26. Suppose A is a 4x4 real matrix and y,y.y,y, are four lincarly independent vectors in R⁴. If Av₁=v₁+v₂, Av₂=v₂+v₃，Av₃=v₃+v₄, Av₄=v₁+v₄,then how many real eigenvalues does A have?
(A) 0
(B) 1
(C) 2
(D) 3
(E)4 

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

5. Which of the following statements is correct? (A) The surface charge density of a perfect conductor must be uniform. (B) The electric field line is the trajectory of a charged particle under electric field. (C) The conductor can be thought of as a dielectric material with dielectric constant k=0. (D) If the electric potential at some position is zero, the electric field at that position must also be zero. (E) The surface of a perfect conductor is an equipotential surface.

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

9. Which quantity cannot be measured by Hall effect? (A) The external magnetic field (B) The sign of the charge carriers (C) Electron's charge to mass ratio (D) The type (N- or P-type) of semiconductor (E) The density of free electrons inside metal

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

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