JOHOKU-2008

Time : 50 minutes

Passing mark : 70%

Answer

Problem 1

(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.