Apply the course version boundaries in notes.md; HSK tasks are transferable language practice, not a claimed HSK 3.0 mock.
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gre_physics:mixed-revision:1
A reversible engine runs between 500 K and 200 K and absorbs 600 J. Find maximum work and rejected heat.
Answer and reasoning · Resposta e raciocínio
Carnot efficiency 0.6 gives work 360 J and rejected heat 240 J.
gre_physics:mixed-revision:2
An infinite well has levels E_n=n²E1. Find the photon energy for 4→2.
Answer and reasoning · Resposta e raciocínio
(16−4)E1=12E1; emitted energy is positive.
gre_physics:mixed-revision:3
For isotropic emission, source distance increases by factor 3. What happens to flux and luminosity?
Answer and reasoning · Resposta e raciocínio
Flux becomes one ninth; intrinsic luminosity remains unchanged.
gre_physics:mixed-revision:4
A 1 kg cart at (6,0) m/s sticks to a 2 kg cart at (0,3) m/s. Find common velocity and kinetic-energy loss.
Answer and reasoning · Resposta e raciocínio
Momentum (6,6), final velocity (2,2) m/s. Initial K=18+9=27 J; final K=12 J, so loss 15 J.
gre_physics:mixed-revision:5
Spring B has four times spring A’s stiffness. Compare stored energies first at equal force, then at equal extension.
Answer and reasoning · Resposta e raciocínio
At equal force U_B/U_A=1/4; at equal extension it is 4. State the fixed quantity before using U.
gre_physics:mixed-revision:6
A rolling solid sphere has total kinetic energy 14 J. How much is rotational? Compare terminal speeds of identical-shape quadratic-drag balls with mass ratio 16, neglecting buoyancy.
Answer and reasoning · Resposta e raciocínio
Rotational fraction 2/7 gives 4 J. Terminal-speed ratio sqrt(16)=4.
gre_physics:mixed-revision:7
A uniform rod of length 2 m is pivoted at one end at angle 30° from vertical. With g=10 m/s², find angular-acceleration magnitude.
Answer and reasoning · Resposta e raciocínio
3g sinθ/(2L)=30·0.5/4=3.75 rad/s², using end-pivot inertia.
gre_physics:mixed-revision:8
A 6 kg table mass connects to a 2 kg hanging mass over an ideal pulley; g=10 m/s². Find acceleration and tension.
Answer and reasoning · Resposta e raciocínio
a=20/8=2.5 m/s² and T=6·2.5=15 N, below hanging weight 20 N.
gre_physics:mixed-revision:9
A level aircraft moves at 90 m/s and releases a load. With no drag and g=10 m/s², find ground velocity, aircraft-relative velocity and vertical drop after 5 s.
Answer and reasoning · Resposta e raciocínio
Ground velocity (90,−50) m/s; relative velocity (0,−50) m/s; drop ½·10·25=125 m.
gre_physics:mixed-revision:10
A centred charge +5Q lies inside a spherical conducting shell of net −2Q and outer radius b. Find induced surface charges and material potential, zero at infinity.
Answer and reasoning · Resposta e raciocínio
Inner surface −5Q; outer +3Q; V=3Q/(4πε0b), constant in the material.
gre_physics:mixed-revision:11
Two attractive forces on an electron have magnitudes 5 N and 12 N at right angles. Find their resultant magnitude; explain why potential would be added differently.
Answer and reasoning · Resposta e raciocínio
Force magnitude sqrt(25+144)=13 N. Potential is a scalar sum, whereas force is a vector sum.
gre_physics:mixed-revision:12
A sinusoidal current has peak 4 A in a 5 Ω resistor. Find rms current and average power.
Answer and reasoning · Resposta e raciocínio
Irms=4/sqrt(2)=2sqrt(2) A; P=Irms²R=8·5=40 W.
gre_physics:mixed-revision:13
A 200-turn coil of area 0.01 m² has perpendicular field decrease from 0.08 to 0.02 T in 0.30 s. Find average emf magnitude and state the Lenz condition.
Answer and reasoning · Resposta e raciocínio
Magnitude=200·0.01·0.06/0.30=0.4 V. Induced current opposes the decrease in original flux.
gre_physics:mixed-revision:14
A −x travelling wave has electric field along +z. Find magnetic and Poynting directions.
