Time : 50 minutes
Passing mark : 70%
Answer : End of the problem
Problem 1
Problem 2
(2)
(3)
(4)
Problem 3
(2)
(3)
(4)
Problem 4
Problem 5
Problem 6
(1) 29- 2 × (17 - 42 / 7) =
(2) 2.15 × 8 + 11.2 × 5 + + 2.85 × 8 + 0.8 × 5 =
(3) Find X
11/4 - X × {(0.75 + 1/4) / 2/3 - 1.25} = 2
Problem 2
(1)
As for the following sequence of numbers, which number from the beginning that 11 comes out for the first time ?
1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4 ......
(2)
List price of certain goods was set to be 20 % increase of the purchase price.
Since 1/4 of the purchases remained unsold, the remaining goods were sold at 20% discount of the list price.
There was profit of 14,000 yen altogether.
Find gross purchase.
(3)
I stood two poles A and B in the pool of a certain depth.
The difference of length of two poles is 18 cm.
As for the pole A, two fifths of the whole length came out of the water surface.
As for the pole B, one third of the whole length came out of the water surface.
Find the depth of this pool.
(4)
During 10:00 and 11:00, find the time when the angle made by the minute hand and the hour hand of a clock is divided by the direction of 12:00 into two equally as shown in a figure.
Problem 3
(1)
There is a square ABCD as shown in a lower figure.
When ∠FEB = 28 degrees, find the degree of angle X.
(2)
In a figure below, D and E are the points dividing AB into three equally and F is the middle point of BC.
When AF = 15 cm, find the length of GH.
(3)
The sector OAB of 90 degrees of central angle and 6cm in radius was moved along with XY 4 cm as shown in a figure below.
Find the area of the portion where the arc AB passed.
(4)
In triangular prism ABC-DEF as shown in a figure below, the points P, Q, and R are points as AP=2cm, BQ=4cm, and CR=3cm.
This triangular prism is cut and divided into two solids by the plane which passes along these three points.
Find the volume of solid PQR-DEF.
Problem 4
There are the ships A and B plying between P town and Q town located along a river.
The graph below expresses that the ship A departs from P town and the ship B departs from Q town simultaneously at 8:00 a.m., respectively, and both ships went back and forth between PQ one time.
The speed of the flow of a river is always constant and the speed of both ships is also constant respectively when there is no flow.
Answer the following questions.
(1) Find the speed of going up the river of the ship B.
(2) Find the first time for two ships A and B to meet.
(3) Find the distance of the point from Q town where two ships A and B meet at the 2nd time.
Problem 5
Six soccer teams of A~F played a game in round robin tournament.
There was no game of a draw and when ranking was decided with the number of victories, it was found the following four fact.
<1> B and E were the 1st place in the same number of victories.
<2> F was the 3rd place independently.
<3> C won E.
<4> C lost A and C was the 4th place independently.
Answer the following questions.
(1) Find all the numbers of games in 6 teams of A~F.
(2) Which team won at the following games?
① C versus D ② A versus D
Problem 6
There is a cube whose length of one side is 10 cm.
Answer the following questions.
(1) As shown in Fig. 1, when hollowing a 10 cm high rectangular prism whose bottom is square with one side 4 cm from this cube and this solid is looked from a top, it looks as it is shown in Fig. 2.
Next, as shown in Fig. 3, a 10 cm high rectangular prism whose bottom is square with one side 4 cm is hollowed from the face A of the solid of Fig. 1 to an opposite face.
When this solid is looked as the face A to be front, it looks as it is shown in Fig. 4.
Find the volume of this solid.
(2) As shown in Fig. 5, when the 10 cm high rectangular prism whose bottom is rectangle with 6 cm in length and 7 cm in width is hollowed from the cube whose length of one side is 10 cm and it is looked from the top, it looks as it is shown in Fig. 6.
Moreover, as shown in Fig. 7, the same rectangular prism is hollowed from the face B of this solid.
Then, when this solid is looked as the face B to be front, it looks as it is shown in Fig. 8.
Find the volume of this solid.
