BSc IT Sem II 2022 2023 2023 NUMERICAL METHOD Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Attempt ANY THREE of the following. 15 marks
   • a) Suppose 1.414 is used as an approximation to V2. Find the absolute and relative errors
   • b) Write short note on Conservation law of engineering problem
   • c) Find the Truncation error in the expansion of f(x) =e?* evaluate first six terms in the series for x = 3.5
   • d) Explain blunders, formulation errors and data uncertainty
   • e) Let f(x) using 3-digit arithmetic and determine the absolute &
   • f) Define -1) Significant digit 2) Error 3) Total numerical error 4) Round -off error
   • 5) Error propagation
2. Q2 Attempt ANY THREE of the following. 15 marks
   • a) Using Secant Method, find the root of f(x) = cosx — xe*= 0 taking the initial approximations
   • b) Find the smallest positive root of f(x) = — 5x + 1= 0 by performing five iterations of
   • c) Perform five iterations of Newton Raphson method to obtain the approximate value of equation, x = 173 starting with the initial approximation xp = 2
   • d) For f(x) = x-e~* = 0 determine the initial approximation to find the smallest positive root Find the root correct to four decimal places using Regula False method up to four iterations
   • e) Construct the divided difference table using Newton’s Interpolation for the given data and hence find the interpolating polynomial
   • f) Solve by Lagrange’s interpolation with the help of given data if f (1) = 3, f (3) = 5, = 9,
3. Q3 Attempt ANY THREE of the following
   • a) Solve the system 6x + y = 20,x + 4y + 5z=7 by using
   • b) Solve the system 5x + 3y +9z=2,7x+2y+Z=3,x+8y +z = 3 by using Gauss
   • c) From the data table given below obtain < and at x = | by Newton divided VCD SEM II FYIT NUMERICALMETHOD 75MARKS 2%HRS \
   • d) Solve by Trapezoidal rule if x? dx dividing into six parts
   • e) Solve by rule if with h= 0.2
   • f) Evaluate f (2), f"(2) by Lagrange’s interpolation differentiation with the help of given data
4. Q4 Attempt ANY THREE of the following C | gy
   • a) Solve by Simple Euler method =x+5y,y(0) = 1, find y at x = 0.2 where h =0.2
   • b) Solve by Runge-Kutta forth order if =x? = 1, find y at x = 0.5 where h
   • c) by Taylor’s method up to fifth order derivative if + 1 = 1, find y at x = 2 where h= 1
   • d) Fit the equation of Straight line by Least Square method with the help of given data
   • e) Fit the equation of 2"" degree of polynomial by least square method with the help of given
   • f) Evaluate equation X on Y and Y on X, , by, if 9x + 3y = + By = 11
5. Q5 Attempt ANY THREE of the following
   • a) Maximize subject to constraints, 2x + 3y < 13, Indicate the feasible region on graph and maximize the function Z = 6x+3y
   • b) Give a mathematical formulation of the following L.P.P. The standard weight of a special are purpose brick is 5 kg and it contains ingredients B, costs Rs 5 per kg. and costs Rs 8 kg. Strength considerations dictate that the brick contains not more than 4 kg of B, least 2 kg of Bz Determine the amount of ingredients and so that the cost of the brick way be minimum Solve the problem graphically
   • c) Find the solution of parabolic equation =2 given u(0,t) =0 , u(4,t) =0 u(x,0) =x (4
   • x) . Assume h =1 Find the values of u up tot = 5
   • d) Using Crank -Nicholson Method, Solve the equation u,, = subject to u(x.0) =0 , u(0.t) =0 and u(1,t) — 1 .Compute t for one time step taking h=1/4
   • e) Classify the following equations in elliptic, parabolic and hyperbolic ii) VCD FYIT NUMERICAL MARKS 2%HRS Solve the elliptic equation Uxx + Uyy = 0 for the following square mesh with boundary values as_ shown in figure by gauss Seidel iteration

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