所屬科目：學士後西醫◆資訊科學
1. A uniform disk of mass m and radius R (so the moment of inertia ) is rotating about its symmetric axis at an angular velocity ω0. A ring of the same mass m and radius R originally at rest suddenly drops on the disk. The coeffieient of kinetic friction between the disk and the ring is μ. How long does it take for the ring to reach a final angular velocity ω and rotate together with the disk as shown in the right figure? The gravitational acceleration is g. (A)(B)(C) (D)(E)
2. Dectermine the height of the center of mass of a solid uniform pyramid that has four triangular faces and a square base with equal sides all of length S as shown in the figure. The height is measured from the bottom of the pyramid.(A)(B)(C) (D)(E)
3. Two speakers both produce sound wave of power = 400πWatt, frequency = 170 Hz, and sound velocity = 340 m/s. They act as point sources and propogate in all directions. The output of the two speakers are synchronized and have the same phase. They are placed at different locations and the distances between the speakers and the observer P are d1 = 10 m and d2 = 5 m, respectively. What is the intensity of the sound wave detected by the observer? (The effect of interference should be considered.)(A) 1 W/m2 (B) 2 W/m2 (C) 3 W/m2 (D) 5 W/m2 (E) 9 W/m2
4. Two thin lenses form a compound lens system. The front focal length (FFL) of this compound lens system is defined as the distance between a point source and the front lens so that the light passing this system will become parallel. In the figure, the two lenses have focal length f1 = 10 cm and f2 = -10 cm, respectively, and are separated by a distance d = 5 cm. What is the FFL of this compound lens system? (A) 5 cm (B) 10 cm (C) 12.5cm (D) 25 cm (E) 30 cm
6. In the circuit, if the switch S is closed for a long time, and is then opened. Just after the switch is opened, what is the potential difference across the inductor L? (A) ε(B) (C) (D)(E)
7. What is the root-mean-square value of a triangular wave with its maximum voltage at ±Vmax and a period of T. (A)(B) (C) (D) (E)
8. A solid rod with length L, resistance R and mass M initially rotates with angular velocity ω in a uniform magnetic field B. The two ends of the rod are connected by a conducting wire without resistance. According to Lenz's Law, the induced current will oppose the change of magnetic flux. In this case, the induced current will produce a torque to slow down the rotation speed. What is the torque? Take the pivot point as the origin.(A)(B)(C)(D) (E)0
10. A cord is wrapped on a pulley (treated as a uniform solid disk) of mass M and radius R as shown in the figure. To one end of the cord, a block of mass M is connected and to other end in ( A ) a force of 2Mg and in ( B ) a block of mass 2M. The cord does not slip relative to the pulley as the block falls. Let angular acceleration of the disk in A and B be aa and ae, respectively. So, M, R M, R 2M1g 2M (A)αA:αB = 3:1 (B)αA: αB = 1:1 (C)αA:αB = 2:1 (D)αA:αB = 5:2 (E)αA:αB = 7:3
11. The average power transferring along the string can be written as P= μvaωb Ac/2, where μ is the mass density of the string, ω the angular frequency of the string wave, A the amplitude, v the speed. What is the value of a+b+c ? (A) 1 (B) 2 (C) 3 로 (D)4 (E) 5
13. Consider two different diatomic ideal gases (N1 and N2) kept in two separate volumes (V1 and V2) at the same temperature T and both volumes thermally isolated to the environment. What is the change of entropy in the mixing process? (A)(B) (C)(D) (E) None of the above.
