Time : 60 minutes
Passing marks : 70%
Answer End of the problem
Problem 1
There are rectangle PQRS and a right-angled triangle ABC whose angle B is right as shown in a figure.
Triangle ABC is rotated for the surroundings of rectangle PQRS as follows.
The point A is at point P and point B is on the side PS at first.
① Triangle ABC is clockwise rotated around point B until the point C and the point S overlaps.
② Next, triangle ABC is clockwise rotated around point C until the point A and the point R overlaps.
③ Next, triangle ABC is clockwise rotated around point A until the point B comes on the side QR.
In this case, find the surrounding length of the portion by which triangle ABC moved.
The answer should round off the 2nd decimal place and should be found to the 1st decimal place.
Problem 2
(2) Find the number which is located 8 grids moved in the upper direction from 0 and 8 grids moved in the left direction further.
(3) Find the number which is next to 555 by 1 girds vertically and horizontally, respectively.
Problem 3
Triangle ABC is rotated for the surroundings of rectangle PQRS as follows.
The point A is at point P and point B is on the side PS at first.
① Triangle ABC is clockwise rotated around point B until the point C and the point S overlaps.
② Next, triangle ABC is clockwise rotated around point C until the point A and the point R overlaps.
③ Next, triangle ABC is clockwise rotated around point A until the point B comes on the side QR.
In this case, find the surrounding length of the portion by which triangle ABC moved.
The answer should round off the 2nd decimal place and should be found to the 1st decimal place.
Problem 2
The number of integer from 0 is written down clockwise in whorl to small order in a grid as shown in a figure as 0, 1, 2, -----, 26, 27, --------.
It is considered eight directions which added slanting direction to the direction of vertical and horizontal.
For example, 27 is located three grids moved in the upper direction from 0. 7 is located 2 grids moved in the direction of upper left from 17.
It is considered eight directions which added slanting direction to the direction of vertical and horizontal.
For example, 27 is located three grids moved in the upper direction from 0. 7 is located 2 grids moved in the direction of upper left from 17.
Answer the following questions.
(1) Find the number which is located 8 grids moved in the downward direction from 0 and 8 grids moved in the right direction further. (2) Find the number which is located 8 grids moved in the upper direction from 0 and 8 grids moved in the left direction further.
(3) Find the number which is next to 555 by 1 girds vertically and horizontally, respectively.
Problem 3
There are three points A, B, and C which move around on the one circumference with a fixed speed, respectively.
A and B move to the same direction and C moves to the opposite direction of A and B.
These three points A, B, and C departed from the same point exactly at 1:00.
A and C met at 1:02 and B and C met for the first time after the start at 1:07.
Moreover, A returned to the original point for the first time at 1:02 30 seconds.
(1) Find the first time for B to return to the original point.
(2) Find the first time for A to catch up with B.
(3) Find the first time for A, B, and C to become three vertex of an equilateral triangle.
(2)-1 Draw as correctly as possible the development view of the triangular pyramid which is made by connecting the four points of A, E, H and K.
< Cautions > Triangle AEH is already drawn.
Draw as not to overflow the frame in which the answer sheet is defined.
(2)-2 Find the area of the quadrangle which is made by connecting the four points A, H, I and J.
There are rectangle PQRS and a right-angled triangle ABC whose angle B is right as shown in a figure.
Triangle ABC is rotated for the surroundings of rectangle PQRS as follows.
The point A is at point P and point B is on the side PS at first.
① Triangle ABC is clockwise rotated around point B until the point C and the point S overlaps.
② Next, triangle ABC is clockwise rotated around point C until the point A and the point R overlaps.
③ Next, triangle ABC is clockwise rotated around point A until the point B comes on the side QR.
In this case, find the surrounding length of the portion by which triangle ABC moved.
The answer should round off the 2nd decimal place and should be found to the 1st decimal place.
Problem 2
Answer the following questions.
(1) Find the number which is located 8 grids moved in the downward direction from 0 and 8 grids moved in the right direction further.
(2) Find the number which is located 8 grids moved in the upper direction from 0 and 8 grids moved in the left direction further.
(3) Find the number which is next to 555 by 1 girds vertically and horizontally, respectively.
Problem 3
A and B move to the same direction and C moves to the opposite direction of A and B.
These three points A, B, and C departed from the same point exactly at 1:00.
A and C met at 1:02 and B and C met for the first time after the start at 1:07.
Moreover, A returned to the original point for the first time at 1:02 30 seconds.
(1) Find the first time for B to return to the original point.
(2) Find the first time for A to catch up with B.
(3) Find the first time for A, B, and C to become three vertex of an equilateral triangle.
Problem 4
There is a rectangular prism as shown in a figure.
I is a point on the side FG.
AB = 1 cm, AD = 6 cm, AE = 2 cm, GI = 1.5 cm.
