114年 - 114 國立清華大學學士後醫學系單獨招生考試_學士後醫學系/智慧資訊科技組：資訊科學#135660

1. A bead of mass m is constrained to move on a massless circular hoop of radius R. The hoop is rotating around its center that is aligned in the z-axis (the direction of gravity g) with a constant angular velocity Ω, see figure below. If the hoop is rotating slowly, this system has a stable equilibrium angular velocity of the hoop exceeds a critical 0 becomes unstable. What is the critical angular velocity?

(A)(B) (C) (D)
(E) None of the above

2. Two masses m₁ and m₂ placed on a flat surface are connected by a spring of a spring constant k aligning in the x direction; see figure below. Initially, the spring is at its natural length (no compression/extension), and m₁ is at rest while m₂ moves with a velocity in the +x direction. Since there is friction, these masses will eventually stop moving. Assume these two masses are identical, m₁ = m₂ = m, and the gravitational strength is g. What is the minimum friction coefficient required so that m₁ never moves during the slowdown motion of m₂? For simplicity, assume the kinetic and static friction coefficients are the same.
(A) (B) (C) (D)
(E) None of the above

3. A pendulum is composed of a bob of mass m₂, a rigid string of length and negligible mass, and a mass m₁ at the point of support, as shown in the figure. The mass m₁ can only move horizontally in the x-direction on a frictionless track, while m₂ swings in the xy plane. The system is placed in a uniform gravitational field =. What is the period of the oscillation when m2 is released at a small angle?
(A) (B) (C) (D)
(E)

4. Jack is pushing a box moving at a constant speed on a horizontal ground. Due to friction between the box and the ground, Jack has to continue to do positive work to keep the box moving, and the work output per unit time is P. If Jack would like to push the box along a slope of an inclined angle of 18 degrees (relative to the horizontal ground) at the same constant speed, make an estimate of the power output required. Assume the coefficient of the kinetic friction remains the same and equals unity.(A) P(B) 1.26P(C) 1.33P(D) 1.42P
(E) 1.51P

5. The membrane serves as an insulator that separates the environments inside and outside the cell. The membrane potential is the difference in electric potential between the interior and the exterior of a biological cell. Assume that the interior of a cell has a lower electric potential compared to the exterior of a cell, and the K+ (potassium ion) concentration inside the cell is higher than that outside the cell. If no ions (such as Na⁺, Ca²⁺, etc.) can enter or leave the cell except the potassium ions through some holes on the membrane, which of the following is true?(A) The potassium ions will enter the cell, which increases the membrane potential(B) The potassium ions will leave the cell, which increases the membrane potential(C) The potassium ions will enter the cell, which decreases the membrane potential(D) The potassium ions will leave the cell, which decreases the membrane potential
(E) None of the above

6. In the double-slit experiments of neutrons, which of the following statements is true?(A) Each neutron simultaneously passes through two slits and interferes with itself to form an interference pattern(B) Different neutrons pass through two slits and interfere with others to form an interference pattern(C) If one tracks the motion of each neutron (by shining a light), the interference pattern becomes more pronounced(D) It will not form the interference pattern because neutrons, unlike electrons, do not carry charges
(E) None of the above

7. An ideal LC circuit with inductor L and capacitor C is initially charged with zero on the capacitor and a maximum instantaneous current of Io. Which of the following is true?(A) The total energy of the LC circuit oscillates with time(B) If one increases the capacitance of the capacitor while keeping the inductance and the initial charges and currents the same, the total energy will not change(C) If one increases the inductance of the inductor while keeping the capacitance and the initial charges and currents the same, the total energy will not change(D) The current will monotonically decay over time and reach zero
(E) None of the above

8. For an RL circuit used as a low-pass or high-pass filter, which of the following statements is true?(A) The inductor blocks high-frequency signals since its reactance increases with the frequency(B) The inductor blocks low-frequency signals since its reactance increases with the frequency(C) The inductor blocks high-frequency signals since its reactance decreases with the frequency(D) The inductor blocks low-frequency signals since its reactance decreases with the frequency
(E) None of the above