Answer and reasoning · Resposta e raciocínio
B is +y, since (−x)×z=+y. E×B=z×y=−x, matching propagation.
gre_physics:mixed-revision:15
A nonmagnetic lossless material has εr=36. Find v/c. At a zero-interior-B boundary, which exterior magnetic component is constrained to zero?
Answer and reasoning · Resposta e raciocínio
v/c=1/6. The component normal to the surface is zero; tangential field may remain.
gre_physics:mixed-revision:16
A 600 Hz moving source approaches a fixed reflecting wall at 20 m/s in still air of speed 320 m/s. Find the echo frequency heard at the source.
Answer and reasoning · Resposta e raciocínio
600·340/300=680 Hz; multiply the moving-source and moving-observer factors.
gre_physics:mixed-revision:17
An ideal both-open pipe has fundamental 90 Hz. Give the first three allowed frequencies after one end is closed; does 90 Hz remain?
Answer and reasoning · Resposta e raciocínio
45,135,225 Hz. The old 90 Hz is twice the new fundamental, an absent even harmonic.
gre_physics:mixed-revision:18
A relaxed-view telescope has 100 cm lens separation and 20 cm eyepiece focal length. Find objective focal length and magnification magnitude.
Answer and reasoning · Resposta e raciocínio
Objective 80 cm; magnification 4.
gre_physics:mixed-revision:19
Evacuating a 4.5 cm Michelson gas cell gives 60 fringes at 450 nm. Find n before evacuation.
Answer and reasoning · Resposta e raciocínio
n−1=60·450×10⁻⁹/0.09=0.0003, so n=1.0003.
gre_physics:mixed-revision:20
For f(x)=1+2cos x+4cos(3x), identify a0, a1, a3 and all b_n under the standard Fourier convention.
Answer and reasoning · Resposta e raciocínio
a0=2, a1=2, a3=4; all b_n=0. The mean is 1.
gre_physics:mixed-revision:21
Two equal coherent waves have individual intensity I0 and phase difference 2π/3. Find combined intensity. Compare the incoherent average.
Answer and reasoning · Resposta e raciocínio
Coherent intensity 2I0(1−1/2)=I0; incoherent average 2I0.
gre_physics:mixed-revision:22
A clockwise triangular P–V cycle has pressure span 6000 Pa and volume span 0.004 m³. Find net work and heat.
Answer and reasoning · Resposta e raciocínio
Work=½·6000·0.004=12 J; net entering heat is also 12 J.
gre_physics:mixed-revision:23
A Carnot refrigerator absorbs 1800 J at 240 K and rejects heat at 300 K. Find COP, work input and rejected heat.
Answer and reasoning · Resposta e raciocínio
COP=240/60=4; work 450 J; rejected heat 2250 J.
gre_physics:mixed-revision:24
A speed-count density has trapezoid area 3av0, N=180 and v0=5 m/s. Find a and explain its units.
Answer and reasoning · Resposta e raciocínio
a=12 particles per unit speed; integrating the density over speed gives a particle count.
gre_physics:mixed-revision:25
Two levels have ground degeneracy 1, excited degeneracy 4 and gap kBT ln4. Find probabilities of the excited level and one specified excited state.
Answer and reasoning · Resposta e raciocínio
Each level has total weight 1, so excited level probability 1/2; one excited state has probability 1/8.
gre_physics:mixed-revision:26
For N=250 independent detector trials and p=0.20, find mean, variance and standard deviation.
Answer and reasoning · Resposta e raciocínio
Mean 50; variance 40; standard deviation sqrt(40)≈6.32.
gre_physics:mixed-revision:27
A kinetic-energy estimate has independent mass uncertainty 8% and speed uncertainty 3%. Find approximate fractional standard uncertainty. Explain why dimensional analysis alone cannot determine a coefficient.
Answer and reasoning · Resposta e raciocínio
sqrt(8²+6²)%=10%. A dimensionless coefficient does not affect unit powers, so it is undetermined by dimensions.
gre_physics:mixed-revision:28
A particle moves at 0.8c. Find γ and kinetic energy as a fraction of rest energy. For a longitudinal wavelength ratio 5, find recession speed.
