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Ibaraki Entrance Math study materials
Ibaraki Entrance Math study resources

Ibaraki Entrance Math practice questions

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Question 1Numbers and Algebraic Expressions

Calculate the value of (-7) + (+3) - (-4).

  1. 8
  2. -14
  3. -8
  4. 0 Correct answer.

Explanation

Subtracting a negative becomes addition: -(-4) = +4. So the expression is -7 + 3 + 4. Working left to right: -7 + 3 = -4, then -4 + 4 = 0.

Question 2Numbers and Algebraic Expressions

Evaluate (-2) × (+5) ÷ (-4).

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

Explanation

Multiply first: (-2)×(+5) = -10. Then divide by -4: -10 ÷ (-4) = 10/4 = 5/2. Negative divided by negative is positive.

Question 3Numbers and Algebraic Expressions

Compute (-3)^2 + (-2)^3.

  1. 17
  2. 1 Correct answer.
  3. -1
  4. 15

Explanation

(-3)^2 = 9 (even power gives positive). (-2)^3 = -8 (odd power keeps the sign). So 9 + (-8) = 1.

Question 4Numbers and Algebraic Expressions

Simplify 2/3 - 5/6 + 1/4 and express as a single reduced fraction.

  1. 1/12 Correct answer.
  2. 1/4
  3. -1/12
  4. 5/12

Explanation

Common denominator 12: 2/3 = 8/12, 5/6 = 10/12, 1/4 = 3/12. So 8/12 - 10/12 + 3/12 = (8-10+3)/12 = 1/12.

Question 5Numbers and Algebraic Expressions

Compute 3/4 ÷ 6/5 × 2/3.

  1. 5/12 Correct answer.
  2. 5/9
  3. 15/24
  4. 12/15

Explanation

Division becomes multiplication by reciprocal: 3/4 × 5/6 = 15/24 = 5/8. Then × 2/3 = 10/24 = 5/12. Working left to right with division and multiplication at equal precedence.

Question 6Numbers and Algebraic Expressions

When x = -2 and y = 3, evaluate 3x^2 - 2xy + 1.

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

Explanation

Substitute: 3(-2)^2 - 2(-2)(3) + 1 = 3(4) - 2(-6) + 1 = 12 + 12 + 1 = 25.

Question 7Numbers and Algebraic Expressions

Expand (x + 4)(x - 5).

  1. x^2 + x - 20
  2. x^2 - 9x - 20
  3. x^2 - x - 20 Correct answer.
  4. x^2 - x + 20

Explanation

FOIL: x·x = x^2, x·(-5) = -5x, 4·x = 4x, 4·(-5) = -20. Combine middle: -5x + 4x = -x. So x^2 - x - 20.

Question 8Numbers and Algebraic Expressions

Factor x^2 - 7x + 12 completely.

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

Explanation

Need two numbers whose product is +12 and sum is -7. Both must be negative: -3 and -4 give product 12 and sum -7. So (x-3)(x-4).

Question 9Numbers and Algebraic Expressions

Factor 2x^2 - 5x - 3.

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

Explanation

Splitting the middle: 2x^2 - 5x - 3. Try (2x+1)(x-3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3. Matches.

Question 10Numbers and Algebraic Expressions

Simplify √72 + √50 - √32.

  1. 9√2
  2. 3√2
  3. 5√2
  4. 7√2 Correct answer.

Explanation

Factor: √72 = √(36·2) = 6√2. √50 = √(25·2) = 5√2. √32 = √(16·2) = 4√2. So 6√2 + 5√2 - 4√2 = 7√2.

Question 11Numbers and Algebraic Expressions

Rationalize and simplify (√3 + √2)(√3 - √2).

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

Explanation

Difference of squares: (a+b)(a-b) = a^2 - b^2. So (√3)^2 - (√2)^2 = 3 - 2 = 1.

Question 12Linear Equations and Functions

Solve the linear equation 3(x - 2) = 2x + 5 for x.

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

Explanation

Expand: 3x - 6 = 2x + 5. Subtract 2x: x - 6 = 5. Add 6: x = 11.

Question 13Linear Equations and Functions

Solve the simultaneous equations: x + y = 5 and 2x - y = 4.

