BCS-054 Solved Assignment 2024-25 (July-January) || IGNOU BCA 5th Semester

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In this video, we will explore key concepts in numerical methods, including fixed-point number representation, round-off errors, and various mathematical operations. We will solve systems of equations using the Gauss elimination method and iterative techniques, and determine the roots of equations using multiple methods. Additionally, we’ll discuss interpolation, population estimation, and derivative calculations.

Question 1:
(a) Explain each of the following concepts, along with at least one suitable example for each:
(i) Fixed-point number representation
(ii) Round-off error
(iii) Representation of zero as a floating-point number
(iv) Significant digits in a decimal number representation
(v) Normalized representation of a floating-point number
(vi) Overflow (6 Marks)

(b) Explain with a suitable example that in computer arithmetic, the multiplication operation (*) may not be distributive over addition. (2 Marks)

(c) Find out how many decimal places the value 22/7 is accurate as an approximation of 3.14159265, where the latter is the value of π, calculated up to 8 decimal places. (6 Marks)

(d) Calculate a bound for the truncation error in approximating f(x) = sin x by sin(x) = x - x³/3! + x⁵/5!, where -1 ≤ x ≤ 1. (3 Marks)

(e) Approximate the value of (3.7)⁻¹ using the first three terms of Taylor’s series expansion. (3 Marks)

Question 2:
(a) Solve the system of equations:
4x₁ + x₂ + 2x₃ = 16
2x₁ + 5x₂ + 3x₃ = 19
3x₁ + 2x₂ - x₃ = 12
using the Gauss elimination method with partial pivoting. (4 Marks)

(b) Perform four iterations (rounded to four decimal places) using:
(i) Jacobi Method
(ii) Gauss-Seidel Method
for the following system of equations:
-5x₁ - 8x₂ + x₃ = -4
1 - 4x₂ + x₃ = -4
-2x₁ + x₂ - 6x₃ = -18
With initial guess (0, 0, 0)ᵀ. (8 Marks)

Question 3:
(a) Determine the smallest roots of the following equation:
f(x) = x²cos(x) + sin(x) = 0
using:
(i) Regula-falsi method
(ii) Newton-Raphson method
(iii) Bisection method
(iv) Secant method (8 Marks)

Question 4:
(a) Explain the role of interpolation in solving numerical problems. (2 Marks)
(b) Express Δ³f₁ as a backward difference. (2 Marks)
(c) Express Δ³f₁ as a central difference. (2 Marks)
(d) For the following data, develop a difference table and find forward differences and backward differences:

Question 5:
(a) Using the given census data, estimate the population for the year 2006 using:
(i) Stirling's central difference formula
(ii) Newton’s forward formula for the year 1992
(iii) Newton’s backward formula for the year 1980 (10 Marks)

Question 6:
(a) Find the values of the first and second derivatives of f(x) at x = 76 from the following table using the 0(h²) forward difference method. Also, find the truncation error (TE) and actual errors. (5 Marks)

Question 7:
(a) Compute the value of the integral:
∫(8.4 to 10.4) (5x + 4x² + 3) dx using:

Conclusion:
Join us as we delve into these important numerical methods and techniques! Don’t forget to like, share, and subscribe for more educational content!
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