14. There is an electric dipole in an external uniform E-field, as shown below. Examine the following expression for the work done on the dipole:
where U(θ)=0 as θ=π/2. (A) This expression is correct (B) The negative sign after the second equal sign is wrong (C) The negative sign after the last equal sign is wrong (D) Both negative signs should be removed (E) The dipole has a maximum potential cnergy when it is parallel to (namely,θ=0)
15. Referring to the DC circuit shown below with the capacitor C been fully charged, the switch S is connected "ON" at t= O so that the crrent I2 starts to flow through the resistor R2. How does the current function I2(t) depend on time? (A) (B)(C) (D)(E)
17. The double-slit experiment also exhibits diffraction effects, as the slits have a finite width D. According to the calculated intensity Iθ = I(θ) of interference pattern, as shown below, with the parameters: d = pD=gλ, where d is the distance between slits and λ is the wavelength of light. Determine what the value of p/q is. (A) 1/12 (B) 1/11 (C) 1/10 (D) 1 (E) 11
18. A particle of mass 1 kg is confined to move in the one-dimensional region x ＞ 0. It is subject to a potential energy function (Joules). If the particle is initially placed at its mechanical equilibrium position and then given a small displacement, what is the period of its subsequent oscillations? (A)(B)(C)(D)(E) None of the above
21. John skates on the ground subject to a friction force. For a stationary observer A relative to the ground, he observes that John has an initial velocity +5 m/s in the x- direction and finally stops after traveling a distance of 12.5 m. This phenomenon is observed by the other observer B, who is riding a bus that moves at a constant speed of +2 m/s in the x-direction relative to the stationary ground. Therefore, the observer B sees that John's velocity changes from +3 m/s to -2 m/s in the x-direction. Consider the kinematic equation that, where △x is the distance travelled； which of the following statements is true? (A) From the observer B's point of view, since , B would measure a different deceleration than what A would have measured. (B) The kinematic equation only works in A's reference frame. (C) The kinematic equation only works in B's reference frame. (D) The inconsistency of the kinematic equation can be amended by the special relativity. (E) None of the above is true.
22. For a 2D monatomic ideal gas, which of the following statements is true? (A) The probability density function of finding particles of the velocity of = is not only lincarly proportional to the Boltzmann factor but also linearly proportional to y. (B) The probability density function of finding particles of the speed of v is only linearly proportional to the Boltzmann factor (C) The probability density function of finding particles of the speed v is not only linearly proportional to the Boltzmann factor but also lincarly proportional to v. (D) The probability density function of finding particles of the speed of v is not only linearly proportional to the Boltzmann factor but also linearly proportional to v2. (E) None of the above is true.
23. For two inductors connected in parallel, see the figure below； which of the following statements is true? (A) The effective inductance of the two inductors is smaller than that of both L1 and L2. (B) The effective inductance of the two inductors is larger than that of both L1and L2. (C) After a sufficiently long time, the voltage across the parallel inductors reaches a maximum steady value. (D) After a sufficiently long time, the voltage across L2 is larger than that across Li. (E) None of the above.
24. For a classical electromagnetic plane wave travelling in a vacuum, which of the following statements is true? (A) The magnetic and electric fields are perpendicular to each other, and there is a phase difference of π/2 between them. (B) The average power delivered by the EM wave increases with the frequency of the EM wave. (C) The traveling direction of the EM wave can be in either the direction of o are the direction of the electric field and the magnetic field, respectively. (D) The power of the EM wave decays as it propagates away from its source. (E) None of the above.
27. Suppose A is a 2x3 real matrix and B is a 3x2 real matrix., then which of the following matrix can be the product of BA?(A)(B)(C)(D)(E)
28. Suppose denote the standard vector norm in R2and R3. What is the smallest possible value for | As:ll if lI x=1? (A) (B) (C) 3 (D) 4(E)
29. Suppos denote the standard vector 110 nomm in R3. What is the smallest possible value for ||x||if Ax=b? (A) 1 (B) (C)(D) 2 (E)
30. Suppose A is a 3✖3 invertible matrix and I3 is the 3✖3 identity matrix. The followings are 5 statements about A that may or may not be true. (1) If p(x) is the characteristic polynomial of A, then the constant term of p(x) cannot be 0.(2) If a 3✖3 matrix B satisties AB+BA=0,then B is singular.(3) For any 3✖5 matrix C, rank(AC)= rank( C ).(4) If a 3✖3 matrix D is diagonalizable,then AD is also diagonalizable.