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

Explanation

Add the equations: (x + y) + (2x - y) = 5 + 4 → 3x = 9, so x = 3. Substitute into eq1: 3 + y = 5, so y = 2. Check eq2: 2(3) - 2 = 4 ✓.

Question 14Linear Equations and Functions

Solve the simultaneous equations: 2x + 3y = 1 and 3x - 2y = 8.

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

Explanation

Multiply eq1 by 2: 4x + 6y = 2. Multiply eq2 by 3: 9x - 6y = 24. Add: 13x = 26, so x = 2. Substitute: 2(2) + 3y = 1 → 4 + 3y = 1 → 3y = -3 → y = -1. Check eq2: 3(2) - 2(-1) = 6+2 = 8 ✓.

Question 15Linear Equations and Functions

Solve x^2 - 5x + 6 = 0.

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

Explanation

Factor: (x-2)(x-3) = 0. So x-2 = 0 → x = 2, or x-3 = 0 → x = 3. Sum of roots = 5, product = 6, matching x^2 - 5x + 6.

Question 16Linear Equations and Functions

Solve x^2 - 4x - 1 = 0 using the quadratic formula.

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

Explanation

Use x = (4 ± √(16+4))/2 = (4 ± √20)/2 = (4 ± 2√5)/2 = 2 ± √5. The discriminant is 16 - 4(1)(-1) = 20.

Question 17Linear Equations and Functions

Solve the inequality 2(x - 3) < x + 1 and express the solution for x.

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

Explanation

Expand: 2x - 6 < x + 1. Subtract x: x - 6 < 1. Add 6: x < 7.

Question 18Linear Equations and Functions

Solve the inequality -3x + 5 ≤ 2x - 10 for x.

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

Explanation

Subtract 2x: -5x + 5 ≤ -10. Subtract 5: -5x ≤ -15. Divide by -5 (flip inequality): x ≥ 3.

Question 19Linear Equations and Functions

A number is doubled, then 5 is subtracted, giving 17. Find the original number.

  1. 6
  2. 8
  3. 11 Correct answer.
  4. 22

Explanation

Let the number be x. Equation: 2x - 5 = 17. Add 5: 2x = 22. Divide by 2: x = 11.

Question 20Linear Equations and Functions

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

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

Explanation

Let larger = x, smaller = y. x + y = 25, x - y = 7. Add: 2x = 32, x = 16. Then y = 9. Check: 16 + 9 = 25, 16 - 9 = 7 ✓.

Question 21Linear Equations and Functions

A shop sells pens at 120 yen each. If you buy a set of 5, the total is 500 yen. How much do you save per pen compared to buying individually?

  1. 40 yen
  2. 25 yen
  3. 20 yen Correct answer.
  4. 100 yen

Explanation

Individual price per pen = 120 yen. Set price per pen = 500 ÷ 5 = 100 yen. Saving = 120 - 100 = 20 yen per pen.

Question 22Linear Equations and Functions

If y is directly proportional to x and y = 12 when x = 4, find y when x = 7.

  1. 21 Correct answer.
  2. 16
  3. 14
  4. 28

Explanation

Direct proportion: y = kx. From y = 12 at x = 4: k = 12/4 = 3. So at x = 7: y = 3·7 = 21.

Question 23Linear Equations and Functions

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

  1. 2
  2. 4.5
  3. 3 Correct answer.
  4. 12

Explanation

Inverse proportion: xy = k. From y = 6 at x = 4: k = 4·6 = 24. At x = 8: y = 24/8 = 3.

Question 24Linear Equations and Functions

A line passes through points (1, 2) and (3, 8). Find the slope.

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

Explanation

Slope = (y2 - y1)/(x2 - x1) = (8 - 2)/(3 - 1) = 6/2 = 3.

Question 25Linear Equations and Functions

Find the y-intercept of the line y = 2x - 5.

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

Explanation

Slope-intercept form y = mx + b: slope m = 2, y-intercept b = -5. Setting x = 0 gives y = -5.

Question 26Linear Equations and Functions

Find the equation of the line passing through (2, -1) with slope -3.

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

Explanation

Point-slope: y - y1 = m(x - x1). y - (-1) = -3(x - 2) → y + 1 = -3x + 6 → y = -3x + 5.

Question 27Linear Equations and Functions

Two lines y = 2x + 3 and y = -x - 3 intersect at point P. Find the x-coordinate of P.