Answer and reasoning · Resposta e raciocínio
γ=5/3 and K/(mc²)=2/3. β=(25−1)/(25+1)=12/13.
gre_physics:mixed-revision:29
A particle has total energy 17 GeV and pc=15 GeV. Find rest energy and explain why E−pc is not the answer.
Answer and reasoning · Resposta e raciocínio
Rest energy sqrt(289−225)=8 GeV. The invariant is quadratic; direct subtraction would give 2 GeV incorrectly.
gre_physics:mixed-revision:30
A two-body one-electron model has nuclear mass 4m_e. Find μ/m_e. A photon has energy 5 eV and work function is 3 eV: find stopping magnitude.
Answer and reasoning · Resposta e raciocínio
Reduced-mass ratio 4/5; maximum electron energy 2 eV, stopping magnitude 2 V.
gre_physics:mixed-revision:31
Franck–Hertz current features are spaced by 6 V. Find the excitation energy and corresponding single-photon wavelength using hc=1240 eV·nm.
Answer and reasoning · Resposta e raciocínio
Energy 6 eV; wavelength 1240/6≈206.7 nm. Repeated features can reflect repeated losses of this same energy.
gre_physics:mixed-revision:32
A step has k2=5k1. Find reflection/transmission probabilities and verify their sum.
Answer and reasoning · Resposta e raciocínio
r=−2/3 gives R=4/9; T=5·(1/3)²=5/9; sum one.
gre_physics:mixed-revision:33
Count spin-zero isotropic oscillator states for N=5, and compute σ_yσ_x.
Answer and reasoning · Resposta e raciocínio
Degeneracy 6·7/2=21; σ_yσ_x=−iσ_z.
gre_physics:mixed-revision:34
Compare B/A for (A,B)=(5,30 MeV) and (15,120 MeV). Is total binding alone the correct comparison?
Answer and reasoning · Resposta e raciocínio
Values 6 and 8 MeV per nucleon. Total binding alone confounds different nucleon counts.
gre_physics:mixed-revision:35
Classify neutrino, kaon and neutron as lepton/meson/baryon in the ordinary constituent model. Explain the sign of R_H for one-carrier holes.
Answer and reasoning · Resposta e raciocínio
Neutrino lepton, kaon meson, neutron baryon. Hole charge +e gives positive R_H under R_H=1/(nq).
gre_physics:mixed-revision:36
Identical satellites have circular orbit-radius ratio 16 around the same dominant mass. Compare angular momentum and period ratios.
Answer and reasoning · Resposta e raciocínio
Angular momentum ratio sqrt(16)=4; period ratio 16^(3/2)=64.
gre_physics:mixed-revision:37
A central-force particle conserves pθ. Radius changes from 4 m to 2 m. Compare angular speeds. For a 1 m pendulum with g=10, give the linear angular frequency and the approximation needed.
Answer and reasoning · Resposta e raciocínio
Angular speed quadruples because mr²θdot is fixed. The pendulum has ω=sqrt(10) rad/s only for small angles in radians.
gre_physics:mixed-revision:38
For L=2(xdot²+ydot²)/2−3x², identify cyclic coordinates, a conserved momentum and the x equation.
Answer and reasoning · Resposta e raciocínio
y is cyclic and p_y=2ydot is conserved. Euler–Lagrange gives 2xddot+6x=0; x is not cyclic.
gre_physics:mixed-revision:39
At r=(3,0,0), Ω=(0,0,2), m=1 and v′=(0,0,0), find apparent forces for constant rotation and the real force required to stay fixed.
Answer and reasoning · Resposta e raciocínio
Centrifugal=(12,0,0) N; Coriolis=Euler=0. Real force=(−12,0,0) N, inward.
gre_physics:mixed-revision:40
An 8 kg passenger accelerates upward at 1.5 m/s² with g=10. Then evaluate Coriolis direction for Ω along +z and relative motion along −x.