<Answer>
Problem 1
Problem 2
(The number of group) = (The first number of each group)
(2)
(3)
(4)
Problem 3
(2)
(3)
(4)
Problem 4
Problem 5
Solution
Problem 6
(1) 29- 2 × (17 - 42 / 7) =
(2) 2.15 × 8 + 11.2 × 5 + + 2.85 × 8 + 0.8 × 5 =
(3) Find X
11/4 - X × {(0.75 + 1/4) / 2/3 - 1.25} = 2
Answer
(1) 7
(2) 100
(3) 3
Problem 2
(1)
As for the following sequence of numbers, which number from the beginning that 11 comes out for the first time ?
1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4 ......
Answer
56th
Solution
The sequence is divided into groups as below.
11 comes out first as the first number of 11 group.
1 + 2 + 3 +------+ 10 = 55
As 11 is next number, it is 55 + 1 = 56th.
(2)
List price of certain goods was set to be 20 % increase of the purchase price.
Since 1/4 of the purchases remained unsold, the remaining goods were sold at 20% discount of the list price.
There was profit of 14,000 yen altogether.
Find gross purchase.
Answer
100,000 yen
Solution
Assuming purchase price = 1 and number of goods = 1.
Number of goods sold at list price is 1 - 1/4 = 3/4
Sales of list price is 1.2 × 3/4 = 0.9.
20% discounted price is 1.2 × (1 - 0.2) = 0.96.
0.96 × 1/4 = 0.24
0.9 + 0.24 = 1.14
Then profit is 1.14 - 1 = 0.14 which is equivalent to 14000 yen.
1 = 14000 / 0.14 = 100000 yen.
(3)
I stood two poles A and B in the pool of a certain depth.
The difference of length of two poles is 18 cm.
As for the pole A, two fifths of the whole length came out of the water surface.
As for the pole B, one third of the whole length came out of the water surface.
Find the depth of this pool.
Answer
108 cm
Solution
1 - 2/5 = 3/5 of A is under the water.
1 - 1/3 = 2/3 of B is under the water.
Assuming the depth of the pool is 1, length of A is 1 / 3/5 = 5/3, length of B is 1 / 2/3 = 3/2.
A : B = 5/3 : 3/2 = 10 : 9
10 - 9 = 1 is equivalent to 18 cm.
Length of A is 18 cm × 10 = 180 cm.
The depth of the pool is 180 cm × 3/5 = 108 cm.
(4)
During 10:00 and 11:00, find the time when the angle made by the minute hand and the hour hand of a clock is divided by the direction of 12:00 into two equally as shown in a figure.
Answer
10 o'clock 120/13 minutes
Solution
According to the figure below, sum total of angle of the minutes hand and the hour hand is 30 × 2 = 60 degrees.
The minutes hand moves 6 degrees per minute and the hour hand moves 0.5 degrees per minute.
60 degrees / ( 6 + 0.5) = 120/13 minutes.
Problem 3
(1)
There is a square ABCD as shown in a lower figure.
When ∠FEB = 28 degrees, find the degree of angle X.
Answer
34 degrees
Solution
△BCF ≡ △DCF
∠FBC = ∠CDF = 180 - (90 + 28) = 62 degrees
X = 62 - 28 = 34 degrees.
(2)
In a figure below, D and E are the points dividing AB into three equally and F is the middle point of BC.
When AF = 15 cm, find the length of GH.
Answer
4.5 cm
Solution
As BF = CF, △BHF = △CHF.
As AE : BE = 2 : 1, △CAH : △CBH = 2 : 1
Assuming △BHF = △CHF = 1, △CAH = 4 as shown in Fig.1.
Thus AH : HF = 4 : 1.
As BF = CF, △BGF = △CGF.
As AD : BD = 1 : 2, △CAG : △CBG = 1 : 2
Assuming △BGF = △CGF = 1, △CAG = 1 as shown in Fig.2.
Thus AG : GF = 1 : 1.
Therefore AG : GH : HF = 5 : 3 : 2.
GH = 15 cm × 3/(5+3+2) = 4.5 cm.
(3)
The sector OAB of 90 degrees of central angle and 6cm in radius was moved along with XY 4 cm as shown in a figure below.
Find the area of the portion where the arc AB passed.
Answer
24 cm2
Solution
(Shadow area) = (Sector OAB + rectangle AOO´A´) - (SectorO´A´B´)
= rectangle AOO´A´ = 4 × 6 = 24 cm2
(4)
In triangular prism ABC-DEF as shown in a figure below, the points P, Q, and R are points as AP=2cm, BQ=4cm, and CR=3cm.
