110年 - [非官方正解]110 國立政治大學_碩士班暨碩士在職專班招生考試_資訊科學系:計算機數學#108089

科目：研究所、轉學考(插大)◆計算機數學 | 年份：110年 | 選擇題數：25 | 申論題數：5

試卷資訊

選擇題 (25)

1. (3%) Let Φ denote an empty set. Which of the following statements is true? (A)(B)(C)(D)|Φ|=1

2. (3%) Let A,B, C be finite sets. Which of the following statements is false? (A) IA ∪B| = |A| + IB -IA∩B (B) P(A ∪B)=P( A ) ∪P( B ) (C) A-(B ∪C)= (A-B)-C (D)

複選題

3. (3%) Let A, B, C be finite sets. Which of the following statements is true? (A) If A∪C = B ∪ C, then A = B (B) If A ∩ C = B∩ C, then A = B (C) If A -C = B -C, then A = B(D)

4. (3%) Let p= 101 be a prime and a = 2 be a primitive root modulo p. Please determine the value of (a⁷)⁹ mod p. (A) 88 (B) 89 (C) 90 (D) 91

5. (3%) Let a,b,k,n be integers. Which of the following statements is not coTect? (A) a ≡b (mod n) ka≡ kb (mod n) (B) a≡b (mod kn) a≡b (mod n) (C) a≡b (mod n) ka≡ kb (mod kn) (D) a≡b (mod n)a≡b (mod kn)

6. (3%) Which of the following statements is true? (A)(B)(C) (D)

7. (3%) Which of the following statements is false? (A) (B) (C) (D)

8. (3%) How many bit strings are there containing exactly five 0s and fourteen Is if every0 must be immediately followed by two ls? (A) 112 (B) 124 (C) 126 (D) 132

9. (3%) Find the number of ways in which nine identical blocks can be given to four children A, B, C, D, where child A gets at most two blocks. (A) 100 (B) 136 (C) 142 (D) 164

10. (3%) How many ways are there to put five temporary employees into four identical offices? (A) 48 (B) 51 (C) 56 (D) 60

11. (3%) Let x= a+ bk be the solution of the system of congrucnces x = 7 (mod 9), x≡ 4 (mod 12),and x≡16 (mod 21), where a and b are the smallest positive integer, k e Z. Which is the correct value of (2a+b) mod 11? (A)9 (B) 7 (C) 4 (D) 6

12. (3%) Let A be a set with n clements. Which of the followings is the number of binary relations on A which are symmetric and antisymmetric, and not irreflexive? (A)2ⁿ(B) 2ⁿ -1 (C)(D)

13. (3%) Let A be a set with n elements. Which of the followings is the number of binary relations on A which are reflexive and not symmetric? (A) (B)(C)(D)

14. (3%) Suppose that someone starts a chain letter. Each person who receives the letter is asked to send it to four other people. Let k be the number of people who have seen the letter, including the first person. Suppose no one receives more than one letter and the chain letter ends after there have been 100 people who read it but did not send it out. which is the correct value of k mod 11? (A) 0 (B) 3 (C) 4 (D) 8

15. (3%) Let G be a connected bipartite planar simple graph with 12 vertices, and let e be the maximum value of the number of edges in G. Which of the followings is the value of e mod 11? (A) 3 (B) 7 (C) 9 (D) 10

16. (3%) Consider the following string aaabbcddddddeeeeeffgggggghhhhiii. Suppose we at least require k bits to encode the string. Which of the followings is the value of k mod 11? (A)1(B) 6 (C) 5(D) 10

17. (3%) Suppose the solution of the recurrence relation with a₀= -2, a₁ = 0, and a₂ =5, is a_n = . Which of the following is the value of(a + b+ c+d₀+d₁+d₂) mod 11? (A) 9 (B) 8 (C) 7 (D) 6

18. (3%) Determine the number of solutions for the linear systcm (A) no solution (B) a unique solution (C) exactly two solutions (D) infinite many solutions

19. (3%) If det ，what is the value of k?(A) 2 (B) 3 (C) 4 (D) 6

20. (3%) Which of the following statements is true? (A) If none of the vectors in the set S = {v₁, v₂,v₃} in R₃ is a multiple of one of the other vectors, then S is linearly independent. (B) If A is a subspace, then its complement is also a subspace. (C) If a square matrix A has independent columns, so does A². (D) Every vector space has at least two distinct subspaces.

21. (3%) Which of the following statements is a linear transformation? (A) L(x,y,z) = (x + y,y²,2z)(B) (C) (D) L(x,y,z) =

22. (3%) Which of the following statements is true? (A) A square matrix with linearly independent column vectors is diagonalizable. (B) If A and B are diagonalizable, so is AB. (C) If A bc diagonalizable and AB = BA, then B is diagonalizable. (D) If A is a 3 ✖3 matrix with distinct eigenvalues 0,1,2, then the matrix (A + I) must be invertible.

23. (3%) Which of the following statements is true? (A) Let S be a subset of a vector space. Then S = (B) Let V be a subset of Rⁿ, and if V is orthogonal to W, then is orthogonal to. (C) If u and v are vectors in Rⁿ, and if distance from u to v equals the distance from u to -v, then and v are orthogonal. (D) If u, v and w are vectors in Rⁿ, and if u is orthogonal to v +w, then u is orthogonal to v and w.

24. (3%) Given the following data points{(-1,1),(1,3),(2,3)}, supposc that the lincar function y =x is the approximation to fit the data in the least squares sensc, please find the value of (a+ b+ c+ d) mod 11. (A) 1 (B) 3 (C) 7(D) 10

25.(3%) Let S=spanf{(1 3 1 1)^T,(1 1 1 1)^T,(-1 5 2 2)^T} be a subspace of R⁴. Suppose w₁= , is aothonormal basis of S. Please find the value of (a₁ + a₂+ a₃+ a₄)+ (b₁ +b₂+b₃ +b₄)+(c₁+c₂ +c₃ +c₄)? (A) 5 (B) 6 (C) 7 (D) 8

申論題 (5)

26. (4%) Use generating functions to prove Pascal's identity:C(n,r)=C(n-1,r)+C(n-1,r-1) when n and r are positive integers with r ＜ n.(Hint: Use the identity (1 +x)ⁿ=.]

27. (5%) Let (X_i, y_i), i = 1,2,3,4,5, be a set of five distinct points with integer coordinates in the xy plane. Show that there is at least one of the midpoints of the segment joining of these points which has integer coordinates.

(a) (6%)Find [T]_βand [T]_γ.

(b) (6%)Find the matrix S such that.