Answer and reasoning · Resposta e raciocínio
Scale force 8(10+1.5)=92 N. z×(−x)=−y, so Coriolis is along +y.
gre_physics:mixed-revision:41
An ideal horizontal incompressible flow speeds up from 2 to 5 m/s with density 800. Find pressure drop. If a laminar tube radius halves at fixed Δp, compare Q.
Answer and reasoning · Resposta e raciocínio
Ideal pressure drop=800(25−4)/2=8400 Pa. At fixed Δp, laminar Q becomes 1/16; this is distinct from fixed-Q continuity.
gre_physics:mixed-revision:42
A liquid has density 900 and viscosity 0.003 Pa·s; mean pipe speed is 0.1 m/s and diameter 0.01 m. Find Re and explain why a steady uniform pipe can still have a pressure drop.
Answer and reasoning · Resposta e raciocínio
Re=900·0.1·0.01/0.003=300. Viscous momentum transfer can dissipate energy at unchanged mean speed; loss-free Bernoulli alone misses it.
gre_physics:mixed-revision:43
A capacitor stays connected to an ideal battery while a dielectric of ε_r=6 completely fills its gap. Compare free charge, E and stored energy with the vacuum case. Contrast the isolated fixed-Q case.
Answer and reasoning · Resposta e raciocínio
At fixed V: free charge and U are six times larger; E=V/d is unchanged. At fixed Q: E, V and U are one sixth. The connected source exchanges energy.
gre_physics:mixed-revision:44
For a uniform linear dielectric with ε_r=2 and free plate density σ, find P/σ. A long solenoid has nI=250 A/m and χ_m=0.2. Find H, M and B/μ₀.
Answer and reasoning · Resposta e raciocínio
P=(ε_r−1)σ/ε_r=σ/2. H=250 A/m, M=50 A/m and B/μ₀=300 A/m under the long-solenoid linear-response assumptions.
gre_physics:mixed-revision:45
An RC discharge has τ=3 s and initial voltage 10 V. Find voltage at 6 s and the time to halve. An RL circuit has L=0.3 H and R=6 Ω; find its time constant.
Answer and reasoning · Resposta e raciocínio
Voltage=10e^(−2)=1.353 V; half-time=3 ln2=2.079 s. RL τ=L/R=0.05 s.
gre_physics:mixed-revision:46
In series RLC, R=10 Ω, L=0.1 H, C=0.001 F, V_rms=20 V and ω=200 rad/s. Find X, impedance magnitude, current, phase direction and mean power.
Answer and reasoning · Resposta e raciocínio
X=20−5=15 Ω; |Z|=sqrt(325)=18.028 Ω; I_rms=1.1094 A. Current lags voltage. P=400·10/325=12.308 W. This is off resonance; ω₀=100 rad/s.
gre_physics:mixed-revision:47
A linear beam of intensity 64 W/m² passes ideal analyser axes at 45° and then 75° relative to its initial direction. Find the two outputs. What would a circular input of the same intensity transmit through one analyser?
Answer and reasoning · Resposta e raciocínio
Outputs 64 cos²45°=32 and 32 cos²30°=24 W/m²; the second relative angle is 30°, not 75°. Circular input gives 32 W/m² through one ideal analyser.
gre_physics:mixed-revision:48
At a fixed point E_x=5 cosωt and E_y=5 cosωt. Classify the field. For ideal incidence from n₁=1.2 into n₂=1.6 find θ_B and the complementary refracted angle.
Answer and reasoning · Resposta e raciocínio
The field is linear at 45°, not circular: the components have equal phase. θ_B=atan(4/3)=53.13°; refracted angle 36.87°.
gre_physics:mixed-revision:49
Two equal finite slits have λ=400 nm, L=2 m, a=0.20 mm and d=1.0 mm. Find fringe spacing, central-envelope width, first missing orders and nominal centres inside the central envelope.
Answer and reasoning · Resposta e raciocínio
Spacing 0.8 mm, width 8 mm, missing ±5 and nine nominal centres −4 through +4. Boundary orders ±5 are dark.
gre_physics:mixed-revision:50
For a uniform far-field single slit, calculate I/I_c at β=π/3. If slit width triples while λ and L remain fixed, compare small-angle central widths.
