1

I 31 9 00 12 00

1. I II

2. I 2 A B 4 A, B,

C, D 2 A B 2 A B10 2 3

3. 1 3 1 2

1 2

4. 1 2

5. I IIA

6. I II6 A B 1 2

I A (50 )

[ 1 ], [ 2 ]

[ 1 ] (1) (4)

(1) n Rn S

(2) 4 R4 S

S = {(x1, x2, x3, x4)T ∈ R4 | x2 = x3, x3 = −x4}

S

(3) 3 R3 S (1, 0, 0)T , (1, 1, 0)T ,(0, 0,−1)T 2

(4) 3 x1,x2,x3 α,β, γ

x1 =

⎛

⎜⎜⎝

α

0

0

⎞

⎟⎟⎠ , x2 =

⎛

⎜⎜⎝

2

β

0

⎞

⎟⎟⎠ , x3 =

⎛

⎜⎜⎝

1

4

γ

⎞

⎟⎟⎠ .

α,β, γ

[ 2 ]

2

[ 2 ] (1) (3)

(1) f : R2 → R2 x = (x1, x2)T ∈ R2( x ̸= 0)

f(x) = − 1(xTx)3/2

x

f

(2) (a) g : R → R− {0} g′(x)

g(x)

(b) g(x) =3x2 + 2x+ 1

x3 + x2 + xx = 0

limϵ→+0

(∫ −ϵ

−1g(x) dx +

∫ 1

ϵg(x) dx

)

(3) limn→∞

n∑

k=1

1

n+ k

3

I B (50 )

[ 1 ] [ 3 ]

[ 1 ] R (1) (3)

(1) p, q

(p → q) ∨ (q → p)

(2) 2 R2

A = {(x, y) ∈ R2∣∣0 < |x|+ |y| ≤ 1}

(3) 2 R2

S = {(x, y) ∈ R2∣∣0 < x < 1, 0 < y < 1}

[ 2 ] α,β ,

α β

(1) (3)

(1)

(2)

(3)

[ 3 ]

4

[ 3 ] α,β > 0

f(x) =

⎧⎪⎨

⎪⎩

(βxβ−1

αβ

)exp

[−(xα

)β](x ≥ 0),

0 (x < 0),

5

6

I A 50 [ ]

[ ] !(#$, #&)

#$ #& ($(&

) > 0

)

) (,$ (,&#̅$ #̅&

($. > (,$) ($. , (,&

#$. #&.

#̅$ ≥ #$. !(#$, #&) = #$

$/2#&3/2 (2) #̅$ #$.

#̅$ > #$.

!(#$, #&) = #$ + 2#& (2) (,$,(’$, (,&

I B (50 )

[ 1 ]

[ 1 ] C I

L

C = C(Y − T ), I = I(r), L = L(r, Y ),

Y r

T > 0

0 < C ′(Y − T )(≡ dC(Y − T )

d(Y − T )

)< 1,

I ′(r)

(≡ dI(r)

dr

)< 0,

Lr(r, Y )

(≡ ∂L(r, Y )

∂r

)< 0, LY (r, Y )

(≡ ∂L(r, Y )

∂Y

)> 0.

C, I, L, Y −T , r (1)(5)

(1) G > 0

Y r

(2)

P > 0 M > 0

Y r

(3) (1), (2) Y r

Y ∗ r∗ (T

) Y ∗ r∗ Y r

7

(4)

r > 0

NX

NX = NX(e), NX ′(e)

(≡ dNX(e)

de

)> 0,

e

G

Y e

Y e

(5) (2) (4)

Y e

Y ∗∗ e∗∗ Y ∗∗ e∗∗

Y e

8

9

I C 50 [ ], [ ]

[ ] 60

A B

) = 78 + 9,9 ∼ &?=) 8 5 A 35 B 25

A B >@A& = 0.50>@D& = 0.30

>A& = >D&

>A& ≠ >D&

60

[ ]

120

10

I D 50 [ ], [ ]

[ ] 2 1 2 U,D

1 2 1 2L,R 1 1 U, 2 L

1 –1 2 3 1 U,2 R 1 4 2 0

1 D, 2 L 12 1 1 D, 2 R

1 # 2 2

# =3

[ ] 2 1 2 A,B

1 A [ ] # = 5 1

B 1 2 25

11

I A 50 [ ], [ ]

[ ]

*

) 9

[ ] (

12

(DM)

