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Time : 50 minutes
Passing mark : 70%
Problem 1
(1)
① (26.5 / 2.5 - 17/3) / 37/24 - 0.96 × 11/6 =
② Find X
53/36 - 11/25 / (27/13 - X × 9/65) = 5/4
③ 3 × 5 × 5 × 5 + 4 × 5 × 5 + 2 × 5 + 1
= A × A × A + B × A × A + C × A + 3.
Find A, B and C, respectively.
A, B and C are the integers from 1 to 9.
(2)
There are 365 cards piled up on which the date of the non-leap year is written.
It is written to the 1st card as January 1, to the 2nd card as January 2, to the 3rd card as January 3rd and to the 365th card as December 31. Answer the following questions.
(2)-1 Remove even-numbered cards counted from the top.
In this case, the date written on the top of the remaining card is January 1 and the date of the 2nd card is January 3.
Find the date of the 28th card.
(2)-2 As a next step, remove odd-numbered cards counted from the top among remaining cards.
In this case, which number of card counted from the top among remaining cards is written as September 12?
(2)-3 Suppose that January 1 is Monday.
Find the day of the week of the 69th card counted from the top among remaining cards.
Problem 2
A figure is a circle with a radius of 3 cm and AB is a diameter of the circle.
All ● in a figure is 15 degrees. Pi is assumed to be 3.14.
(1) Find the sum of the area of the slash portion of P and Q.
(2) Find the area of the slash portion of Q.
Problem 3
When the cube of Fig. 1 is opened up by cutting by the bold line of Fig. 2, the development view become Fig. 3.
How should it be cut respectively, in order for the development view of each cube to become as shown in (1) and (2)?
As shown in Fig. 2, write a bold line in the sketch of the cube.
Problem 4
As shown in a figure, the square card was equally divided into four and the integer was written to each portion sequentially from 1.
A lot of these cards were made.
All the sizes of the card are same.
Answer the following questions.
(1) Cards were arranged according to the direction in the figure from left to right sequentially from the 1st card.
Which card is the first card for the sum of the two number of the upper rows of one card to become larger than 120?
(2) The card was piled up in order by direction in the figure from the 1st card to the 50th card.
Find the sum total of 50 numbers which has overlapped with the number 1 of the 1st card including 1 of the 1st card.
(3) Rotate 90 degrees of the 1st card counterclockwise and pile up on the 2nd card.
Next, rotate 90 degrees counterclockwise both of 1st and 2nd cards piled up and 2 cards are piled up on the 3rd card.
Thus, when you pile up 50 cards, find the sum totals of 50 numbers including 3 which has overlapped with the number 3.
Problem 5
There is a rectangle ABCD with 20 cm in length and 10 cm in width as shown in a figure.
The point P leaves the point A and moves in order of A→B→C→D→A→-------- on the side of this rectangle.
The point Q leaves the point B simultaneously with the point P and moves in order of B→C→D→A→ ----.
The point P moves at the speed of 1 cm/s and the point Q moves 2 cm/s in the first round.
Then, the point P increases the speed per 1 second for every round as 2 cm/s in 2nd round, 3 cm/s in 3rd round.
The point Q also increases the speed per 1 second for every round as 3 cm/s in 2nd round, 4 cm/s in 3rd round.
Answer the following questions.
(1) Find the time after leaving when the point Q overlaps with the point P for the first time.
(2) Regarding from 1st round to 3rd round, write the time when two point P and Q take moving one round of rectangle ABCD, respectively in a lower table.
Let the measure be second.
(3) 100 seconds after leaving, where are two point P and Q?
Write the position of point P in Fig. 1 and write the position of the point Q in Fig. 2. according to the (Example) below.
(Example) When the point P is at the position of 7 cm from A on the side AD in 100 seconds.
(4) In 100 seconds after leaving, find the last time of seconds when the point Q overlaps with the point P.
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