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平成３０年度　九州大学大学院システム情報科学府情報学専攻情報知能工学専攻電気電子工学専攻

修士課程　入学試験問題

数学 (Mathematics)（7枚中の 1）

解答上の注意 (Instructions):

1. 問題用紙は，『始め』の合図があるまで開いてはならない．Do not open this cover sheet until the start of examination is announced.

2. 問題用紙は表紙を含め 7枚，解答用紙は 3枚つづり (1分野につき 1枚)である．You are given 7 problem sheets including this cover sheet, and 3 answer sheets (1 sheet for

each field).

3. 以下の 6分野から 3分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．Select 3 fields out of the following 6 fields and answer the questions. You must use a separate

answer sheet for each of the fields you selected.

分野 field page1 線形代数 Linear algebra 22 微分方程式 Differential equation 33 ベクトル解析 Vector analysis 44 複素関数論 Complex function theory 55 確率・統計 Probability and statistics 66 記号論理学 Symbolic logic 7

4. 解答用紙の全部に，専攻名，コース名（情報学専攻を除く），選択分野番号（○で囲む），受験番号および氏名を記入すること．Fill in the designated blanks at the top of each answer sheet with the department name,

course name (except the department of informatics), the selected field number (mark with a

circle), your examinee number and your name.

5. 解答は解答用紙に記入すること．スペースが足りない場合は裏面を用いても良いが，その場合は，裏面に解答があることを明記すること．Write your answers on the answer sheets. You may use the backs of the answer sheets when

you run out of space. If you do so, indicate it clearly on the sheet.




6分野のうちから 3分野を選び解答すること．選んだ分野毎に解答用紙を別にすること．Select 3 fields out of the 6 fields and answer the questions. Use a separate answer sheetfor each field.

1. 【線形代数 (Linear algebra)分野】正方行列Aが交代的であるとは，A� = −Aを満たすことである．ここで，A�はAの転置を表す．以下の各問に答えよ．

(1) 任意のn次正方行列A ∈ Rn×nと整数 i, j (1 ≤ i, j ≤ n)に対し，ベクトルの組x, y ∈ Rn

が存在し，x�Ay = Aijを満たすことを示せ．ただし，AijはAの (i, j)成分である．

(2) n次正方行列A ∈ Rn×nが交代的であるための必要十分条件は，

y�Ax = −x�Ay (∀x, y ∈ Rn)

が成り立つことであることを示せ．

(3) 交代的な行列Aが固有値 λを持つとき，Aは−λも固有値として持つことを示せ．

ヒント：任意の正方行列Xの行列式 |X|について，|X| = |X�|が成り立つ．

A square matrix A is said to be alternative if A satisfies A� = −A, where A� denotes the

transpose of A. Answer the following questions.

(1) For any n by n square matrix A ∈ Rn×n and any integers i, j (1 ≤ i, j ≤ n), find a pair

of vectors x, y ∈ Rn that satisfies x�Ay = Aij, where Aij is the (i, j) entry of A.

(2) Show that an n by n square matrix A ∈ Rn×n is alternative if and only if

y�Ax = −x�Ay (∀x, y ∈ Rn).

(3) Assume that an alternative matrix A has an eigenvalue λ. Then, show that A has the

eigenvalue −λ as well.

Hint: Any square matrix X satisfies |X| = |X�|, where |X| is the determinant of X.





2. 【微分方程式 (Differential equation)分野】次の微分方程式の一般解を求めよ．なお，y′は関数 y(x)の xに関する 1階導関数を表している．

(1) y′ =9(y2 + 1)

x3 − 3x + 2

(2) y′′ − 3y′ + 2y = e2x

Find general solutions to the following differential equations. Here, y′ denotes the derivative

of first order with respect to x for a function y(x).