9. A ball of mass m is connected by a string of length L to a pivot fixed in place. A. The ball is originally at rest. A force F applied horizontally to the right on the ball. The force barely balances the gravity and the string tension so that the ball has negligible kinetic energy when it slowly swings up. The string finally makes an angle θ with the vertical line. What is the work done by the force F during this process?
(A) mgL(sinθ)(B) mgL(1-cosθ)(C) mgL(cosθ-1)(D) mgL(cosθ)
(E) 0

10. Two uniform ladders, both of length L, are hinged at point A and tied together by a horizontal rope at its center. The angle between the ladders is 90 degree. The ladders have mass m₁ and m₂ respectively. Assume the floor is frictionless. Find the tension in the rope.(A)(B) (C) (D) 2m₁g + 2m₂g
(E) 0

11. One mole of real gas with interaction between molecules obeys van der Waals equation of state, (P + )(V - b) = RT. The internal energy function can be calculated to be E_int =RT - . What is Cv, the molar heat capacity at constant volume of this gas?(A)(B) (C) (D)
(E)

12. A heat engine using monatomic ideal gas undergoes a cyclic process BACB, where BA is an isothermal process. In which process(es) does the gas absorb heat from the environment?
(A) AC (B) AC and CB (C) AC and BA (D) BA and CB (E) All three processes

13. A cylinder has insulating walls and is divided into two parts by a frictionless insulating piston. Initially, the two parts have equal volumes V₀, equal temperatures To and equal pressure P₀. The gas is ideal with Cv = R, Cp = . A small heating bar is now inserted on the left part and heat is supplied slowly to the gas on the left until its pressure reaches 32P₀. As a result, the left part will expand and the right part will be compressed. How much heat is supplied to the gas on the left part? (A) 32 P₀V₀ (B) PoVo (C) P₀V₀ (D) 59 P₀V₀ (E) 93 P₀V₀

14. A particle with mass m and positive charge q starts from rest at the origin under a uniform electric field in the +y-direction and a uniform magnetic field directed out of the page, so . Neglecting gravity, the particle is accelerated by the E field, but the magnetic force bends it to the right and the resulting trajectory is shown in the figure. At point P, the particle reaches its maximum height y_max. What is the velocity of the particle at point P?(A)(B) (C) (D)
(E)

15. The Maxwell's equations related to static magnetic field are and . Which of the following magnetic field line illustrated in the figures clearly violates Maxwell's equation?(A)(B) (C) (D)
(E)

16. Consider the n = 4 states of hydrogen, what is the minimum angle θ between its angular momentum vector and the z-axis?(A) (B)(C) (D)
(E) 0

17. Two blocks (with the same mass m) connected by a spring of negligible mass are at rest on a frictionless table, as shown in the upper part of figure. The separation distance of the blocks when the spring is relaxed is L. During a time interval △t, a constant force of magnitude F is applied horizontally to the left block, moving it through a distance x₁ as shown in the lower part of figure. During this time interval, the right block moves through a distance x₂. At the end of this time interval when the force is removed, what is the maximum energy could be momentarily stored in the spring?
(A) F(x₁+x₂) (B) F(x₁+x₂)/2 (C) F(x₁-x₂)/2 (D) F(x₁-x₂) (E) It cannot be determined.

18. A block of mass M is attached to a spring of mass m and force constant K. What is the time period T of oscillation of the block-spring system?(A) 2π√(M/k)(B) 2π√((M/k) + √(m/k))(C) 2π√((k/M) + √(k/m))^(-1)(D) 2π√((M+m)/k)
(E) 2π√((M+(m/3))/k)

19. Imagine that an ideal monatomic gas is taken from its initial state B to state D by an isobaric process, from D to A by an isochoric (i.e., isovolumetric) process, and from A back to its initial state B by an isothermal process. What is the thermal efficiency of this BDAB cycle? Note that ln2 = 0.6931 and ln10 = 2.3026.
(A) 0.144(B) 0.192(C) 0.240(D) 0.288
(E) None of the above