This triangular prism is cut and divided into two solids by the plane which passes along these three points.
Find the volume of solid PQR-DEF.
Answer
216 cm3
Solution
(Volume of prism) = (Bottom area) × (Average of each height of vertical side)
= ( 8 × 6 / 2) × (10 + 8 + 9)/3 = 216 cm3
Problem 4
There are the ships A and B plying between P town and Q town located along a river.
The graph below expresses that the ship A departs from P town and the ship B departs from Q town simultaneously at 8:00 a.m., respectively, and both ships went back and forth between PQ one time.
The speed of the flow of a river is always constant and the speed of both ships is also constant respectively when there is no flow.
Answer the following questions.
(1) Find the speed of going up the river of the ship B.
(2) Find the first time for two ships A and B to meet.
(3) Find the distance of the point from Q town where two ships A and B meet at the 2nd time.
Answer
(1) 12 km/h
(2) 9:30
(3) 8.4 km
Solution
(1)
As for A, from P to Q, 12:00 - 8:00 = 4 hours and from Q to P, 15:00 - 12:00 = 3 hours.
36 / 4 = 9 km/h
36 / 3 = 12 km/h
The flow speed of the river is (12 -9) /2 = 1.5 km/h
As for B, from Q to P, 10:24 - 8:00 = 12/5 hours
35 / 12/5 = 15 km/h
15 - 1.5 × 2 = 12 km/h.
(2)
36 / (9 + 15) = 1.5 hours
8:00 + 1.5 hours = 9:30
(3)
12:00 - 10:24 = 8/5 hours
12 × 8/5 = 19.2 km
36 - 19.2 = 16.8 km
Down speed of A = up speed of B = 12 km/h
16.8 / 2 = 8.4 km
Problem 5
Six soccer teams of A~F played a game in round robin tournament.
There was no game of a draw and when ranking was decided with the number of victories, it was found the following four fact.
<1> B and E were the 1st place in the same number of victories.
<2> F was the 3rd place independently.
<3> C won E.
<4> C lost A and C was the 4th place independently.
Answer the following questions.
(1) Find all the numbers of games in 6 teams of A~F.
(2) Which team won at the following games?
① C versus D ② A versus D
Answer
(1) 15 games
(2) ①C ②D
(1)
5 × 6 / 2 = 15 games
(2)
If E and B won only three, total number of win of A,C,D,F is 15 - 3 × 2 = 9 which means there is at least one team which won three.
Then it is found that E and B won four and we can get Table 2.
Total number of win of A,C,D,F is 15 - 4 × 2 = 7.
As F is 3rd, C is 4th independently, F won three, C won two.
A and D won one.
The we can get Table 3.
Problem 6
There is a cube whose length of one side is 10 cm.
Answer the following questions.
(1) As shown in Fig. 1, when hollowing a 10 cm high rectangular prism whose bottom is square with one side 4 cm from this cube and this solid is looked from a top, it looks as it is shown in Fig. 2.
Next, as shown in Fig. 3, a 10 cm high rectangular prism whose bottom is square with one side 4 cm is hollowed from the face A of the solid of Fig. 1 to an opposite face.
When this solid is looked as the face A to be front, it looks as it is shown in Fig. 4.
Find the volume of this solid.
(2) As shown in Fig. 5, when the 10 cm high rectangular prism whose bottom is rectangle with 6 cm in length and 7 cm in width is hollowed from the cube whose length of one side is 10 cm and it is looked from the top, it looks as it is shown in Fig. 6.
Moreover, as shown in Fig. 7, the same rectangular prism is hollowed from the face B of this solid.
Then, when this solid is looked as the face B to be front, it looks as it is shown in Fig. 8.
Find the volume of this solid.
Answer
(1) 744 cm3
(2) 370 cm3
Solution
(1)
Volume of two rectangular prisms = 4 × 4 × 10 × 2 = 320 cm3.
Volume of overlapping is 4 × 4 × 4 = 64 cm3.
Volume to be found is 10 × 10 × 10 - (320 - 64) = 744 cm3.
(2)
The figure below is graphic of overlapping.
Volume of two rectangular prisms = 6 × 7 × 10 × 2 = 840 cm3.
Volume of overlapping is 6 × 5 × 7 = 210 cm3.
Volume to be found is 1000 - (840 - 210) = 370 cm3.