1 65.0 100 2 45.0 120 3 75.0 95 4 90.0 110 5 92.0 100 6 97.0 100

100

0 20 S

1

2

1

0 5

0 1 9

2

3

4 5 9

I B (50 )

[ 1 ], [ 2 ]

[ 1 ] (1) (2)

(1)

x1 − x2 + x32x1 + x2 − x3 = 2,−x1 + 2x3 ≤ 5,x2 ≥ 0, x3 ≥ 0

(2) 3 (0.2, 0.5, 0.3)

P =

⎡

⎢⎢⎣

0.5 0.3 0.2

0.3 0.4 0.3

0.2 0.4 0.4

⎤

⎥⎥⎦

[ 2 ] x̄−R

n = 6

x̄−R

13

14

I A 50 [ ], [ ]

[ ]

1

9 [ ]

1

15

I B 50 [ ], [ ]

[ ] X

1,022 300

500 8

10 25

195 1,000

S 150

400

700 S 500

300

130

% % %

[ ]

16

[ ] (1) (5)

MM

rE rDD/E

rU 0.3

0.2 26

2 4 13

1 12

WACC1

5 5 1 100

100 NPV

1

31 9 00 12 00

I I

1. I II 25

2. II 2 2 A B 2 4 A, B, C, D 2 2 A B 2 2 A B 3

10 2 2 3

3. 1 2 3 1 2 1 2

3

4. 1 1 5 21

5. I IIA

6. I I 2 II2 ) I A B 1 1 2

II A (50 )

[ 1 ] [ 3 ]

[ 1 ] (1) (2)

(1) X rank(X) X

m× n A,B

rank(A) + rank(B) ≥ rank(A+B)

X rank(X) (P)

(P) X

rank(X)

(a) A,B A+B m× 3nC rank(A) + rank(B) ≥ rank(C)

(b) C rank(C) ≥ rank(A+B)

(2) n A n

[ 2 ],[ 3 ]

2

[ 2 ] f(x) x̄

∀ϵ > 0, ∃δ > 0, ∀x ∈ R, |x− x̄| < δ → |f(x)− f(x̄)| < ϵ.

(1) (2)

(1) f(x) x̄ !

!ϵ > 0, !δ > 0, !x ∈ R, .

!⎧⎪⎪⎨

⎪⎪⎩

∀, ∃, ∨, ∧, ≡, →, ↔,|x− x̄| < δ, |x− x̄| ≥ δ,|f(x)− f(x̄)| < ϵ, |f(x)− f(x̄)| ≥ ϵ

⎫⎪⎪⎬

⎪⎪⎭

(2) f(x)

f(x) =

{x (x ≤ 0),x− 1 (x > 0).

f(x) x = 0 (1)

[ 3 ] f(x) ak, bk (k = 1, 2, · · · , n)

f(x) =n∑

k=1

(ak cos kx+ bk sin kx)

(1) (2)

(1) f(x) ak, bk (k = 1, 2, · · · , n) f(x)ak, bk (k = 1, 2, · · · , n) f(x)

(2) f(x)

f ′′(x) + 2f ′(x) + f(x) =n∑

k=1

1

k + 1cos kx

ak, bk (k = 1, 2, · · · , n)

3

II B (50 )

[ 1 ] [ 3 ]

[ 1 ] N M = {2, 4, 6, 8, 12} Nn ≺ m m ∈ N n ∈ N

(1) (3)

(1) n ≺ m(2) (N,≺) M

(3) (N,≺) M

[ 2 ] X ϵ > 0

Prob(X ≥ ϵ) ≤ E(X)ϵ

.

[ 3 ] α,β > 0

f(x) =

⎧⎪⎨

⎪⎩

(βxβ−1

αβ

)exp

[−(xα

)β](x ≥ 0),

0 (x < 0),

Γ(x) =

∫ ∞

0ux−1 exp(−u) du (x > 0).