20. To explain the phenomenon of condensation of real gases to liquids, two constants A and B for different gases are introduced in the van der Waals equation of state

Which of the following expression is correct?(A) A ＜ 0 and B ＜ 0(B) A ＞ 0 and B ＜ 0(C) A ＞ 0 and B ＞ 0(D) A ＜ 0 and B ＞ 0
(E) Whether A is larger or smaller than 0 depends on gases

21.A solenoid consisting of n turns per unit length with length 4a, equal to four times its radius a, carries a steady current I. About the magnitude of magnetic B-field at the center (Bcenter) and at the point of each end (Bend-point) along its axis, which statement in the following is correct?
(A) B_end-point is larger than one half of B_center, i.e., B_end-point ＞ B_center/2(B) B_end-point is equal to one half of B_center, i.e., B_end-point = B_center/2(C) B_end-point is smaller than one half of B_center, i.e., B_end-point ＜ B_center/2(D) Along the axis and in free space, B_center is maximal and equal to μ₀nI
(E) None of the above

23. An antiproton has the same rest mass as a proton which is created in the reaction p+p→p+p+p+. If two initial protons are moving with equal speeds in opposite directions and continue to exist after a head-on collision that also produces an extra pair of p-. If all four particles are at rest after the collision, what is the initial speed of the protons in their center-of-momentum frame? (Hint: Energy is conserved in the collision.)(A) 1.2c(B)(C)(D) 0.6c
(E) (2-)c

24. For getting a reasonable sensitivity to differentiate the Compton wavelength shift, what type of electromagnetic wave should be taken in the measurement?(A) microwaves(B) infrared waves(C) visible light(D) ultraviolet waves
(E) x-rays

25. As measured in an order of magnitude, which quantity of the following objects is about 1 nm?(A) the Bohr radius(B) the Compton wavelength of the electron(C) the de Broglie wavelength of a 1.5-GeV electron(D) the wavelength of the 1.24-keV soft x-ray
(E) the wavelength of Mo-Kα emission, that is, hf = 17444 eV

26. What is the determinant of the following matrix?
(A) el - kf(B) 0(C) bcgiy - bdghy(D) adfyh + bcxdk – cazej
(E) bcy +gi-bd - aghy

27. Let L(x) = Ax represent a linear transformation from R² to R², where:
A =
What kind of geometry operation is L?(A) L is the linear operator that rotates each x in R² by 45° in the clockwise direction(B) L is the linear operator that reflects each vector x in R² about the x₁ axis and then rotates it 90° in the counterclockwise direction(C) L doubles the length of x and then rotates it 30° in the counterclockwise direction(D) L reflects each vector x about the line x₂ = x₁ and then projects it onto the x₁-axis
(E) L rotates x 270° in the counterclockwise direction

28. Consider the system of linear equations:
x + y + az = b
x - 2y + z = 1
x - 8y - 11z = -7
For what values of a and b will the system have infinitely many solutions?(A) a = 7, b = 1(B) a = 11, b = 7(C) a = 11, b = 5(D) a = 5, b = 7
(E) a = 7, b = 5

29. Let {v₁, v₂, v₃} = {[1, 1, 0]ᵀ, [0, 1, 1]ᵀ, [1, 0, 1]ᵀ} and {u₁, u₂, u₃} = {[1, 2, 4]ᵀ, [1, 3, 5]ᵀ, [1, 1, 1]ᵀ} be two ordered bases for the vector space R³. If x = v₁ - 2v₂ + 3v₃, what is the coordinates of x with respect to the basis {u₁, u₂, u₃}?(A) [7, -6, 3]ᵀ(B) [1, -2, 3]ᵀ(C) [4, -1, 1]ᵀ(D) [-2, 0, 3]ᵀ
(E) [2, -3, 0]ᵀ

30. For the matrix

what is the corresponding eigenvalue with the multiplicity 2?(A) 1(B) 2(C) 3(D) 4
(E) 5

