Tiêu đề 26 Mathematical Science * Paper-III.pdf ✅

K-2614 2 Paper III Total Number of Pages : 16 ( ) 1. n 3 n .... 3 2 1 n 1 lim n + + + + ∞ → is equal to (A) 0 (B) 1 ∞ (C) (D) 2 1 dx If a > 0 then 2. a x x log 2 2 + 0 ∫ ∞ is equal to (A) a2 a log π (B) 2 a log π (C) a a log π a g lo π (D) Which one of the following statements is 3. false ? (A) Every sequentially compact metrizable space is compact (B) Every locally compact Hausdorff space is regular (C) Every limit point compact space is compact (D) If X is locally compact Hausdorff space and A is open in X, then A is locally compact The sturm-Liouville problem 4. 0 y dx y d 2 2 0 =) π( y , y (0) = 0, = λ + 0 ≤ λ (A) has a non-trivial solution if (B) has a non-trivial solution if .... , 3, 2,1 = n, n = λ (C) has no non-trivial solutions if .... , 3, 2,1 = n , n2 = λ (D) has non-trivial solutions if ..... , 3, 2,1 n, n2 = = λ The solution to the heat equation 5. ut = 3uxx, 0 < x < 2, t > 0, ux (0, t) = ux (2, t) = 0, u (x, 0) = 3x, is ∑ (A) ∞ = π + = 1 n 12 2 3 )t , x( u n )1 ( 1 + n − 4 t n3 e 2 2π − ⎟ ⎠ ⎞ ⎜ ⎝ π ⎛ 2 x n cos ∑ (B) u(x, t) = ∞ = π + 1 n 12 2 3 n )1 ( 1+n − 4 t n3 e 2 2π − ⎟ ⎠ ⎞ ⎜ ⎝ π ⎛ 4 x n cos + (C) u(x, t) = 2 3 n 1 12 1 n ∑ ∞ = 4 π t n3 e 2 2π − ⎟ sin ⎠ ⎞ ⎜ ⎝ π ⎛ 2 x n (D) u (x, t) = n 1 4 12 2 3 1 n ∑ ∞ = + 4 t n3 e 2 2π − ⎟ cos ⎠ ⎞ ⎜ ⎝ π ⎛ 2 x n MATHEMATICAL SCIENCE PAPER – III question Each objective type questions. seventy-five (75) This paper contains Note : . compulsory questions are All marks. two (2) carries