Tiêu đề K-2616 Mathematical Science (Paper - II).pdf ✅

Total Number of Pages : 8 Paper II 4 K-2616 22. In X = [0, 2] ∪{3} with the topology given by d(x, y) = x − y , Int.( ) [1, 2] ∪{3} = (A) (1, 2) (B) (1, 2)∪{3} (C) (1, 2] ∪{3} (D) [1, 2] ∪{3} 23. The only compact subsets of the real line IR with the usual distance metric are (A) φ and IR (B) Closed intervals (C) Finite union of closed intervals (D) Closed and bounded sets 24. The number of linearly independent solutions of the form y = xr of the differential equation 12y 0 dx dy 6x dx d y x 3 3 3 − + = is (A) 0 (B) 1 (C) 2 (D) 3 25. The partial differential equation + ∂ ∂ + − ∂ ∂ ∂ − − ∂ ∂ 2 2 2 2 2 2 2 2 y z y(y 1) x y z (y 1)x x z x 0 y z y x z x = ∂ ∂ + ∂ ∂ is hyperbolic in a region in the xy – plane if (A) x ≠ 0 and y = 1 (B) x = 0 and y ≠ 1 (C) x ≠ 0, y ≠ 1 and y ≠ 2 (D) x = 0 and y = 1 26. The partial differential equation of the set of all right circular cones whose axes coincide with z-axis is (A) y z y x z x ∂ ∂ = ∂ ∂ (B) y z x x z y ∂ ∂ = ∂ ∂ (C) 2 2 2 2 2 2 y z y x z x ∂ ∂ = ∂ ∂ (D) 2 2 2 2 2 2 y z x x z y ∂ ∂ = ∂ ∂ 27. In the motion of a two particle system, if two particles are connected by a rigid weightless rod of constant length, then the number of degrees of freedom of the system is (A) 5 (B) 2 (C) 3 (D) 6 28. Extremal for the variational problem dx dx dy 1 x dx dy I [y(x)] 2 x x 1 0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + ∫ is a solution of the differential equation (A) 0 dx dy dx dy x2 + = (B) 0 dx dy 2 dx d y 2 2 + = (C) 0 dx dy 2 dx d y x 2 2 + = (D) 0 dx dy 2x dx d y 2 2 + =