Title 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
Paper III 3 K-2614 Total Number of Pages : 16 The extremal for the functional 6. dx ) y3 y x( 2 b a ′ + + ∫ is given by (A) y (x) = x 0 ≠ (B) y (x) = c, where c is any real constant (C) y (x) = 0 (D) y (x) = e Consider the Fredholm integral equation 7. ) y( e λ +) x(f =) x( φ y x 1 0 φ − ∫ dy .... (1) for a ,1 ≠ λ . If )1 ≤ x ≤ 0( given real function f(x) then which one of the following is the solution of (1) ? (A) 1 ) x(f ) x( − λ λ ) y(f e e dy − = φ y 1 0 − x ∫ ⎟ (B) ⎠ ⎞ ⎜ ⎝ ⎛ − λ + λ − = φ 1 1 dy ) y(f e e ) x(f ) x( y 1 0 − x ∫ (C) 1 ) x(f ) x( − λ λ dy ) y( e e + = φ y 1 0 x φ − ∫ (D) 1 ) x(f ) x( 2 2 − λ λ dy ) y(f e e + = φ y 1 0 − x ∫ The value of y (0.2) obtained by 8. Runge-Kutta method of fourth order, 1 ) 0( y, y x given that dx dy with = + = increment h = 0.2 is (A) 1.2426 (B) 1.2425 (C) 1.2428 (D) 1.2424 Consider the homogeneous linear 9. system n n1 1 11 1 x)t( a ... x)t( a dt dx + + = n n2 1 21 2 x)t( a ... x)t( a dt dx + + = • • • • • • • • • • • • . x)t( a ... x)t( a dt dx n nn 1 1n n + + = Let to be any point of [a, b] and let ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ φ ⋅ ⋅ ⋅ φ φ = φ n 2 1 be a solution of the above 0 ) t( system such that = 0 . Then φ for atleast one value of t in 0 ≠)t( φ (A) [a, b] ] b, a[ ∈t ∀ 0 =)t( φ (B) for all values 0 =)t(′ φ and 0 ≠)t( φ (C) of t0 ] b, a[ ∈t ≠ [a, b] with ∈ for all t 0 ≠)t( φ (D) 0t ≠t If A, B, C, D are nonempty sets, then 10. which one of the following statements is true ? ) D ∪B( ×) C ∪A( =) D× C( ∪) B× A( (A) ) D ∪B( ×) C ∪A( ⊃) D× C( ∪) B× A( (B) ) D ∪B( ×) C ∪A( ⊂) D× C( ∪) B× A( (C) ) D ∪B( ×) C ∪A( =) D× C( ∩) B× A( (D)