Tiêu đề K-2517 (Physical Science).pdf ✅

Total Number of Pages : 8 K-2517 3 Paper II 9. A 3 × 3 matrix has eigen values 0, 2 + i and 2 – i. Which of the following statements is correct ? (A) The matrix is Hermitian (B) The matrix is unitary (C) The matrix is singular (D) The matrix is non-singular 10. There are 40 persons in a room all born in the month of April. The probability that at least two of them share the same birthday is closest to (A) 0.4 (B) 0.7 (C) 0.8 (D) 0.9 11. A particle of mass m is thrown upwards from the surface of the earth with initial velocity components ux , uy, along the horizontal and vertical directions, respectively. The trajectory therefore is given by x(t) = ux t, y(t) = uyt – 2 1 gt2. The instant of time at which the acceleration and velocity vectors are perpendicular to each other is given by (A) uy g (B) ux g (C) ux − uy g (D) (ux + uy ) g 12. The phase curve of a freely falling body under gravity is (A) ellipse (B) hyperbola (C) stable node (D) stable star 13. The fixed point of a one-dimensional harmonic oscillator obeying the dynamical equation x w x 0 2 - + = , is (A) unstable spiral (B) saddle (C) stable spiral (D) elliptic 14. A planet is revolving around a star in an elliptic orbit. The ratio of the farthest distance to the closest distance of the planet from the star is 1.5. The eccentricity of the orbit is given by (A) 1.5 (B) 0.2 (C) 0 (D) 0.4 15. If a particle experiences a force F r r = (where r denotes the instantaneous position vector of the particle) the force is identified to be (A) central and conservative (B) non-central and non-conservative (C) non-central and conservative (D) central and non-conservative 16. The Lagrangian is 2 dx dy L 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + , where y(x) is a function of x. The stationary condition on the action, ∫ = x2 x1 S L dx with fixed end points x1, x2 leads to one of the following expressions with a, b constants. The correct option is (A) y = ax + b (B) y = ax2 + b (C) y = a/x + b (D) y = a/x2 + b 17. A particle of mass m is subjected to one-dimensional potential ( ) 2 4 x 4 kx 2 1 V x λ = − + . If the particle executes small amplitude oscillations about the minimum of the potential the frequency of oscillation is, [ λ and k are positive constants.] (A) m k (B) m λ (C) k λ (D) m 2k 18. The elements of a moment of inertia tensor of a rigid body expressed as a 3 × 3 matrix in the body fixed co-ordinate system as, ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = 0 0 0 0 0 0 2 0 2 0 0 I I I I I I , where I0 > 0 the rigid body is (A) spherical (B) symmetric (C) asymmetric (D) hyperboloid
Total Number of Pages : 8 Paper II 4 K-2517 19. Which of the following gives valid upper and lower bounds on the angle θ between the angular momentum L and the angular velocity ω of a rigid body ? (A) 0 ≤ θ ≤ π (B) 4 3 0 π ≤ θ ≤ (C) 6 7 0 π ≤ θ ≤ (D) 2 0 π ≤ θ ≤ 20. Volume of a sphere is V0 in its rest frame. Its volume V in an inertial frame moving with a speed c/2 along x-axis relative to the rest frame is (A) V < V0 (B) V = V0 (C) V = 2V0 (D) V → ∞ 21. The electric flux through any closed surface is a measure of (A) the total charge inside the surface (B) the total charge outside the surface (C) the total charge at the surface (D) the total charge, both, inside and outside the surface 22. Which of the following Maxwell’s equations need modification on assumption of the existence of the magnetic monopoles ? (A) ∇ . D = ρ (B) ∇ .B = 0 (C) t B E ∂ ∂ ∇ × = − (D) t D H J ∂ ∂ ∇ × = + 23. Match the following : P) Biot-Savart law 1. Ο = ∫ E . dl 0 Q) Gauss’ law 2. 2 cr d r dB × = l I R) Ampere’s law 3. E 0 ∇ . = ρ ∈ S) ∇ ×E = 0 4. B J0 ∇ × = μ (A) P – 2, Q – 1, R – 3, S – 4 (B) P – 1, Q – 3, R – 4, S – 2 (C) P – 2, Q – 3, R – 4, S – 1 (D) P – 4, Q – 2, R – 3, S – 1 24. A plane polarized light is refracted by a dielectric medium of refractive index, μ . If the angle of incidence is i=tan–1μ , the angle of refraction is (A) r = π 2 (B) = μ −1 r tan (C) − μ π = −1 tan 2 r (D) = π − μ −1 r tan 25. A parallel plate capacitor is filled with insulating material of dielectric constant, ∈r . As a result, the following effects are observed (A) the dielectric will reduce electric field and potential difference (B) the dielectric will reduce electric field and increase potential difference (C) the dielectric will increase electric field and the potential difference (D) the dielectric will not alter the electric field and the potential difference 26. The electric field in a source free region is given by j byˆ i ˆ E = x + . The value of b is (A) –1 (B) 0 (C) 1 (D) ∞ 27. Consider three charges 2q, –q, –q kept at (0, 0, a), (0, 0, 0) and (0, 0, –a). The electric dipole moment is (A) zero (B) 3qa along the z-axis (C) qa along the z-axis (D) dependent on the origin 28. An electrostatic field is said to be conservative, when (A) divergence of the electric field is zero (B) curl of the electric field is zero (C) curl of the electric field is equal to t B ∂ ∂ − (D) the Laplacian of the electric field is zero