(1) y′ =9(y2 + 1)

x3 − 3x + 2

(2) y′′ − 3y′ + 2y = e2x





3. 【ベクトル解析 (Vector analysis)分野】

直交座標系において，x, y, z 軸方向の単位ベクトルをそれぞれ i, j, kとする．S を以下の面とし，C をその外周とするとき，ベクトル場A = yi− 2xj + xzk に対し，線積分

∮C

A · dr，および面積分

∫S(∇× A) · dSをそれぞれ計算せよ.なお，線積分は z軸正方向からみて反時

計回りに沿って行うものとし，S の法線ベクトルの z成分は非負とする．

S : x2 + y2 + z2 = 1 (z ≥ 0)

Let i, j and k denote the unit vectors on x, y and z axes of Cartesian coordinates, respectively.

Calculate the line integral∮

CA · dr and the surface integral

∫S(∇× A) · dS for the vector

field A = yi − 2xj + xzk, where S is the following surface and C is its boundary. Here, the

path of the line integral is oriented counterclockwise viewed from the z-axis positive side, and

z component of the normal vector of S is non-negative.

S : x2 + y2 + z2 = 1 (z ≥ 0)





4. 【複素関数論 (Complex function theory)分野】

図に示す曲線Cに沿った複素積分∮

C

(ln z)2

z2 + 1dzを考える．ただし，R > 1, ε < 1とする．次

の各問に答えよ．

(1)

∮C

(ln z)2

z2 + 1dzの値を求めよ．

(2)

∮C

(ln z)2

z2 + 1dzの値を用いて，

∫ ∞

0

(ln x)2

x2 + 1dx =

π3

8を示せ．

Consider the integral of the complex function

∮C

(ln z)2

z2 + 1dz, where C is a curve as shown in

the figure, R > 1, and ε < 1. Answer the following questions.

(1) Find the value of

∮C

(ln z)2

z2 + 1dz.

(2) Using the value of

∮C

(ln z)2

z2 + 1dz, prove that

∫ ∞

0

(ln x)2

x2 + 1dx =

π3

8.





5. 【確率・統計 (Probability and statistics)分野】n (n ≥ 1)個の独立した連続確率変数X1, . . . , Xnは，開区間 (0, 1)の一様分布に従うものとする．また Y = min{X1, . . . , Xn}とする．以下の各問に答えよ．

(1) 実数 y (0 < y < 1)に対して，確率 Pr[Y > y]を求めよ．

(2) 期待値 E[Y ]を求めよ．

(3) 分散Var[Y ]を求めよ．

(4) Z = 1/Y とする．Zの期待値が存在する場合は期待値を求め，存在しない場合はその理由を述べよ．

Let X1, . . . , Xn (n ≥ 1) be independent continuous random variables uniformly distributed

over the open interval (0, 1). Then, let Y = min{X1, . . . , Xn}. Answer the following questions.

(1) Find the probability Pr[Y > y] for a real y (0 < y < 1).

(2) Find the expectation E[Y ].

(3) Find the variance Var[Y ].

(4) Let Z = 1/Y . Find the expectation of Z if it exists, otherwise explain the reason why

it does not exist.





6. 【記号論理学 (Symbolic logic)分野】

(1) ψ1 = (p → (q ∨ r)) ∧ (q → (r ∨ p)) ∧ (r → (p ∨ q)), ψ2 = ((p ∧ q) → r) ∧ ((q ∧ r) →p) ∧ ((r ∧ p) → q) , ψ3 = (p ∧ q ∧ r) ∨ ¬(p ∨ q ∨ r) とする．ψ1と ψ2の前提から ψ3が帰結することを resolution法により示せ．

(2) 人を要素とする集合を議論領域とし，四つの文 S1 =「x, y, zが誰であれ，xが yを好きで，かつ xが zを好きならば，yが zを信用する」，S2 =「yが誰であれ，yを信用する誰かが存在するのは yを好きな誰かが存在するときに限る」，S3 =「『全員が全員を信用しない』ということはない」，S4 =「x, yが誰であれ，xが yを信用するならば，yが x

を信用する」について考える．

(a) S1, S2, S3, S4から翻訳された述語論理式をそれぞれφ1, φ2, φ3, φ4とする．「xが yを好きである」を表す述語 L(x, y)と「xが yを信用する」を表す述語B(x, y)を用いてφ1, φ2, φ3, φ4を書け．

(b) φ1, φ2, φ3の前提から φ4が帰結するか否かについて，理由とともに述べよ．

(1) Let ψ1 = (p → (q ∨ r)) ∧ (q → (r ∨ p)) ∧ (r → (p ∨ q)), ψ2 = ((p ∧ q) → r) ∧ ((q ∧ r) →p) ∧ ((r ∧ p) → q), and ψ3 = (p ∧ q ∧ r) ∨ ¬(p ∨ q ∨ r). Show that ψ3 follows from the

premises ψ1 and ψ2 by the resolution method.