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

Explanation

Set equal: 2x + 3 = -x - 3. Add x: 3x + 3 = -3. Subtract 3: 3x = -6. Divide by 3: x = -2.

Question 28Linear Equations and Functions

For the linear function y = -2x + 7, find the rate of change of y with respect to x.

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

Explanation

In y = mx + b, the rate of change (slope) is m. Here m = -2, so y decreases by 2 per unit increase in x.

Question 29Quadratic Functions and Equations

For y = ax^2 with a = -2, find y when x = 3.

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

Explanation

Substitute: y = -2(3)^2 = -2·9 = -18. The negative coefficient makes the parabola open downward.

Question 30Quadratic Functions and Equations

For y = 3x^2, find the rate of change as x increases from 1 to 2.

  1. 3
  2. 9 Correct answer.
  3. 6
  4. 12

Explanation

At x = 1: y = 3(1)^2 = 3. At x = 2: y = 3(4) = 12. Rate of change = (12 - 3)/(2 - 1) = 9/1 = 9.

Question 31Quadratic Functions and Equations

For y = -x^2, find the average rate of change of y as x goes from -2 to 1.

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

Explanation

At x = -2: y = -(-2)^2 = -4. At x = 1: y = -(1)^2 = -1. Average rate = (-1 - (-4))/(1 - (-2)) = 3/3 = 1.

Question 32Linear Equations and Functions

A point P(3, -4) lies on a graph. What is the distance from P to the origin?

  1. 7
  2. 25
  3. 1
  4. 5 Correct answer.

Explanation

Distance formula from origin: √(x^2 + y^2) = √(9 + 16) = √25 = 5.

Question 33Linear Equations and Functions

A car travels at a constant speed. The distance y (km) from a starting point after x hours satisfies y = 60x. How far is the car from the start after 2.5 hours?

  1. 60 km
  2. 75 km
  3. 120 km
  4. 150 km Correct answer.

Explanation

Substitute x = 2.5: y = 60·2.5 = 150 km. The slope 60 km/h is the constant speed.

Question 34Linear Equations and Functions

Water is poured into a tank at a constant rate. After x minutes the volume y in liters is y = 8x. How many minutes does it take to fill a 200-liter tank?

  1. 20
  2. 12.5
  3. 32
  4. 25 Correct answer.

Explanation

Set y = 200: 8x = 200, so x = 25 minutes. The slope 8 L/min is the fill rate.

Question 35Linear Equations and Functions

Two lines y = ax + 1 and y = -x + 7 intersect at x = 2. Find the value of a.

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

Explanation

At the intersection both y-values are equal. Substitute x = 2 into the second line: y = -2 + 7 = 5. So in the first line, 5 = 2a + 1, giving 2a = 4 and a = 2.

Question 36Geometry: Plane Figures and Angles

Triangle ABC has AB = 5, BC = 12. If angle B is a right angle, find AC.

  1. 17
  2. 13 Correct answer.
  3. 7
  4. 60

Explanation

By Pythagorean theorem, AC^2 = AB^2 + BC^2 = 25 + 144 = 169, so AC = 13.

Question 37Geometry: Plane Figures and Angles

A ladder 10 m long leans against a wall with its foot 6 m from the wall. How high up the wall does the ladder reach?

  1. 10 m
  2. 8 m Correct answer.
  3. 4 m
  4. 6 m

Explanation

Wall height = √(10^2 - 6^2) = √(100 - 36) = √64 = 8 m. The ladder forms the hypotenuse.

Question 38Geometry: Plane Figures and Angles

Two parallel lines are cut by a transversal. If one interior angle on the same side is 110°, find the other interior angle on the same side.

  1. 140°
  2. 110°
  3. 70° Correct answer.
  4. 20°

Explanation

Same-side interior angles are supplementary (sum to 180°). So the other is 180° - 110° = 70°.

Question 39Geometry: Plane Figures and Angles

In triangle ABC, angle A = 60° and angle B = 80°. Find angle C.

  1. 20°
  2. 60°
  3. 100°
  4. 40° Correct answer.

Explanation

Angles of a triangle sum to 180°. So angle C = 180° - 60° - 80° = 40°.

Question 40Geometry: Plane Figures and Angles

An isosceles triangle has a vertex angle of 50°. Find each base angle.

