BCA Semester 2 – Mathematics II – Unit 6: Computational Method (10Hrs)

Question 1

Solve the following system by Gauss–Seidel method: \[ -4x + y - z = -8 \] \[ 3x + 6y + 2z = 1 \] \[ x - y + 3z = 2 \]

Marks: 5

Year: 2022 Final TU FOHSS

$\textbf{Step 1: Rearrange equations for Gauss–Seidel form}$ Rewriting each equation to express one variable: From $ -4x + y - z = -8 $: \[ x = \frac{8 + y - z}{4} \] From $ 3x + 6y + 2z = 1 $: \[

Question 2

Using simplex method, find the optimal solution of \[ Z = 7x1 + 5x2 \] subject to \[ x1 + 2x2 \le 6 \] \[ 4x1 + 3x2 \le 6 \] \[ x1 \ge 0,\; x2 \ge 0 \]

Marks: 10

Year: 2022 Final TU FOHSS

Answer $\textbf{Step 1: Convert the problem into standard form}$ Introduce slack variables $ s1 $ and $ s2 $: \[ x1 + 2x2 + s1 = 6 \] \[ 4x1 + 3x2 + s2 = 6 \] Objective function: \[ Z = 7x1 + 5x2

Question 3

Using Simpson’s $ \tfrac{1}{3} $ rule, evaluate \[ \int0^1 \frac{1}{1+x}\,dx \] with 3 points of intervals. Find the error of approximation. How many points are to be considered to make the ap

Marks: 10

Year: 2022 Final TU FOHSS

$\textbf{Part (a): Simpson’s } \tfrac{1}{3} \textbf{ Rule}$ Simpson’s $ \tfrac{1}{3} $ rule for $ n = 2 $ (i.e., 3 points) is: \[ \inta^b f(x)\,dx \approx \frac{h}{3}\left[f(x0) + 4f(x1) + f(x2)\r

Question 4

Evaluate \[ \int0^2 \sqrt{1+x^3}\,dx \] by using Simpson’s $ \tfrac{1}{3} $ rule, taking $ n=4 $.

Marks: 5

Year: 2023 Final TU FOHSS

Step 1: Simpson’s $ \tfrac{1}{3} $ Rule For $ n $ even, $ n+1 $ points $ x0, x1, ..., xn $: \[ \inta^b f(x)\,dx \approx \frac{h}{3} \Big[ f(x0) + f(xn) + 4 \sum{\text{odd } i} f(xi) + 2 \sum{\

Question 5

Define $$\text{pivot element}$$, $$\text{pivot column}$$, and $$\text{consistency }$$in a system of equations. Using the simplex method, maximize \[ \quad F = 5x - 3y \] subject to \[ 3x + 2y \l

Marks: 10

Year: 2023 Final TU FOHSS

(a) Definitions 1. Pivot Element: The element of a simplex table used to perform row operations in order to introduce a new basic variable into the solution. It is located at the intersection of the

Question 6

Compute the approximate value of the integral \[ \int1^2 \frac{1}{1+x^2}\,dx \] by using the $$\textbf{composite trapezoidal rule}$$ with three points, and compare the result with the actual value.

Marks: 10

Year: 2023 Final TU FOHSS

Step 1: Composite Trapezoidal Rule For $ n $ subintervals ($ n+1 $ points): \[ \inta^b f(x)\,dx \approx \frac{h}{2} \Big[ f(x0) + 2\sum{i=1}^{n-1} f(xi) + f(xn) \Big] \] where \( h = \frac{b-a}{n}

Question 7

Using the trapezoidal rule, compute $ \int0^2 (2x^2 - 1)\,dx $ with 4 intervals. Find the absolute error of approximation from its actual value.

Marks: 5

Year: 2024 Final TU FOHSS

Step 1: Composite Trapezoidal Rule For $ n $ subintervals: \[ \inta^b f(x) \, dx \approx \frac{h}{2} \left[ f(x0) + 2 \sum{i=1}^{n-1} f(xi) + f(xn) \right] \] where $ h = \frac{b-a}{n} $, \( xi =

Question 8

Using Newton–Raphson method, find a root of $ x^3 - x - 4 = 0 $ between 1 and 2 correct to three decimal places.

Marks: 5

Year: 2024 Final TU FOHSS

Step 1: Newton–Raphson formula \[ x{n+1} = xn - \frac{f(xn)}{f'(xn)} \] Given: \[ f(x) = x^3 - x - 4 \] \[ f'(x) = 3x^2 - 1 \] --- Step 2: Choose initial guess The root lies between 1 and 2. Check \(

Question 9

Using the simplex method, find the optimal solution of the following linear programming problem. Maximize \[ z = 15x + 12y \] Subject to \[ 2x + 3y \le 21 \] \[ 3x + 2y \le 24 \] \[ x \ge 0,\; y

Marks: 10

Year: 2024 Final TU FOHSS

Step 1: Convert inequalities to standard form Add slack variables $ s1, s2 \ge 0 $: \[ 2x + 3y + s1 = 21 \] \[ 3x + 2y + s2 = 24 \] Objective function: \[ z = 15x + 12y \quad \Rightarrow \quad Z -