Japan Sholarship Mathematic Examination

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    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

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    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

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    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

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    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