Mathematical Science 基礎数理

(1)

For each question, choose one correct answer and write its symbol (A–E) in the box.

以下の各問に対して正しい答えを選び，その記号

(A–E)

を枠の中に書いてください．

Q1 Let X and Y be arbitrary n × n non-singular matrices. Which equation is correct? Here the inverse and the transposition of a non-singular matrix A are denoted by A ⁻¹ and A ^⊤ , respectively.

X

および

Y

を任意の

n × n

非特異行列とする．正しい等式を選べ．ただし

A ⁻ ¹

および

A ^⊤

はそれぞれ

A

の逆行列および転置行列を表している．

(A) (X ^⊤ + Y ⁻ ¹ ) ^⊤ = X + (Y ^⊤ ) ⁻ ¹ (B) (X ^⊤ + Y ⁻ ¹ ) ^⊤ = (X ⁻ ¹ ) ^⊤ + Y ^⊤ (C) (X ^⊤ + Y ⁻¹ ) ^⊤ = X ^⊤ + (Y ^⊤ ) ⁻¹ (D) (X ^⊤ + Y ⁻ ¹ ) ^⊤ = (X ^⊤ ) ⁻ ¹ + (Y ⁻ ¹ ) ^⊤ (E) None of these

Q2 The gradient vector of a function f : R ⁿ → R is defined as ∇f (x) = h _∂f(x)

∂x

₁

, . . . , ^∂f(x) _∂x

_n

i ⊤

. Let f (x) = kx + bk ² , where x ∈ R ⁿ , k · k is the Euclidean norm and b ∈ R ⁿ is a constant. Which equation is correct?

関数

f : R ⁿ → R

の勾配ベクトルは

∇f (x) = h _∂f(x)

∂x

₁

, . . . , ^∂f(x) _∂x

n

i ⊤

と定義される．いま，f

(x) = kx + bk ²

とおく．ただし，x

∈ R ⁿ , k · k

はユークリッドノルム，b

∈ R ⁿ

は定数とする．次の選択肢のうち正しい等式を選べ．

(A) ∇f (x) = x + b

(B) ∇f ( x ) = x − b

(C) ∇f (x) = 2x + 2b

(D) ∇f (x) = 2x − 2b

(E) None of these

(2)

Q3 Let x and b be a random variable and a constant scalar, respectively. Denote the expectation operator by E(·). Which equation is correct?

x

を確率変数，bを定数とする．期待値演算子を

E(·)

で表すとする．次の選択肢のうち正しい等式を選べ．

(A) E ((x + b) ² ) = (E (x)) ² + 2bE(x) + b ² (B) E ((x + b) ² ) = E(x ² ) + 2bE(x) + b ²

(C) E ((x + b) ² ) = E(x ² ) + (E(x)) ² + 2bE(x) + b ² (D) E ((x + b) ² ) = E(x ² ) − (E(x)) ² + 2bE(x) + b ² (E) None of these

Q4 Let k and n be integers such that k ≥ 1, n ≥ 1. How many combinations of non-negative integers i 1 , i 2 , . . . , i k are there such that i 1 + i 2 + · · · + i k = n?

k

と

n

を

1

以上の整数とする．非負の整数

i 1 , i 2 , . . . , i k

で

i 1 + i 2 + · · · + i k = n

となる組み合わせはいくつあるか．

(A) ⁿ _k

(B) ^n+k−1 _n

(C) ^n+k+1 _n

(D) ⁿ⁻¹ _k

(E) None of these

Q5 Let us consider a function f such that y = f (x 1 , x 2 ), where x 1 , x 2 , y ∈ {0, 1}. How many different functions f are there?

y = f (x 1 , x 2 )

となる関数

f

を考える．ただし，x

1 , x 2 , y ∈ {0, 1}

とする．相異なる関数

f

はいくつあるか．

(A) 2 (B) 4 (C) 8 (D) 16 (E) None of these

Q6 If the binary (1.101) 2 is converted to the decimal, which of the followings is correct?

