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Pune University - SY BSc (Sem - I) MATHEMATICS - II (B), April 2010

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Total No. of Questions : 4] [Total No. of Pages : 2 [3717] - 102 P259 S.Y. B.Sc. (Sem. - I) MATHEMATICS (B) : Numerical Analysis (New Course) Paper - II (B) Time : 2 Hours] Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. 3) Use of non - programmable calculator is permitted. [Max. Marks : 40 Q1) Attempt the following: [10] a) An approximate value of is given by 3.1427852 and it s true value is 3.14159265. Find absolute and relative error. b) Use Newton - Raphson method for evaluating 4 1 3 . c) Prove that - . d) If f(x) = e) Round - off following numbers to two decimal places. 1 , find f(a, b). x2 61.0561, 105.7845. f) Is it possible to solve the system of equations:3x + 2y = 5; 7x 5y = 8 by Gauss - Seidel iterative method? Justify. g) 2 Evaluate x3, take h = 1. h) Find the number of changes of signs of the polynomialf(x) = 2x7 6x4 + 8x3 15 = 0. i) State Sturm s theorem. j) Construct the divided difference table for the data. x: 1 f(x) : 21 1 15 2 12 P.T.O. 3 Q2) Attempt any two of the following: a) [10] Solve the system of equations; by Gauss - Seidel iterative method: 10x + y + z = 12; 2x + 10y + z = 13, 2x + 2y + 10z = 14. b) Find the number and position of real roots of the equation: f(x) = x4 x3 4x2 + 4x + 1 = 0. c) Solve the equation : logx = cosx by Newton - Raphson method. (Root lies between 1 and 2). Q3) Attempt any two of the following: a) [10] Determine least square polynomial of second degree for the data: x: 1 2 3 y: b) 0 1 6 17 34. Use Lagrange s interpolation formula, to express the function as sums of partial fractions. x3 + x 3 f(x) = 3 . x 2 x2 x + 2 1 c) 1 1 1 rd 3 th dx with h = by Simpson s s and rule. Compare 6 3 8 0 1+ x the result with actual value. Evaluate Q4) Attempt any one of the following: a) [10] Use Runge - Kutta second order formula to find y(0.5); Given dy 1 = dx x + y with y(0) = 1. b) Given : dy = x2 + y with y(0) = 1. dx Determine y(0.02), y(0.04) using Euler s modified method. Y [3717] - 102 2 4

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Additional Info : S.Y. B.Sc. Mathematics (B) (Semister - I) : Numerical Analysis (Paper - II(B)) (New Course), Pune University
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