Answer and reasoning · Resposta e raciocínio
(sin(π/3)/(π/3))²=27/(4π²)≈0.6839. Central width becomes one third. Amplitude and intensity ratios are different.
gre_physics:mixed-revision:51
A diverging lens has f=−20 cm and a real object 60 cm away. Find signed image distance and magnification. Contrast a concave mirror f=20 cm with the same object distance.
Answer and reasoning · Resposta e raciocínio
Lens s′=−15 cm and M=+1/4, virtual upright reduced. Concave mirror s′=30 cm and M=−1/2, real inverted in front.
gre_physics:mixed-revision:52
A converging lens at x=0 has f=10 cm and real object x=−30 cm. A second f=10 cm lens is at x=10 cm, before the first predicted image. Find its signed object distance and final image coordinate.
Answer and reasoning · Resposta e raciocínio
First predicted image x=15 cm. At the second lens incoming rays converge toward a point 5 cm beyond it, so s₂=−5 cm. 1/s₂′=1/10+1/5=3/10, giving s₂′=10/3 cm and final x=40/3 cm. M₁=−1/2, M₂=+2/3 and total −1/3.
gre_physics:mixed-revision:53
Two parallel heat paths have resistances 0.5 and 1.5 K/W between 50°C and 20°C. Find each rate, total rate and equivalent resistance.
Answer and reasoning · Resposta e raciocínio
Rates 60 W and 20 W; total 80 W; equivalent R=30/80=0.375 K/W. They share the temperature difference rather than the heat rate.
gre_physics:mixed-revision:54
Ice already at its melting point has mass 0.04 kg and specified latent heat 3×10^5 J/kg. It receives 9000 J with no losses. Find melted mass and explain why mcΔT is insufficient.
Answer and reasoning · Resposta e raciocínio
Melted mass=Q/ℓ=0.03 kg; 0.01 kg ice remains. Temperature stays at the melting point during this ideal phase conversion; latent energy is not a temperature rise.
gre_physics:mixed-revision:55
A two-state fixed spectrum has energies 0 and Δ with Δ/(kBT)=ln4. Calculate U/Δ, Var(E)/Δ² and C/kB.
Answer and reasoning · Resposta e raciocínio
p_exc=1/5, so U/Δ=1/5, variance=1/5−1/25=4/25; C/kB=4(ln4)²/25≈0.307490. Differentiate at fixed Δ.
gre_physics:mixed-revision:56
A canonical two-level model has a unique ground state. Describe U, S and C at low temperature and at arbitrarily high temperature; then shift both levels by fixed ε.
Answer and reasoning · Resposta e raciocínio
Low T: U→0, S→0, C→0 for energies 0,Δ. High T: U→Δ/2, S→kB ln2, C→0. Offset ε shifts U/F by ε but leaves S, variance and C unchanged.
gre_physics:mixed-revision:57
At (ε−μ)/(kBT)=ln9, compute single-state Fermi, Bose and Maxwell–Boltzmann means. Count two identical bosons in four complete states and two identical fermions in the same states.
Answer and reasoning · Resposta e raciocínio
Fermi 1/10, Bose 1/8, dilute approximation 1/9. Bosonic patterns choose(5,2)=10; fermionic choose(4,2)=6.
gre_physics:mixed-revision:58
An oscillator has fixed spacing ε and ε/(kBT)=ln4. Find thermal energy above zero point divided by ε and C/kB. State the classical high-T and frozen low-T heat-capacity limits.
Answer and reasoning · Resposta e raciocínio
Energy/ε=1/3; C/kB=4(ln4)²/9≈0.854139. High T C→kB; low T C→0. The fixed zero-point ε/2 does not change these derivatives.
gre_physics:mixed-revision:59
A normalised well state in 0<x<5 nm receives weak γx with γ=0.012 eV/nm. Find its first-order shift. Explain what changes if the perturbation is γ(x−2.5 nm).
Answer and reasoning · Resposta e raciocínio
The first shift is γL/2=0.03 eV. The centred odd perturbation has zero first-order diagonal expectation; off-diagonal mixing and higher-order shifts can remain.
gre_physics:mixed-revision:60
Within a two-state exact degeneracy, a perturbation is ε[[3,−1],[−1,3]], ε>0. Find first-order shifts and the symmetric/antisymmetric combinations.