2

進数の

(1.101) 2

を

10

進数に変換したとき，次のうちどれが正しいか？

(A) (1.125) 10 (B) (1.375) 10 (C) (1.625) 10 (D) (1.750) 10 (E) None of these

Q7 Define the sequence {T n } by T 1 = 1, T 2 = 1, T 3 = 2 and T n = T _n−1 + T _n−2 + T _n−3 for n ≥ 4. What is the smallest n such that T n ≥ 100?

数列

{T n }

を以下の通り定義する．T

1 = 1，T 2 = 1，T 3 = 2，T n = T _n−1 + T _n−2 + T _n−3 (n ≥ 4). T n ≥ 100

となる最小の

n

はいくつか．

(A) 6 or smaller (B) 7 (C) 8 (D) 9 (E) 10 or larger

(3)

Q8 What is the number of integers between 1 and 100 whose binary representation has one or two 1’s?

1

から

100

までの整数のうち，2進数で書いたときに

1

の個数が

1

個または

2

個になるものは全部でいくつあるか．

(A) 20 or less (B) Between 21 and 25 (C) Between 26 and 30 (D) Between 31 and 35 (E) 36 or more

Q9 Suppose that x + 1

x = 3. What is x ³ + 1 x ³ ? x + 1

x = 3

とする．x

³ + 1

x ³

はいくつか．

(A) 12 (B) 16 (C) 18 (D) 27 (E) None of these

Q10 How many permutations of the letters A, B, C, D, E, F and G contain the string ABCD?

文字

A, B, C, D, E, F, G

の順列で文字列

ABCD

を含むものはいくつか.

(A) 6 or less (B) Between 7 and 12 (C) Between 13 and 24 (D) Between 25 and 48 (E) 49 or more

Q11 Let (x,y) be the point with x-coordinate x and y-coordinate y. How many ways are there to travel from the point (3,2) to the point (6,6) by taking steps one unit in the positive x direction, or one unit in the positive y direction?

x

座標が

x

であり, y座標が

y

である点を

(x,y)

とする. 点

(3,2)

から点

(6,6)

まで移動する方法は何通りあるか. ただし移動はステップからなり、各ステップは

1

だけ正の

x

方向に進む,もしくは、1だけ正の

y

方向に進む,のいずれかとする.

(A) 6 or less (B) Between 7 and 12 (C) Between 13 and 24 (D) Between 25 and 48 (E) 49 or more

Q12 Let A = {n ∈ N | 2 ⁿ < 10n ² }, where N is the set of natural numbers {1, 2, 3, . . .}. We write |A| for the cardinality of A. Choose a correct one.

A = {n ∈ N | 2 ⁿ < 10n ² }

とする．ただし

N

は自然数の集合

{1, 2, 3, . . .}

とする．集合

A

の要素の個数を

|A|

と書く．正しいものを選択せよ．

(A) |A| ≤ 8

(B) 8 < |A| ≤ 10

(C) 10 < |A| ≤ 12

(D) 12 < |A| ≤ 14

(E) 14 < |A|

(4)

Q13 Let x and y denote real numbers. Which one is correct?

x, y

を実数とする．正しいものを選べ．

(A) ∀x∃y x = sin y (B) ∃x∀y x ² > y − 1 (C) ∃x cos 2x = 1 (D) ∀x∀y x ² + y ² > 0 (E) None of these

Q14

Suppose that log ₂ x + log ₂ y = 3. Which one is correct?

log ₂ x + log ₂ y = 3

とする．正しいものを選べ．

(A) 2x + y ≤ 8 (B) |2x + y| ≤ 8 (C) 2x + y = 8 (D) 2x + y ≥ 8 (E) None of these

Q15 A fair die is rolled 5 times. Let x/6 ⁵ be the probability of getting exactly 2 fives. Which equation is correct?

正しいサイコロを

5

回投げる．ちょうど

2

回，5が出る確率を

x/6 ⁵

とする．どの式が正しいか.