  1. 50°
  2. 80°
  3. 65° Correct answer.
  4. 130°

Explanation

Base angles are equal; sum of angles is 180°. So each base angle = (180° - 50°)/2 = 65°.

Question 41Geometry: Plane Figures and Angles

In a parallelogram, consecutive angles have measures x and 3x. Find x.

  1. 30
  2. 90
  3. 60
  4. 45 Correct answer.

Explanation

Consecutive angles of a parallelogram are supplementary: x + 3x = 180°, so 4x = 180, x = 45.

Question 42Geometry: Congruence, Similarity and Measurement

Triangles ABC and DEF are similar with a similarity ratio of 2:3. If the area of ABC is 16 cm², find the area of DEF.

  1. 48 cm²
  2. 32 cm²
  3. 24 cm²
  4. 36 cm² Correct answer.

Explanation

For similar figures, area ratio is the square of the side ratio: (3/2)^2 = 9/4. So area of DEF = 16·(9/4) = 36 cm².

Question 43Geometry: Congruence, Similarity and Measurement

Triangles PQR and P'Q'R' are similar with linear ratio 1:4. If the perimeter of PQR is 12, find the perimeter of P'Q'R'.

  1. 36
  2. 3
  3. 24
  4. 48 Correct answer.

Explanation

Perimeters scale linearly with the similarity ratio. So perimeter of P'Q'R' = 12·(4/1) = 48.

Question 44Geometry: Congruence, Similarity and Measurement

In a proof, to show triangles ABC and DEF are congruent by SAS, which must hold?

  1. Two sides and the included angle of one equal the corresponding parts of the other Correct answer.
  2. Three sides of one equal three sides of the other
  3. The hypotenuse and a leg of one equal those of the other
  4. Two angles and a side of one equal the corresponding parts of the other

Explanation

SAS requires two pairs of sides and the included angle (the angle between those sides) to be equal in both triangles.

Question 45Geometry: Congruence, Similarity and Measurement

Two triangles are both right triangles and have one pair of equal acute angles. Which similarity condition applies?

  1. SAS similarity
  2. AA similarity Correct answer.
  3. HL similarity
  4. SSS similarity

Explanation

Both triangles are right (one 90° angle) and share an equal acute angle, so two pairs of angles are equal. By AA (angle-angle) similarity, the triangles are similar.

Question 46Geometry: Plane Figures and Angles

A central angle in a circle measures 80°. What is the measure of the inscribed angle subtending the same arc?

  1. 20°
  2. 40° Correct answer.
  3. 160°
  4. 80°

Explanation

The inscribed angle is half the central angle subtending the same arc. So it is 80°/2 = 40°.

Question 47Geometry: Plane Figures and Angles

A tangent line touches a circle at point P. The radius OP is perpendicular to the tangent. If OP = 5 and the tangent segment from an external point Q to P has length 12, find OQ.

  1. 17
  2. 7
  3. 13 Correct answer.
  4. 60

Explanation

Triangle OPQ is right at P (tangent ⟂ radius). So OQ^2 = OP^2 + PQ^2 = 25 + 144 = 169, OQ = 13.

Question 48Geometry: Plane Figures and Angles

In a circle of radius 13, a chord is 5 cm from the center. Find the chord length.

  1. 24 Correct answer.
  2. 12
  3. 26
  4. 20

Explanation

Half the chord, the radius to an endpoint, and the perpendicular distance form a right triangle. Half chord = √(13^2 - 5^2) = √(169-25) = √144 = 12. Full chord = 2·12 = 24.

Question 49Geometry: Plane Figures and Angles

In a right triangle, sin A = 5/13. Find cos A.

  1. 1
  2. 5/12
  3. 13/12
  4. 12/13 Correct answer.

Explanation

With opposite = 5 and hypotenuse = 13, the adjacent leg = √(13^2 - 5^2) = √(169-25) = √144 = 12. So cos A = adjacent/hypotenuse = 12/13.

Question 50Geometry: Plane Figures and Angles

A right triangle has hypotenuse 10 and one leg 6. Find the value of cos of the angle opposite the leg 6.

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

Explanation

Other leg = √(100 - 36) = 8. The angle opposite leg 6 has adjacent leg 8, hypotenuse 10. So cos = 8/10 = 4/5.