BCA Semester 4 – Numerical Methods () Questions & Answers | Past TU Exam Papers

Question 1

Compute the root of equation \(x2−5x+6=0\) in the vicinity of \(x=5\) using Newton-Raphson method.

Marks: 5

Year: 2021 Final TU FOHSS

Newton–Raphson Method Given: The equation is \[ f(x) = x^2 - 5x + 6 \] We are asked to compute the root in the vicinity of \(x = 5\) using the Newton–Raphson method. --- Step 1: Formula of Newton–

Question 2

Write an algorithm and program to compute root of nonlinear equation using bisection method.

Marks: 10

Year: 2021 Final TU FOHSS

(A) Algorithm: Bisection Method Step 1: Define the nonlinear function \(f(x)\). Step 2: Choose initial guesses \(a\) and \(b\) such that \[ f(a)\cdot f(b) < 0 \] (This ensures a root lies between \(

Question 3

If the true value of π is 3.1415926 and its approximate value is given by 3.1428571. Find the absolute and relative errors.

Marks: 5

Year: 2022 Final TU FOHSS

Given: - True value of \( \pi \): \[ \pi{\text{true}} = 3.1415926 \] - Approximate value of \( \pi \): \[ \pi{\text{approx}} = 3.1428571 \] --- (a) Absolute Error \[ \text{Absolute Error} = \left| \p

Question 4

When would we not use N-R method? Find the root of the equation \[ x3−4x+1=0, \] lying in (0, 1) using Bisection method performing 10 iterations.

Marks: 10

Year: 2022 Final TU FOHSS

Part 1: When We Do Not Use Newton-Raphson Method Newton–Raphson (N-R) method may fail or is not recommended when: 1. Derivative is Zero or Near Zero - If \(f'(x) = 0\) at or near the root, iterat

Question 5

Define error. Explain the Taxonomy Errors.

Marks: 5

Year: 2023 Final TU FOHSS

1. Definition of Error In numerical computation, error is the difference between the true value and the approximate value of a quantity: \[ \text{Error} = \text{True Value} - \text{Approximate Value}

Question 6

Explain absolute and relative error. Find the relative error of number 5.6 if both of its digits are correct.

Marks: 5

Year: 2024 Final TU FOHSS

1. Definitions (a) Absolute Error (AE): The difference between the true value \(x{\text{true}}\) and the approximate value \(x{\text{approx}}\) of a quantity: \[ \text{Absolute Error} = |x{\text{tru

Question 7

On what type of equations Newton's methods can be applicable. Justify.

Marks: 5

Year: 2024 Final TU FOHSS

1. Type of Equations Applicable Newton's method (or Newton-Raphson method) is applicable to nonlinear equations of the form: \[ f(x) = 0 \] where: 1. \(f(x)\) is differentiable in the neighborhood of

Question 8

How can x we use Laterpolation techniques (methods) to approximate the value of the root for the functions whose derivative can't be found? Explain. Write a program to solve \(sin\space x - 2x + 1 = 0

Marks: 10

Year: 2024 Final TU FOHSS

1. Interpolation Techniques for Root Finding When the derivative \(f'(x)\) cannot be found, derivative-based methods like Newton-Raphson are not applicable. In such cases, interpolation techniques c