(A) x ≤ 100 (B) 100 < x ≤ 500 (C) 500 < x ≤ 1000 (D) 1000 < x ≤ 2000 (E) None of these

(5)

Mathematical Science 基 礎 数 理

For each question, choose one correct answer and write its symbol (A–E) in the box.

(A–E)

Q1 Let X and Y be arbitrary n × n non-singular matrices. Which equation is correct? Here the inverse and the transposition of a non-singular matrix A are denoted by A −1 and A ⊤ , respectively.

X

Y

n × n

A − 1

A ⊤

A

(A) (X ⊤ + Y − 1 ) ⊤ = X + (Y ⊤ ) − 1 (B) (X ⊤ + Y − 1 ) ⊤ = (X − 1 ) ⊤ + Y ⊤ (C) (X ⊤ + Y −1 ) ⊤ = X ⊤ + (Y ⊤ ) −1 (D) (X ⊤ + Y − 1 ) ⊤ = (X ⊤ ) − 1 + (Y − 1 ) ⊤ (E) None of these

Q2 The gradient vector of a function f : R n → R is defined as ∇f (x) = h ∂f(x)

∂x

, . . . , ∂f(x) ∂x

i ⊤

. Let f (x) = kx + bk 2 , where x ∈ R n , k · k is the Euclidean norm and b ∈ R n is a constant. Which equation is correct?

f : R n → R

∇f (x) = h ∂f(x)

∂x

, . . . , ∂f(x) ∂x

i ⊤

(x) = kx + bk 2

∈ R n , k · k

∈ R n

(A) ∇f (x) = x + b

(B) ∇f ( x ) = x − b

(C) ∇f (x) = 2x + 2b

(D) ∇f (x) = 2x − 2b

(E) None of these

Q3 Let x and b be a random variable and a constant scalar, respectively. Denote the expectation operator by E(·). Which equation is correct?

x

E(·)

(A) E ((x + b) 2 ) = (E (x)) 2 + 2bE(x) + b 2 (B) E ((x + b) 2 ) = E(x 2 ) + 2bE(x) + b 2

(C) E ((x + b) 2 ) = E(x 2 ) + (E(x)) 2 + 2bE(x) + b 2 (D) E ((x + b) 2 ) = E(x 2 ) − (E(x)) 2 + 2bE(x) + b 2 (E) None of these

Q4 Let k and n be integers such that k ≥ 1, n ≥ 1. How many combinations of non-negative integers i 1 , i 2 , . . . , i k are there such that i 1 + i 2 + · · · + i k = n?

k

n

1

i 1 , i 2 , . . . , i k

i 1 + i 2 + · · · + i k = n

(A) n k

(B) n+k−1 n

(C) n+k+1 n

(D) n−1 k

(E) None of these

Q5 Let us consider a function f such that y = f (x 1 , x 2 ), where x 1 , x 2 , y ∈ {0, 1}. How many different functions f are there?

y = f (x 1 , x 2 )

f

1 , x 2 , y ∈ {0, 1}

f

(A) 2 (B) 4 (C) 8 (D) 16 (E) None of these

Q6 If the binary (1.101) 2 is converted to the decimal, which of the followings is correct?

2

(1.101) 2

10

(A) (1.125) 10 (B) (1.375) 10 (C) (1.625) 10 (D) (1.750) 10 (E) None of these

Q7 Define the sequence {T n } by T 1 = 1, T 2 = 1, T 3 = 2 and T n = T n−1 + T n−2 + T n−3 for n ≥ 4. What is the smallest n such that T n ≥ 100?

{T n }

1 = 1，T 2 = 1，T 3 = 2，T n = T n−1 + T n−2 + T n−3 (n ≥ 4). T n ≥ 100

n

(A) 6 or smaller (B) 7 (C) 8 (D) 9 (E) 10 or larger

Q8 What is the number of integers between 1 and 100 whose binary representation has one or two 1’s?