The intersection of the plane which passes along the three points A, H, and I, and the side BF is set to J.
The intersection of the plane which passes along the three points A, H, and I, and the line which extended the side EP is set to K.
I is a point on the side FG.
AB = 1 cm, AD = 6 cm, AE = 2 cm, GI = 1.5 cm.
The intersection of the plane which passes along the three points A, H, and I, and the side BF is set to J.
The intersection of the plane which passes along the three points A, H, and I, and the line which extended the side EP is set to K.
Answer the following questions.
(1) Find the length of FJ and FK. (2)-1 Draw as correctly as possible the development view of the triangular pyramid which is made by connecting the four points of A, E, H and K.
< Cautions > Triangle AEH is already drawn.
Draw as not to overflow the frame in which the answer sheet is defined.
(2)-2 Find the area of the quadrangle which is made by connecting the four points A, H, I and J.
<Answer>
Problem 1There are rectangle PQRS and a right-angled triangle ABC whose angle B is right as shown in a figure.
Triangle ABC is rotated for the surroundings of rectangle PQRS as follows.
The point A is at point P and point B is on the side PS at first.
① Triangle ABC is clockwise rotated around point B until the point C and the point S overlaps.
② Next, triangle ABC is clockwise rotated around point C until the point A and the point R overlaps.
③ Next, triangle ABC is clockwise rotated around point A until the point B comes on the side QR.
In this case, find the surrounding length of the portion by which triangle ABC moved.
The answer should round off the 2nd decimal place and should be found to the 1st decimal place.
Answer
54.1 cm
Solution
5 × 2 × 3.14 × 1/4 + 5 × 2 × 3.14 × 120/360 2 + 7 + 5 + 4 + 3
= 5/2 × 3.14 + 20/3 × 3.14 + 4 × 2 × 3.14 × 1/4 + 7 + 5 + 4 + 3
= (15/6 + 40/6 + 2) × 3.14 + 19
= 67/6 × 3.14 + 19
= 54.06----
Problem 2
The number of integer from 0 is written down clockwise in whorl to small order in a grid as shown in a figure as 0, 1, 2, -----, 26, 27, --------.
It is considered eight directions which added slanting direction to the direction of vertical and horizontal.
For example, 27 is located three grids moved in the upper direction from 0. 7 is located 2 grids moved in the direction of upper left from 17.
It is considered eight directions which added slanting direction to the direction of vertical and horizontal.
For example, 27 is located three grids moved in the upper direction from 0. 7 is located 2 grids moved in the direction of upper left from 17.
(1) Find the number which is located 8 grids moved in the downward direction from 0 and 8 grids moved in the right direction further.
(2) Find the number which is located 8 grids moved in the upper direction from 0 and 8 grids moved in the left direction further.
(3) Find the number which is next to 555 by 1 girds vertically and horizontally, respectively.
Answer
(1) 256
(2) 288
(3) 554, 556, 464, 654
(2) 288
(3) 554, 556, 464, 654
Solution
(1)
Below 1 and right 1 = 4 = 2 × 2 = (2 × 1) × (2 × 1)
Below 2 and right 2 = 16 = 4 × 4 = (2 × 2) × (2 × 2)
Below 3 and right 3 = 36 = 6 × 6 = (2 × 3) × (2 × 3)
Below 8 and right 8 = (2 × 8) × (2 × 8) = 16 × 16 = 256
(2)
Above 1 and left 0 = 1 = 1 × 1 (1 = 2 × 1 - 1)
Above 2 and left 1 = 9 = 3 × 3 (3 = 2 × 2 - 1)
Above 3 and left 2 = 25 = 5 × 5 (5 = 2 × 3 - 1)
Above 8 and left 7 = 225 = 15 × 15 (15 = 2 × 8 - 1)
Above 9 and left 8 = 289 = 17 × 17 (17 = 2 × 9 - 1)
Then
Above 1 and left 1 = 9 - 1 = 8
Above 2 and left 2 = 25 - 1 = 24
Above 7 and left 7 = 225 - 1 = 224
Above 8 and left 7 = 289 - 1 = 288
(3)
555 is the number between 23 × 23 = 529 and 24 × 24 = 576.
Since (23 +1) / 2 = 12, 529 is a position of above 12 left 11.
The number of right corner is 529 + (11 + 12) = 552.
555 is the 3rd number from 552.
The number of above 11 left 10 is 21 × 21 = 441.
The number of right corner is 441 + (10 + 11) = 462.
The left side of 555 is 462 + 2 = 464.
The number of above 13 left 12 is 25 × 25 = 625.
The number of right corner is 625 + (12 + 13) = 650.
The right side of 555 is 650 + 4 = 654.
Problem 3
There are three points A, B, and C which move around on the one circumference with a fixed speed, respectively.