(5)If E is a 3x3 matrix and AE is singular,then E is also singular.Which of the above statements about A is False? (A)(1) (B) (2) (C)(3) (D)(4) (E) (5)
31. If ||u||= 8 and ||v|| = 3, then what are the largest possible values of the inner product (u + v,v)? (A) 24 (B) 12 (C) 15(D)33(E) 45
32. Consider the system of linear equations:
For what value of C will the system be singular? (A)-10 (B)-6 (C) -3 (D) 4 (E) 10
33. Given the row-reduced echelon form of a matrix A. What is the closest vector to b = [0,3,0, 1]T in the row space of A? (A) [1,1,2,0]T (B) [0,-2,-1,1]T (C) [-1, 2,0, -1]T (D) [-1,2,1,0]T (E) [1,1,3,1]T
34. Let ， which of the following statement is incorrect? (A) The rank of A is 3. (B) The inverse matrix is (C) If we define the characteristic polynomial of A is f(t)=det(A-tIn), then f(1)=-t3+3t2+3t-1. (D) One of the eigenvalues is -1 and the corresponding eigenvector is (-2, 1, 2)T. (E) The matrix cannot be diagonalized over the real field.
36. A vector space is spanned by {1,cos(t),sin(t)} for -π≤t≤π . If a vector: v=a.1+b.sin(t)+cㆍcos(t) is the closest vector in this vector space to a continuous function: f(t)=t for -π ≤t≤π, what is this closest v? The inner product of two continuous functions: fand g within [a, b] is defined by You may need the following integral:. (A)v=1+2sin(t)-2cos(t) (B) v=1-2cos(t) (C) v= 2sin(r) (D) v=1+2sin(t) (E) v=-2cos(t)
37. A quadratic equation is described as: x2 +8xy + 7y2 = 225. Which of the following statement is incorrect? (A) The curve is centered at the origin (B) This quadratic curve is an ellipse (C) One of the principal axes is (D) The other principal axis is (E) The shortest distance from this quadratic curve to the origin is 5
38. Which of the following statements is true? (A) Let A be an arbitrary matrix of size m X n. If m ＞ n, then rank( A ) ＜ rank(At). (B) Let A be an arbitrary matrix of size m ✖n. If m ＞ n,then rank( A ) ＞ rank(At). (C) If W1, W2 are two subspaces of V, then the union W1U W2 is also a subspace of V. (D) Let W be a subspace of a vector space V. Then, dim(W) ≤ dim(V). (E) Let A be a square matrix, k be a constant. Then, det(kA) = kdet( A ).
39. Consider the matrix . How many different mattices are there such that B2 = A? (A) 0 (B) 1 (C) 2 (D) 4 (E) 16
40. Consider the Fibonacci series a0 = 1, a1 = 2,and an = an-1 + an-2 for any n≥ 2. Therefore, we have {an} = {1,2,3,5,8, ...}. The general expression for an as a function of n can be found via the following method； first, definc , and an define the state vector . Thus, we have the initial condition ，and the relation Vn+1 = Avn. The following general expression can be derived: an = are the eigenvalues of A. How many ofthe following statements are true? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4
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) is rotating about its symmetric axis at an angular velocity ω0. A ring of the same mass m and radius R originally at rest suddenly drops on the disk. The coeffieient of kinetic friction between the disk and the ring is μ. How long does it take for the ring to reach a final angular velocity ω and rotate together with the disk as shown in the right figure? The gravitational acceleration is g. 