(2) Let the domain of discourse be a set of persons. Consider the four sentences: S1 =“For

any x, y, and z, if x likes y and x likes z then y believes in z,” S2 =“For any y, there

exists someone who believes in y only if there exists someone who likes y,” S3 =“It is

not the case that everyone believes in no one,” and S4 =“For any x and y, if x believes

in y then y believes in x.”

(a) Let φ1, φ2, φ3, and φ4 be the predicate logic formulas translated from S1, S2, S3, and

S4, respectively. Write φ1, φ2, φ3, and φ4 using the predicate L(x, y) for “x likes y”

and the predicate B(x, y) for “x believes in y.”

(b) State, with a reason, whether or not φ4 follows from the premises φ1, φ2, and φ3.

30

(Special subjects I)

(Instructions):

1. Do not open this cover sheet until the start of examination is announced.

2. 7 3 You are given 7 problem sheets including this cover sheet, and 3 answer sheets.

3. 3 1 Select 1 out of the following 3 fields and answer the questions.

field page 1 Circuit theory 2 2 Electronic circuits 4 3 Control engineering 6

4.

Your answers should be written on the answer sheets. Use one sheet for each question. You may continue to write your answer on the back of the answer sheets if you need more space. In such a case, indicate this clearly.

5.

6. Fill in the designated blanks at the top of each answer sheet with the course name, your examinee number, your name and the question number, and circle your selected field name,

電気回路

4問中 3問を選び，解答用紙欄に解答した問題番号を記入すること．

【問 1】図 1の回路について，以下の問いに答えよ．ただし，電源電圧Eの角周波数を ωとする．

(1) 電源から見たインピーダンスZを求めよ．

(2) 電流 Iを求めよ．

(3) 位相差 arg(I/E)が π/2となる条件を求めよ．図 1

【問 2】 2端子対回路Nと電圧Eの電源，インピーダンス ZGからなる図 2(a)の回路と，図 2(b)の回路が等価であるとき，以下の問いに答えよ．

(1) 2端子対回路Nのインピーダンス行列Zが，

Z =

[z11 z12z21 z22

]

で与えられるとき，E0およびZ0を求めよ．

(2) 2端子対回路Nが図 2(c)で与えられるとき，E0およびZ0を求めよ．

図 2

【問 3】図 3の回路について，以下の問いに答えよ．ただし，電源電流 J の角周波数を ωとする．(1) 抵抗RLの電流 Iと消費電力 P を求めよ．

(2) 次の 3つの場合についてそれぞれ，消費電力P が最大となる条件を求めよ．

(a) X1, X2 がともに可変である．ただし，RL < R0とする．

(b) X1が固定で，X2が可変である．

(c) X1が可変で，X2が固定である．図 3

【問 4】図 4の回路について，以下の問いに答えよ．ただし，R = 2 Ω，C = 1/6 F，L = 3 Hとする．

(1) 電源電圧 E が 0 Vで回路が定常状態に達した後，時刻 t = 0で Eを 2 Vに変化させた．t > 0における電流 i(t)を求めよ．

(2) 電源電圧 E が 4 Vで回路が定常状態に達した後，時刻 t = 0で Eを 8 Vに変化させた．t > 0における電流 i(t)を求めよ．図 4

2

Circuit Theory

Choose three out of the four questions and write the chosen question number on each answer sheet.

【Q1】Consider the circuit shown in Fig. 1, where the source voltage

E has the angular frequency ω. Answer the following questions.

(1) Find the impedance Z seen from the source.

(2) Find the current I.

