11. (10%) How many zero entries are there in the inverse of the following matrix?__________

12. (10%) Give a basis for the vector space of the linear transformations from the vector space R3 of real triples to the vector space R? of real pairs:_____________.

1. (10%) Given a position vector below, determine the velocity, speed, acceleration, the tangential and normal component of acceleration, the curvature, and the unit tangent vector, unit normal vector. F-3ti-2j-2k

2. (10%) Find the Laplace transform of At)-[sin(t)-cos(t)]2

3. (10%) Solve the differential equation, y"-ty'+y=1, y(0)=1, y'(0)=2

4. (10%) Produce a matrix P that diagonalizes the matrix below:  

5. (10%) Find the general solutions of

6. (25%) Let u-u(x，y，t) be a function of the space (x.y) and the time t. S=S(s,f) is a function of x and t, and cl and co are constant. Determine whether the method of separation of variables is applicable to the following partial differential equations or not (each has 5 %). The rationale behind your answer needs be given; otherwise, it will not be credited.

7. (25%) Let u=u(xy) be a function of x and y. (x), g(x), f1(y), and g1(y) are given; K and L are positive constant. Solve the problem:  

Please select all the correct answer(s) to question 1 to 10. Note that there could be O to 5 correct answ wers. If non answer is correct, answer "none". If you do not wish to answer a question, leave it blank. All 10 answer rs m must be written on the first page of your answer book, and the answer to question 1 must be in the first line, the answer to question 2 must be in the second line, and so on. If you fail to follow these rules then your answers will be ignored. ne of the a 1. (3 points) Let f(n) be the number of additions in the following algorithm.  1.f(n)=2. f(n)=O(n2logn) 3. f(n)=O(n3) 4. f(n)=O(n3logn) 5. f(n)=

2. (3 points) Given two sorted list A and B (in increasing key order), the following recursive algorithn merges A and B into a sorted list C (also in increasing key order). ㆍ If either A or B is empty then the result is the other list. ㆍ If both A or B are not empty, we compare the keys of the first nodes of A and B, and select the smaller one (denoted as s), and remove it from the list. Then we recursively merge the remaining parts of A and B into a new sorted list C', then we concatenate s with C' into the iral sorted list Let f(n, m) be the minimum number of comparisons of this algorithm, where n and m are the numbers of nodes in A and B respectively. f(n,m) is? 1.Ω(n) 2. Ω(m) 3. Ω(n +m) 4. Ω(mn) 5.Ω(mar.(logm +logn))

110年 - 110 國立臺灣大學_碩士班招生考試_資訊工程學研究所:數學(含線性代數、離散數學)#102151

109年 - 109 國立臺灣大學_碩士班招生考試_資訊工程學研究所:數學(含線性代數和離散數學)#105828

108年 - 108 國立臺灣大學_碩士班招生考試_資訊工程學研究所:數學(含線性代數和離散數學)#106050