(A)
(B)
(C) 
(D)
(E)
(A)
(B)
(C)
(D)
(E) 
(A) 1 W/m2 (B) 2 W/m2 (C) 3 W/m2 (D) 5 W/m2 (E) 9 W/m2
(A) 5 cm (B) 10 cm (C) 12.5cm (D) 25 cm (E) 30 cm
(A) ε(B) 
(C) 
(D)
(E)
(A)
(B) 
(C)
(D) 
(E)
(A)
(B)
(C)
(D) 
(E)0
(A)αA:αB = 3:1 (B)αA: αB = 1:1 (C)αA:αB = 2:1 (D)αA:αB = 5:2 (E)αA:αB = 7:3
(B) 
(C)
(D)
(E) None of the above. 
(namely,θ=0)
(A)
(B)
(C) 
(D)
(E)
(A) 1/12 (B) 1/11 (C) 1/10 (D) 1 (E) 11
(Joules). If the particle is initially placed at its mechanical equilibrium position and then given a small displacement, what is the period of its subsequent oscillations? (A)
(B)
(C)
(D)
(E) None of the above
, where △x is the distance travelled； which of the following statements is true? (A) From the observer B's point of view, since 
, B would measure a different deceleration than what A would have measured. (B) The kinematic equation only works in A's reference frame. (C) The kinematic equation only works in B's reference frame. (D) The inconsistency of the kinematic equation can be amended by the special relativity. (E) None of the above is true.
is not only lincarly proportional to the Boltzmann factor 
but also linearly proportional to y. (B) The probability density function of finding particles of the speed of v is only linearly proportional to the Boltzmann factor 
(C) The probability density function of finding particles of the speed v is not only linearly proportional to the Boltzmann factor 
but also lincarly proportional to v. (D) The probability density function of finding particles of the speed of v is not only linearly proportional to the Boltzmann factor 
but also linearly proportional to v2. (E) None of the above is true.
(A) The effective inductance of the two inductors is smaller than that of both L1 and L2. (B) The effective inductance of the two inductors is larger than that of both L1and L2. (C) After a sufficiently long time, the voltage across the parallel inductors reaches a maximum steady value. (D) After a sufficiently long time, the voltage across L2 is larger than that across Li. (E) None of the above.
o
are the direction of the electric field and the magnetic field, respectively. (D) The power of the EM wave decays as it propagates away from its source. (E) None of the above.
, then which of the following matrix can be the product of BA?(A)
(B)
(C)
(D)
(E)
denote the standard vector norm in R2and R3. What is the smallest possible value for | As:ll if lI x=1? (A) 
(B) 
(C) 3 (D) 4(E) 
denote the standard vector 110 nomm in R3. What is the smallest possible value for ||x||if Ax=b? (A) 1 (B) 
(C)
(D) 2 (E) 
of a matrix A. What is the closest vector to b = [0,3,0, 1]T in the row space of A? (A) [1,1,2,0]T (B) [0,-2,-1,1]T (C) [-1, 2,0, -1]T (D) [-1,2,1,0]T (E) [1,1,3,1]T
， which of the following statement is incorrect? (A) The rank of A is 3. (B) The inverse matrix is 
(C) If we define the characteristic polynomial of A is f(t)=det(A-tIn), then f(1)=-t3+3t2+3t-1. (D) One of the eigenvalues is -1 and the corresponding eigenvector is (-2, 1, 2)T. (E) The matrix cannot be diagonalized over the real field.
You may need the following integral:
. (A)v=1+2sin(t)-2cos(t) (B) v=1-2cos(t) (C) v= 2sin(r) (D) v=1+2sin(t) (E) v=-2cos(t)
(D) The other principal axis is
(E) The shortest distance from this quadratic curve to the origin is 5
. How many different mattices 
are there such that B2 = A? (A) 0 (B) 1 (C) 2 (D) 4 (E) 16
, and an define the state vector 
. Thus, we have the initial condition 
，and the relation Vn+1 = Avn. The following general expression can be derived: an = 
are the eigenvalues of A. How many ofthe following statements are true?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4