(3) Determine the condition when the phase difference arg(I/E)

is π/2.Fig. 1

【Q2】 Consider the circuit shown in Fig. 2(a) consisting of 2-port

circuit N, source with voltage E and impedance ZG, and the circuit

shown in Fig. 2(b) equivalent to Fig. 2(a). Answer the following

questions.

(1) Find the voltage E0 and the impedance Z0 when the

impedance matrix for 2-port circuit N is defined by

Z =

[z11 z12z21 z22

].

(2) Find the voltage E0 and the impedance Z0 when the 2-port

circuit N is given by Fig. 2(c).Fig. 2

【Q3】 Consider the circuit shown in Fig. 3, where the source current J has the angular frequency ω.

Answer the following questions.

(1) Find the current I and the effective power P in

the resistance RL.

(2) In the following cases, find the requirements under

which the effective power P is maximized.

(a) Both X1 and X2 are variable for RL < R0.

(b) X1 is fixed, and X2 is variable.

(c) X1 is variable, and X2 is fixed.

Fig. 3

【Q4】 Consider the circuit shown in Fig. 4, where R = 2 Ω, C = 1/6 F and L = 3 H. Answer the

following questions.

(1) The source voltage E changes to 2 V at the time

t = 0 after the steady state for E = 0 V. Find the

current i(t) for t > 0.

(2) The source voltage E changes to 8 V at the time

t = 0 after the steady state for E = 4 V. Find the

current i(t) for t > 0.Fig. 4

3

ELECTRONIC CIRCUITSAug.17, 2017

All answers should be written on the answer sheets.

5

[1] [2] [3]

[1] 1 PID

(1) 1 PID

(2) 1 PI PID

dB

[2] u(t) y(t) v(t), w(t), z(t)

a, b, c, d, e, f, g

0

d

dtw(t) = −aw(t) + bu(t), v(t) = cw(t) + du(t),

d2

dt2z(t) = ev(t), y(t) = f v(t) + gz(t)

(1) u y

(2)

[3] s

(1) t ≥ 0 u(t) = 1 e1(t)

u(t) = t, u(t) = sin t

e2(t), e3(t) t < 0 0

0

(2) e1(t), e2(t), e3(t) K

|ei(t)| < ε

|ei(t)| < ε

ε

6

Control engineering

Answer the following questions ([1], [2] and [3]).

[1] Answer the following questions regarding a first-order lag element and a PID controller.

(1) Show transfer functions of a first-order lag element and a PID controller, respec-

tively. Use appropriate symbols to denote parameters in the transfer functions.

(2) Sketch, by broken-line approximation, frequency-gain characteristics of the first-

order lag element and the PID controller in a PI operation mode, respectively. Set

the horizontal axis as angular frequency in logarithmic scale, and the vertical axis

as gain in dB. Write down, in the sketches, the slope of each line segment and the

angular frequency at which the lines break.

[2] The input u(t) and the output y(t) of a system and the other signals in the system,

v(t), w(t) and z(t), have the following relationships:

d

dtw(t) = −aw(t) + bu(t), v(t) = cw(t) + du(t),

d2

dt2z(t) = ev(t), y(t) = f v(t) + gz(t).

Here the signals including the input and the output are all scalars, and a, b, c, d, e, f and

g are non-zero constants. Answer the following questions.

(1) Determine the transfer function from the input u to the output y.

(2) Define state variables appropriately and express the system with a state equation

and an output equation.

[3] Consider a system depicted by a block diagram below, where s denotes the Laplace

operator.

Answer the following questions.

(1) Suppose that the input u(t) = 1 is fed to the system when t ≥ 0. Find the error e1(t)

when a sufficiently long time has passed. Similarly, find the errors e2(t) and e3(t)

when a sufficiently long time has passed for the inputs u(t) = t and u(t) = sin t,

respectively. Here every input is 0 when t < 0 and all the initial values are 0.

(2) For each of e1(t), e2(t) and e3(t) obtained above, determine if |ei(t)| < ε is achieved

by adjusting the parameter K in the system. In addition, if it is achievable, describe

how the parameter should be adjusted. If not, explain its reason. If |ei(t)| < ε is

already satisfied without adjusting the parameter, state so. Here ε is an arbitrary

positive constant.