4

5

I II A 50 [ ] 0

[ ] A B X Y2 0 A3%&'(,&*(+ = (5, 0) A 3

2((3(, 4() = 3( +1334( −

124(:

B 3%&';,&*;+ = (0, 5) B 3

2;(3;, 4;) =1333; −

123;: + 4;

0 &'( X A &*( Y A

3( X A 4( Y A &'; X B

&*; Y B 3; X B 4; Y B

X

II B (50 )

[ 1 ]

[ 1 ] 2 i ci (i = 1, 2)

U(c1, c2) = u(c1) +1

1 + ρu(c2).

u(c) 2 u′(c) > 0 u′′(c) < 0 ρ > 0

i pi > 0 yi r > 0

I = y1 + y2/(1 + r) I > 0 1

s 2 p2c2 = (1+ r)s+ y2

i c∗i (i = 1, 2) (1)

(5)

(1) 1 1 2

s

(2) c∗i (i = 1, 2)

c1 c2

(3) c∗1 = c∗2

(4) i c∗i (i = 1, 2)

(a) u(c) = ln(c) ln

(b) u(c) = ca 0 < a < 1

(5) i yi

pi r

(a) u(c) = ln(c)

(b) u(c) = ca 2 (y2 = 0)

0 < a < 1

6

7

I II C 50 [ ] [ ] 0

[ ] 1

160

[ ]

4 = => + ?,? ∼ A(0B, C:Σ),Σ > 0

E F,G

F = E4

G = E= Σ3 0 4 E

60

8

I II D 50 [ ] 0

[ ] A H ⊂ 2J K ∈ H

M(K) 1 0

(A, M)

(i) M(A) = 1, M(∅) = 0 (ii) M(K) = 12 K ⊆ P M(P) = 1 (iii) M(K) = 1 M(A ∖ K) = 0

4 0

(A, MR) (a)2 (d) 0

4 V, W, X, Y2 0 A = {V, W, X, Y}

3 3 2 V3

6

(a) (A, MR) HR (b) 4

[^;&(, &;,&`, &a] ^3 ^

c ∈ A &d 1 (c) 3

(d) fd(MR)1

>d(MR) c = V, W, X, Y)

H g (A, M)

>d(M) = |{K ⊆ A:K ∈ H, K ∖ {c} ∉ H}|

2BkR

|P| T

9

(A, M:) (a)2 (d) 0

3 V, W, X, Y 4 1 A =

{V, W, X, Y} V 37%, W 36%, X

14%, Y 13%

3 51% 3

(a) (A, M:) H:

(b) 4

[^;&(, &;, &`, &a] ^3 ^

c ∈ A &d 1

(c) 3

(d) fd(M:)1

>d(M:) c = V, W, X, Y)

II A (50 )

[ 1 ], [ 2 ]

[ 1 ]

(1) (3)

(1) x (x > 0) Px 1

p h s

E(s)

(2) (1) E(s) s

(3) 2

X 1 X

X 100 0.06

X 550 1,000

X 1 200 X

1

(a) (c)

1: X( / ) 100 110 120 130 140 150

0.06 0.17 0.25 0.28 0.15 0.09

(a) 1

X

(b) 1

X

10

(c)

(a) (b)

[ 2 ] A (a) (e)

B (i) (v)

A

B 1

A

(a)

(b)

(c)

(d)

(e)

B

(i) Match between system and the real world

(ii) User control and freedom

(iii) Error prevention

(iv) Recognition rather than recall

(v) Aesthetic and minimalist design

11

II B (50 )

[ 1 ], [ 2 ]

[ 1 ] A K1, K2, K3 B S1, S2,

S3, S4 K1, K2, K3 B 100

80 120 S1, S2, S3, S4

B 60 70 80 90

B

BS1 S2 S3 S4

K1 4 3 5 4

K2 5 2 3 5

K3 6 4 4 5

(1) (3)

(1)

(2)

(3)

[ 2 ]

12

[ 2 ]

A B

y L

A B (Ai : i = 1, 2, 3;Bj : j = 1, 2, 3)

k = 1, 2

µ αi A βj B (αβ)ij A

B ϵ

(1) yijk = µ+ αi + βj + (αβ)ij + ϵijk

(2) yijk = µ+ αi + ϵ(1)ik + βj + (αβ)ij + ϵ

(2)ijk

(3) yijk = µ+ βj + ϵ(1)jk + αi + (αβ)ij + ϵ

(2)ijk

13

14

I II A 50 [ ] [ ] 0

[ ]

3

0

2

3 0

2 0

30

2

[ ]

0 3

3 22 3 0

6

15

I II B 50 [ ] [ ] 0

[ ] 2 0

3 22 , 2 4

2 2

3

2

[ ]

16

[ ] (1)2 (4) 0

2017 12 3 40 2

6 3 0

2

X Y 50

2

X Y

8 12

10 12

0.5

Z 1001 2

8

3 1/3 33.3333(%

LBO 2