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102 年 - 102 國立中山大學_碩士班招生考試_電機系(甲、丙、丁、戊、己組):工程數學甲#110510 

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1. (7%) Find the Laurent series representation of a function
with center at a = j in the domain 1 <|z -j |< 2, j = 63083aecc3d28.jpg.


2. (8%) Evaluate the following integral:
63083b107848b.jpgwhere C denotes a counterclockwise simple closed contour |z| = 3.


3. (15%) Compute the Fourier transform63083b4180d6f.jpg of a signum function f(t) defined as  63083ca694321.jpgEach calculation step is required for obtaining the credit. 


4.(15%)下面的問題共有三個子題,只要寫出每個子題的答案即可(不需寫出計算過程),例如: (a) β=1,γ=2。 Let a1 = [1 0 1 α]T,a2=[I β 2 2]T, and a3 = [-2 3 γ -4]T be three vectors in R4, where a, β, and γ are three real parameters, and denote A := [a1 a2 a3] 63083db0c8d30.jpg.


(a) (4%) Suppose a is not a positive integer. Find real β and γsuch that{a1, a2, a3} is a linearly dependent set.


(b) (5%) Now let a = 2, β = -1,γ = -5, and let x be a nonzero vector in the null space N
(A) of A. Find the value of k to satisty ||x|| 1 + 2||x||∞ + k||x||2 = 0.


(c) (6%) Now let a = 2, β = -1, γ= -5, and let d denote the distance between vector [1 4 0]T and R(AT), the range space of AT. Compute the value of d.


5.(10%)下面的問題共有二個子題,只要寫出每個子題的答案即可(不需寫出計算過程),例如: (a)θ=30或θ=π/6。
Consider the inner product space C[0, 1] with (f, g) :=63083f211fb9e.jpg f(x)g(x)dc and the nomm 63083f3ecf7c0.jpg. Denote S := span (1, x} as a subspace of C[0, 1].


(a) (4%) Compute the angle θ, taken value in [0, π /2), between 1 and x.

8.【題組】(b) (6%) Find a vector u(x) in C[0, 1], so that {1, u(x)} forms an orthonormal basis for S.


6. (20%) Consider the following system of differential equations: 630841e36cfdc.jpg where630841fd16afb.jpgare constant coefficients.


(a) (5%) Suppose u ≡ 0 and the equations are driven by non-zero initial conditions. Determine the conditions on the coefficients63084230a6682.jpg such that 6308427c67d82.jpg


(b) (10%) Let the initial conditions be equal to zero. For the values 630842af69c46.jpg, 630842c799822.jpg= -1, and
calculate the response y(t) = 2x1(t) - x2(t). Determine at what time the peak value of y occurs.


(c) (5%) For the values 6308433211dbc.jpg = -2, and u(t) = sin(t), calculate the steady-state response of y(t)=x1(t)-x2(t).


7. (15%) Consider the region R enclosed by the x-axis, x = I and y = x3, as illustrated below


(a) (3% + 2%) Find the flux of 630843a0cff2f.jpg = (1 + y2)j out of R through the two sides C1 (the horizontal segment) and C2 (the vertical segment).


(b) (10%) Find the flux of 630843cf71245.jpg= (1 + y2)j out of the third side C3.

8.(10%) Consider the following Lyapunov equation
 where A is a (n-dimensional) real square matrix, and X, Q are real symmetric matrices. 


(a) (5%) Suppose all eigenvalues of A have negative real parts. Show that X =63084478514fe.jpgdτ a solution to the Lyapunov equation.

15.【題組】(b) (5%) Suppose Q is positive definite and the Lyapunov equation has a positive definite solution X. Show that all eigenvalues of A have negative real parts.



一、請根據題目的敘述,辨別臺灣地形的特色,選出正確的答案。 【題組】4、中間低平,四周較高的地形。(A)山地 (B)丘陵 (C)平原 (D)盆地 ...

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102 年 - 102 國立中山大學_碩士班招生考試_電機系(甲、丙、丁、戊、己組):工程數學甲#110510-阿摩線上測驗

102 年 - 102 國立中山大學_碩士班招生考試_電機系(甲、丙、丁、戊、己組):工程數學甲#110510