7

30

(Special subjects )

(Instructions):

1. Do not open this cover sheet until the start of examination is announced.

2. 19 3 You are given 19 problem sheets including this cover sheet, and 3 answer sheets.

3. 3 1 Select 1 out of the following 3 fields and answer the questions.

field page 1 Electromagnetism 2 2 Semiconductor device 8 3 Computer engineering 16

4.

Your answers should be written on the answer sheets. Use one sheet for each question. You may continue to write your answer on the back of the answer sheets if you need more space. In such a case, indicate this clearly.

5.

Fill in the designated blanks at the top of each answer sheet with the course name, your examinee number, your name and the question number, and circle your selected field name,

a, b ( b > a )

V ε

C u u

U

σr I

a < r < b

a b

V

a b

V

2

.(1)(2)

σ.(3)

a < r < b(4)

a b

V

a b

V

3

S

V

Tt

V t

V td

a

rJ, J

6

S

V

Tt

V t

V td

a

rJ, J

7　

3 1(a) C B E

1(b) (P)

(B) (As)

(1) (8)

1

1

τpE 10−9 s

μpE 10−2 m2/Vs

τnB 4× 10−8 s

μnB 0.1 m2/Vs

WB 500 nm

kT 0.025 eV

ni 1.5× 1016 m−3

(1) C B E

(2) nB0

(3)

αE 3

(4) n′B(x)

∂2n′B(x)

∂x2=

n′B(x)

L2nB

10

LnB x

1(a)

WB n′B(0) = KnB0 (K � 1)

n′B(x)

n′B(WB) = 0 sinh

(5) LnB � WB

(6) (4) αB

y 0 cosh y ≈ 1 + y2

2

(7) ( )α

1 3

(8) β

11　

3 Consider a bipolar junction transistor shown in Fig. 1(a). The regions C, B, and E are emitter,

base, and collector, respectively. These regions were doped with phosphorous, boron, and arsenic.

The doping profiles of these dopant are shown in Fig. 1(b). Table 1 shows physical parameters of

the regions. Answer the following questions (1) - (8).

Fig. 1

Table 1 Physical parameters of the transistor

Parameter Symbol Value

Hole lifetime in the emitter τpE 10−9 s

Hole mobility in the emitter μpE 10−2 m2/Vs

Electron lifetime in the base τnB 4× 10−8 s

Electron mobility in the base μnB 0.1 m2/Vs

Width of the base neutral region WB 500 nm

Thermal energy at operation temperature kT 0.025 eV

Intrinsic carrier concentration at operation temperature ni 1.5× 1016 m−3

(1) Describe the majority carrier and its density in each of the three regions, C B and E.

(2) Evaluate the minority carrier density nB0 in the base region.

(3) Assuming that the electron current and the hole current flowing through the emitter/base

junction are proportional only to the carrier densities in the regions which inject the carriers,

evaluate the emitter injection efficiency αE . Answer to the third decimal place.

(4) When the transistor is in the normal operation, the minority carrier distribution in the base

is obtained by solving the diffusion equation

∂2n′B(x)

∂x2=

n′B(x)

L2nB

,

14

where LnB is the diffusion length of minority carriers in the base, x is the distance in the base

from the edge of the emitter side along the axis toward the collector as shown in Figs. 1(a)

and 1(b). Assume the width of the base to be WB. Obtain a solution n′B(x) of the above

diffusion equation for the case where the emitter/base junction is biased to have n′B(0) =

KnB0 (K � 1). You may assume n′B(WB) = 0. Use hyperbolic function, sinh, to answer.

(5) Prove LnB � WB for this transistor.

(6) Using the answer for the question (4) obtain an expression of the base transport efficiency

αB of the transistor. You may assume that the electric field in the base is zero. Use the

approximation cosh y ≈ 1 + y2

2 when y is close to zero to simplify the expression.

(7) Evaluate the current transport efficiency (i.e., the base-common current-amplification factor)

α of the transistor. Answer to the third decimal place. Assume that the collector efficiency

is unity.

(8) Evaluate the emitter-common current-amplification factor β of this transistor.

15

計算機工学

16

計算機工学

17

Computer engineering

18

Computer engineering

19

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