SUBJECT: - Numerical Methods and Computer Programming. Max Marks:- 80 Duration: - 3 HRS Q 1(a) Write a C-program to find root of equation using Newton-Raphson method. Q 4(a) Solve the system of equation by matrix inversion method. (7M) (7M) Q 4(b) Use Gauss-Elimination to solve: Q 1(b) Find the value of θ, when amplitude of sine wave becomes half of its peak positive value. Initial interval is (0, 1). (6M) OR Q 2(a) Explain the “Newton-Raphson” to find the roots of the given equation. (6M) Q 2(b) Find a real root of equation , correct to four decimal places by using the iteration method. (7M) Q 3(a) Solve the following system of equations by Gauss – Siedel Iteration method. (6M) Q 5(a) Derive the Newton‟s Forward difference interpolation formula. (7M) Q 5(b) From the following data, find the value of corresponding to using Lagrange‟s Interpolation formula: 1 1 1.2 1.095 1.3 1.14 1.4 1.183 1.5 1.225 (7M) Q 3(b) Solve the system of equations by Gauss – elimination method (7M) OR Q 6(a) From the following table, calculate the number of students who obtained marks more than 65: (6M) OR 30-40 40-50 50-60 60-70 70-80 31 42 51 35 (7M) 31 Q 6(b) Following values of x and y are given: 1 1 2 5 3 11 4 8 (7M) Q 9(a) If , using modified Euler‟s method find given , taking step size of Q 9(b) Solve the simultaneous differential equations: for Using Runge – Kutta fourth order method. Initial values are at OR Q 10(a) Write down the „C‟ program to find the solution of ordinary differential equation using fourth order Runge-Kutta method. (7M) Q 10(b) Estimate y (0.1) correct to four decimal places using Taylor‟s series method if, Using Cubic Spline, show that Q 7(a) The velocity of a car running on a straight road at an interval of 2 minutes are given below: 0 0 2 22 4 30 6 27 8 18 10 7 (7M) 12 0 Find the distance covered by the car. Q 7(b) Evaluate: ∫ using Romberg‟s method. (7M) Q 11(a) What is object oriented programming? And what are the advantages of OOP paradigm? (6M) Q 11(b) Explain various control statements with suitable examples? (7M) OR Q 12(a) Explain the benefits of object oriented programming. (6M) Q 12(b) What do you mean by dynamic binding? How it is useful in OOP? (7M) OR Q 8(a) Derive the formula for numerical integration using Trapezoidal rule. (7M) Q 8(b) The population of a certain town is given below. Find the rate of growth of the population in 1931 and 1941. 1971 1931 1941 1951 1961 40.62 60.80 79.95 103.56 132.65