1

100

1

1

2

(A) 20 or less (B) Between 21 and 25 (C) Between 26 and 30 (D) Between 31 and 35 (E) 36 or more

Q9 Suppose that x + 1

x = 3. What is x 3 + 1 x 3 ? x + 1

x = 3

3 + 1

x 3

(A) 12 (B) 16 (C) 18 (D) 27 (E) None of these

Q10 How many permutations of the letters A, B, C, D, E, F and G contain the string ABCD?

A, B, C, D, E, F, G

ABCD

(A) 6 or less (B) Between 7 and 12 (C) Between 13 and 24 (D) Between 25 and 48 (E) 49 or more

Q11 Let (x,y) be the point with x-coordinate x and y-coordinate y. How many ways are there to travel from the point (3,2) to the point (6,6) by taking steps one unit in the positive x direction, or one unit in the positive y direction?

x

Mathematical Science 基礎数理

Q1 Let X and Y be arbitrary n × n non-singular matrices. Which equation is correct? Here the inverse and the transposition of a non-singular matrix A are denoted by A ⁻¹ and A ^⊤ , respectively.

A ⁻ ¹

A ^⊤

(A) (X ^⊤ + Y ⁻ ¹ ) ^⊤ = X + (Y ^⊤ ) ⁻ ¹ (B) (X ^⊤ + Y ⁻ ¹ ) ^⊤ = (X ⁻ ¹ ) ^⊤ + Y ^⊤ (C) (X ^⊤ + Y ⁻¹ ) ^⊤ = X ^⊤ + (Y ^⊤ ) ⁻¹ (D) (X ^⊤ + Y ⁻ ¹ ) ^⊤ = (X ^⊤ ) ⁻ ¹ + (Y ⁻ ¹ ) ^⊤ (E) None of these

Q2 The gradient vector of a function f : R ⁿ → R is defined as ∇f (x) = h _∂f(x)

, . . . , ^∂f(x) _∂x

. Let f (x) = kx + bk ² , where x ∈ R ⁿ , k · k is the Euclidean norm and b ∈ R ⁿ is a constant. Which equation is correct?

f : R ⁿ → R

∇f (x) = h _∂f(x)

, . . . , ^∂f(x) _∂x

(x) = kx + bk ²

∈ R ⁿ , k · k

∈ R ⁿ

(A) E ((x + b) ² ) = (E (x)) ² + 2bE(x) + b ² (B) E ((x + b) ² ) = E(x ² ) + 2bE(x) + b ²

(C) E ((x + b) ² ) = E(x ² ) + (E(x)) ² + 2bE(x) + b ² (D) E ((x + b) ² ) = E(x ² ) − (E(x)) ² + 2bE(x) + b ² (E) None of these

(A) ⁿ _k

(B) ^n+k−1 _n

(C) ^n+k+1 _n

(D) ⁿ⁻¹ _k

Q7 Define the sequence {T n } by T 1 = 1, T 2 = 1, T 3 = 2 and T n = T _n−1 + T _n−2 + T _n−3 for n ≥ 4. What is the smallest n such that T n ≥ 100?

1 = 1，T 2 = 1，T 3 = 2，T n = T _n−1 + T _n−2 + T _n−3 (n ≥ 4). T n ≥ 100

x = 3. What is x ³ + 1 x ³ ? x + 1

³ + 1

x ³

Q12 Let A = {n ∈ N | 2 ⁿ < 10n ² }, where N is the set of natural numbers {1, 2, 3, . . .}. We write |A| for the cardinality of A. Choose a correct one.

A = {n ∈ N | 2 ⁿ < 10n ² }

(A) ∀x∃y x = sin y (B) ∃x∀y x ² > y − 1 (C) ∃x cos 2x = 1 (D) ∀x∀y x ² + y ² > 0 (E) None of these

Suppose that log ₂ x + log ₂ y = 3. Which one is correct?

log ₂ x + log ₂ y = 3

Q15 A fair die is rolled 5 times. Let x/6 ⁵ be the probability of getting exactly 2 fives. Which equation is correct?

x/6 ⁵