8/14/2019 Japan Sholarship Mathematic Examination

1/4

A 1

AUndergraduate Students

I. 1. 4 6 2 5 3 x x x

x

2. 2x 3 2 2 f x x ax bx f x 1x 3

a b

3. 2 1 y x ax a a

4. 3log 52 3log 3 log 4 3

5. tan 2A sin cossin cos

A A

A A

II. 2 2: 1 1 0C x y kx k y k k1. 2. C

III. 7 7, 0 , 0, 7 , , 02 6

A B C

2 y x ax b BA

BC

1. a b 2. BA

8/14/2019 Japan Sholarship Mathematic Examination

2/4

A 2

I.

1. 2 4 6 2 64 6 2 5 3 1 32 5 3 3 3

x x x x x x x

x x x

2.

2x

f x

2x

0f x

8 4 2 2 0 2 5 1a b a b

3 2

22 11 11 1 1

f x x ax bx a b x a x b a

x x x

1

f x

x 3 1 3 4 2a b a b

1 2 2 5 3

41

a b a

a bb

3. 2 1 y x ax 0y 2 1 0 x ax 0 2 4 0 2a a 2a

4. 3log 52 3 2log 3 log 4 3 log 4 5 2 5 7 5.

cos tan 1sin cos 2 12 1 2 2 3 2 2

sin cos cos tan 1 2 1

A AA A

A A A A

II.1.

2 2: 1 1 0C x y kx k y k k 2 2 1 1 0 x y y x y k k

2 22 2 1 11 0

1 21 0

x y y x y y

x yx y

2

0

1 2 1 2 0 1

2

y

x y xy y xy yx

0y 1x

12

x 12

y

C 1,0 1 1,2 2

2. C

2 2 2 2

2 2 1 1: 1 1 0

2 2 2 2

k k k k C x kx y k y k

8/14/2019 Japan Sholarship Mathematic Examination

3/4

A 3

2 2 2 2 2

1 4 4 1 2 2 6 5:

2 2 4 4

k k k k k k k k C x y

2

2 22 6 5 1 2 6 54 2

k k R R k k

22 6 5 f k k k 4 6 f x k

4 0f x

3

02

f x k

2

min

3 3 9 12 6 5 9 5

2 2 2 2f

min min

1 1 1 2

2 2 2 4R f

III.1. a b

BA BC BA y ax b

7 27

2

ya

x

2 y x b 0,7B 7b

: 2 7 BA y x 22 7 x x ax b

2 2 7 0 x a x b 0 22 4 7 0 1a b BC y a x b

7 67

6

ya

x

6 y x b 0,7B 7b : 6 7 BC y x 26 7 x x ax b

2 6 7 0 x a x b 0 26 4 7 0 2a b

1 2

2

2

2 4 7 0

6 4 7 0

a b

a b

2 6 2a a a 16 28 4 0 3b b

8/14/2019 Japan Sholarship Mathematic Examination

4/4

A 4

2, 3a b 2.

2 2 3 y x x : 2 7 BA y x

2

2

02 7 2 3S x x x dx

22 3 2

0

2 7 2 32 3 2

x x xx x

84 14 4 6

3

83

S

-1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

x

y