31. Consider {0} and {1} as the basis for a vector space in C², then the natural basis for the tensor product space C² ⊗ C² has the ordered bases {00}, {10}, {01}, {11}. The tensor product of two vectors is defined from their decomposition on the bases.
What is the corresponding matrix formed by

wheredenotes the transpose.(A)(B)(C)(D)
(E)

32. Let R⁴ˣ⁴ denote the collection of all 4 × 4 real matrices. Define the linear
transformation T: R⁴ˣ⁴ → R⁴ˣ⁴ by T( A ) = Aᵀ + 2A for any 4 × 4 matrix A.
What is the rank of T?(A) 4(B) 8(C) 10(D) 12
(E) 16

33. Let R⁶ denote the 6-dimensional real vector space and ||•|| denote the standard vector norm in R⁶. How many 6 × 6 real matrices U satisfy || Ux || = || x || for all x ∈ R⁶ and |trace(U)| = 6?(A) 1(B) 2(C) 3(D) 6
(E) Infinitely many

34. Let R_θ denote the rotation matrix through an angle 0＜θ＜counterclockwise about the origin in R². Let e₁ = [1,0], e₂ = [0,1]ᵀ be the standard basis of R². Suppose A is a 2x2 matrix satisfying Ae₁ = R_θe₁ and R_θAe₂ = e₂. The followings are 5 statements about A that may or may not be true.
(1) A²e₁ = R₂θe₁
(2) det( A ) = 1
(3) trace( A ) = 2cos θ
(4) A is a positive definite matrix
(5) AR_θ = R_θA
Which of the following combination of statements about A are all true?(A) (1)(2)(B) (1)(3)(C) (2)(3)(D) (3)(4)
(E) (4)(5)

35. Given a matrix
Find the eigenva lue of A².(A) 1 and 2(B) -1 and -2(C) 2 and 4(D) 1 and 8
(E) 1 and 16

36. What are the dimensions of the row space, column space, and null space for the matrix?
(A) 3, 5, 5(B) 3, 5, 2(C) 3, 3, 2(D) 2, 3, 3
(E) 2, 2, 3

37. A linear equation system Ax = b consists of m linear equations with n variables. Which of the following statements is correct?(A) We can apply elementary column operations to solve this linear equation system.(B) If A and B matrices are similar, then Ax=b and Bx=b have the same solution set.(C) If this linear equation system has solutions, the solutions form a subspace.(D) If m ≠ n and rank( A ) = m, then there exist infinite solutions.
(E) If m ≠ n, we can use "Cramer's rule" to find its solution.

38. If a linear transformation T: R³→R³ is defined by:
T(x, y, z) = (3x+2y, -2x+3y, 5z), which of the following statement is incorrect?(A) The basis of the kernel (null space) of T is {0}.(B) T is one to one.(C) T is onto.(D) T is invertible.
(E) T is not diagonalizable over the real field.

39. Which of the following statements is true?(A) Let S = {v₁, v₂, ..., vk} be a set of vectors in Rⁿ. Let A ∈ Mmxn(R). If S is linearly independent, then S' = {Av₁, Av₂, ..., Avk} is also linearly independent(B) Let S be a set of vectors in a finite-dimensional vector space V. Then, span(S) is the intersection of all subspaces of V that contains S(C) Assume that S = {v₁, v₂, ..., vn} is a subset of a vector space V. If S is linearly dependent, then there exists a proper subset of S that is a basis for V(D) The basis of any vector space is unique
(E) Assume that S = {v₁, v₂, ..., vn} is a subset of a vector space V. If S is linearly independent, then there exists a proper subset of S that is a basis for V

40. Consider the continuous function f(x) = 3 cos 2x + 4 sin 2x, and let g(x) = a cos x + b sin x be a linear combination of the functions cos x and sin x.
Define inner product of two functions as (f, g) = f(x)g(x)dx. Please find a and b such that ||f - g||² = |f(x) – g(x)|² dx is minimized. Which of the following shows the correct answer?(A) a = 4 and b = 3(B) a = 3 and b = 4(C) a = 0 and b = 0(D) a = 3 and b = -4
(E) a = -3 and b = 4