A and B move to the same direction and C moves to the opposite direction of A and B.
These three points A, B, and C departed from the same point exactly at 1:00.
A and C met at 1:02 and B and C met for the first time after the start at 1:07.
Moreover, A returned to the original point for the first time at 1:02 30 seconds.
(1) Find the first time for B to return to the original point.
(2) Find the first time for A to catch up with B.
(3) Find the first time for A, B, and C to become three vertex of an equilateral triangle.
Problem 4
(2)-1 Draw as correctly as possible the development view of the triangular pyramid which is made by connecting the four points of A, E, H and K.
< Cautions > Triangle AEH is already drawn.
Draw as not to overflow the frame in which the answer sheet is defined.
(2)-2 Find the area of the quadrangle which is made by connecting the four points A, H, I and J.
A and B move to the same direction and C moves to the opposite direction of A and B.
These three points A, B, and C departed from the same point exactly at 1:00.
A and C met at 1:02 and B and C met for the first time after the start at 1:07.
Moreover, A returned to the original point for the first time at 1:02 30 seconds.
(1) Find the first time for B to return to the original point.
(2) Find the first time for A to catch up with B.
(3) Find the first time for A, B, and C to become three vertex of an equilateral triangle.
Answer
(1) 1 : 23 20 seconds
(2) 1 : 02 48 seconds
(3) 1 : 04 40 seconds
(2) 1 : 02 48 seconds
(3) 1 : 04 40 seconds
Solution
(1)
2 minutes = 120 seconds, 7 minutes = 420 seconds, 2 minutes and 30 seconds = 150 seconds.
The length of the circumference is set to be LCM of 120, 420 and 150, which is 4200.
The speed of A = 4200 / 150 = 28.
The speed of C = 4200 / 120 - 28 = 7.
The speed of B = 4200 / 420 - 7 = 3.
4200 / 3 = 1400
1400 / 60 = 23 and 20 = 23 minutes 30 seconds
(2)
4200 / (28 - 3) = 168 = 2 minutes and 48 seconds
(3)
The time of the difference between A and B should be 1400 or 2800 is
56 seconds = 1400 / (28 - 3)
112 seconds = 2800 / 25
224 seconds = (4200 + 1400) / 25
280 seconds = (4200 + 2800) / 25
The time of the difference between A and C should be 1400 or 2800 is
40 seconds = (4200 - 2800) / (28 + 7 )
80 seconds = (4200 - 1400) / 35
160 seconds = (4200 + 1400) / 35
200 seconds = (4200 + 2800) / 35
Next is to be 200 + 80 = 280 seconds.
280 / 60 = 4 minutes and 40 seconds
Problem 4
There is a rectangular prism as shown in a figure.
I is a point on the side FG.
AB = 1 cm, AD = 6 cm, AE = 2 cm, GI = 1.5 cm.
The intersection of the plane which passes along the three points A, H, and I, and the side BF is set to J.
The intersection of the plane which passes along the three points A, H, and I, and the line which extended the side EP is set to K.
I is a point on the side FG.
AB = 1 cm, AD = 6 cm, AE = 2 cm, GI = 1.5 cm.
The intersection of the plane which passes along the three points A, H, and I, and the side BF is set to J.
The intersection of the plane which passes along the three points A, H, and I, and the line which extended the side EP is set to K.
Answer the following questions.
(1) Find the length of FJ and FK. (2)-1 Draw as correctly as possible the development view of the triangular pyramid which is made by connecting the four points of A, E, H and K.
< Cautions > Triangle AEH is already drawn.
Draw as not to overflow the frame in which the answer sheet is defined.
(2)-2 Find the area of the quadrangle which is made by connecting the four points A, H, I and J.
Answer
(2)-2 49/8 cm2
(1) FJ = 1.5 cm, FK = 3 cm
(2)-1
(2)-1
(2)-2 49/8 cm2
Solution
(1)
Reference to FIg.1, △AEH∽△FJI.
Ratio = 6cm : 6cm - 1.5cm = 4 : 3
FJ = AE × 3/4 = 2 × 3/4 = 1.5 cm
Reference to Fig.3, FK : (FK+1) : 1.5 : 2 = 3 : 4
Therefore FK = 3 cm.
(2)-1
The development view is as shown in the answer.
(2)-2
The area of triangle KAH
= 6 × 6 - (2 × 6 / 2) - (2 × 4 / 2) - (4 × 6 / 2)
= 36 - 6 - 4 - 12
= 14 cm2
KI : KH = 3 : 4
The area ratio of △KJI : △KAH = 3 × 3 : 4 × 4 = 9 : 16.
The area ration of △KAH : AHIJ = (16 - 9) : 16 = 7 : 16.
AHIJ = 14 cm2 × 7/16 = 49/8 cm2
