2 + 6x, where x is the time in seconds after launch and y is the height in meters. What is the maximum height of the ball?

  1. 18 m
  2. 36 m
  3. 9 m Correct answer.
  4. 3 m

Explanation

Complete the square: y = −(x^2 − 6x) = −(x − 3)^2 + 9. Since the parabola opens downward, the vertex (3, 9) is the maximum point, so the maximum height is 9 m.

Question 48Linear Equations and Functions

A rectangle has a perimeter of 30 cm. If one side has length x cm and the adjacent side has length y cm, which equation expresses y as a function of x?

  1. y = 15 − x Correct answer.
  2. y = x − 15
  3. y = 30 − x
  4. y = 30 − 2x

Explanation

The perimeter satisfies 2x + 2y = 30. Dividing by 2 gives x + y = 15, so y = 15 − x.

Question 49Linear Equations and Functions

Find the area of the triangle formed by the line y = −2x + 6, the x-axis, and the y-axis.

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

Explanation

The line meets the x-axis where 0 = −2x + 6, giving (3, 0), and meets the y-axis at (0, 6). The triangle has base 3 and height 6, so its area is (1/2) × 3 × 6 = 9.

Question 50Quadratic Functions and Equations

Point A lies on the parabola y = 2x^2 and has x-coordinate 2. Find the slope of the line through the origin O and point A.

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

Explanation

At x = 2, y = 2 × 2^2 = 8, so A = (2, 8). The slope of OA is rise over run: 8/2 = 4.