Answer and reasoning · Resposta e raciocínio
Symmetric (1,1)/√2 gives 2ε; antisymmetric (1,−1)/√2 gives 4ε. The two diagonal entries 3ε are not the first-order energy shifts.
gre_physics:mixed-revision:61
Two electrons can access four spatial orbitals with two spin states each. Count complete-state occupations for two electrons. Which spin sector can put both in one spatial orbital?
Answer and reasoning · Resposta e raciocínio
Eight complete states give choose(8,2)=28 patterns. Same-orbital double occupation needs antisymmetric spin, the singlet.
gre_physics:mixed-revision:62
For distinct orthonormal orbitals a,b, classify the exchange symmetry of Ψ_+χ_triplet and Ψ_−χ_singlet. Are either allowed two-electron total states?
Answer and reasoning · Resposta e raciocínio
Both products are symmetric overall: (+)(+) and (−)(−). Neither is allowed for two identical electrons. Allowed pairings are Ψ_+ with singlet or Ψ_− with triplet.
gre_physics:mixed-revision:63
A finite-well ground root is ρ=2, z=1.02986653. Find E/V₀ and κa. Why should the well-edge amplitude not be set to zero?
Answer and reasoning · Resposta e raciocínio
E/V₀=z²/4−1≈−0.734844; κa≈1.714461. Finite boundaries match the nonzero oscillatory interior to decaying tails; zero-edge conditions belong to infinite walls.
gre_physics:mixed-revision:64
For a one-electron Z=2 ion at n=4, find approximate bound energy and spatial degeneracy. For l=3,m_l=−2, give L magnitude and L_z.
Answer and reasoning · Resposta e raciocínio
E≈−13.6·4/16=−3.4 eV; spatial degeneracy 16. L=√12ℏ and L_z=−2ℏ. Spin/fine structure and reduced mass have been excluded.
gre_physics:mixed-revision:65
An electron in the uncoupled magnetic regime has m_l=−1,m_s=+1/2. Find its linear shift. A weak LS-coupled L=1,S=1/2,J=3/2,m_J=−3/2 level is instead placed in 0.5 T: find its shift using μ_B=5.788×10⁻⁵ eV/T.
Answer and reasoning · Resposta e raciocínio
Uncoupled shift is zero. In the other regime g_J=4/3, so g_J m_J=−2 and the shift is −5.788×10⁻⁵ eV. Regime-specific labels cannot be interchanged.
gre_physics:mixed-revision:66
A degenerate opposite-parity pair has supplied real dipole matrix [[0,d],[d,0]], d=1.5 e·nm. For F=2×10⁵ V/m find the level shifts and splitting. What would a nondegenerate parity-state diagonal expectation alone predict at first order?
Answer and reasoning · Resposta e raciocínio
Shifts ±0.0003 eV, separation 0.0006 eV. A nondegenerate parity-state expectation is zero at first order; it cannot describe the off-diagonal splitting in this degenerate model.
gre_physics:mixed-revision:67
For a 5000 K grey surface with ε=0.4,A=3×10⁻⁵ m² and T_env=1000 K, find wavelength peak using b=2.898×10⁻³ m·K and net power with σ=5.670×10⁻⁸. State the wavelength-independent-emissivity assumption.
Answer and reasoning · Resposta e raciocínio
Peak 579.6 nm under the grey spectral model. Net power 0.4·3×10⁻⁵·5.670×10⁻⁸(5000⁴−1000⁴)=424.5696 W. Emissivity is independent of wavelength and the large surroundings fill the relevant view.
gre_physics:mixed-revision:68
A heavy-nucleus one-electron Z=2 ion emits n=4→2. Find photon energy/wavelength, n=4 Bohr radius, and ionisation energy from n=4, using 13.6 eV and 1240 eV·nm constants. Distinguish the Bohr nℏ from wave angular momentum.
Answer and reasoning · Resposta e raciocínio
Photon 10.2 eV; λ≈121.5686 nm; Bohr radius 8a₀; n=4 ionisation 3.4 eV. Wave orbital L=√[l(l+1)]ℏ with l=0,…,3; it is not fixed to 4ℏ for every n=4 orbital.