@cyberian said in MTH603 Assignment 1 Solution and Discussion:

Question 1 5 Marks
Find a root of the equation 𝑥3 − 3𝑥 − 5 = 0, in the interval (2,3) using Bisection Method after three Iterations.
Note: Accuracy up to four decimal places is required.

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MTH603 - Assignment 1 - Spring 2024 - Nabeela Wali - Solution file.pdf

@asad-saab
Fall 2023
MTH603
Assignment # 1
Section In charge: Husna Muzaffar Total Marks 20
Instructions

To solve this assignment you need to have a good grip on lectures 1-15. The course is segmented into four sections, each of which is supervised by a different faculty member. Information regarding the section in charge can be
found in the course information section on the LMS. A distinct assignment file has been given to each section, resulting in a total
of four separate assignment files. The relevant assignment file can be downloaded from the announcement section of the course. It is important to note that students can only view the announcements relevant to their respective sections. You will prepare the solution of assignment on Word file and upload at the assignment interface on LMS as per usual practice. Plagiarism in the submitted assignment will lead to a zero grade. Additionally, any student who submits a solution file that is not applicable to their section will also get a zero grade.
𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧# 𝟏: Marks 10 Solve the system of equations by using Crout’s method.
2𝑥 + 5𝑦 + 3𝑧 = 16

𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧# 𝟐:
Marks 10
3𝑥 + 𝑦 + 2𝑧 = 11 −3𝑥 + 7𝑦 + 8𝑧 = 10
Solve the following system of equations by using Jacobi′s iterative method for the first three iterations by taking initial starting of solution vector as (0,0,0). 8𝑥 − 2𝑦 − 2𝑧 = 3
−2𝑥 + 6𝑦 + 𝑧 = 9
−2𝑥+𝑦+7𝑧= 6

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@zareen said in MTH603 Assignment 1 Solution and Discussion:

Question #2: Solve the system of linear equations with the help of Gaussian elimination method.
2x + y + z = 9;3x −2y + 4z = 9;x +y-2z = 3

System of Linear Equations entered :

[1] 2x + y + z = 9
[2] 3x - 2y + 4z = 9
[3] x + y - 2z = 3

Solve by Substitution :

// Solve equation [3] for the variable y

[3] y = -x + 2z + 3

// Plug this in for variable y in equation [1]

[1] 2x + (-x +2z+3) + z = 9
[1] x + 3z = 6
// Plug this in for variable y in equation [2]

[2] 3x - 2•(-x +2z+3) + 4z = 9
[2] 5x = 15
// Solve equation [2] for the variable x

[2] 5x = 15

[2] x = 3
// Plug this in for variable x in equation [1]

[1] (3) + 3z = 6
[1] 3z = 3
// Solve equation [1] for the variable z

[1] 3z = 3

[1] z = 1
// By now we know this much :

x = 3
y = -x+2z+3
z = 1
// Use the x and z values to solve for y
y = -(3)+2(1)+3 = 2

Solution :
{x,y,z} = {3,2,1}

MTH603 MCQ’s for Final Term.pdf

Solution idea

@zaasmi said in MTH603 Grand Quiz Solution and Discussion:

A series 16+8+4+2+1 is replaced by the series 16+8+4+2, then it is called

Each number in the sequence is half the value of the number receding it. So the common difference in the series is dividing by two.

16÷2=8

8÷2=4

4÷2=2

2÷2=1

1÷2=½

The answer is ½ or 0.5

When you keep dividing by two, you will notice an interesting pattern: the denominator continues to increase by two, while the numerator value remains the same. That’s fascinating because in natural, whole numbers the numbers in the series would decrease by two.

1/4 , 1/8 , 1/16 etc.

@zaasmi said in MTH603 Quiz 2 Solution and Discussion:

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Spring 2020_MTH603_1_SOL.docx

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

which of the following points the max value of 2nd deruvative of function f(x)=-(2/x) in the inteval : [1,4] exits

To find where the maximum value of the second derivative of the function
𝑓
(
𝑥
)

−
2
𝑥
f(x)=−
x
2
​
exists in the interval
[
1
,
4
]
[1,4], we will first find the second derivative of the function and then determine where it achieves its maximum value within the given interval.

how many eigenvalues will exit corresponding to the function exp(ax)

@zainab-ayub said in MTH603 Quiz 2 Solution and Discussion:

Mth603 ka koi student hai tu plz yeh question bta dy kis trha solve ho ga Given the following data x:1 2 5 y:1 4 10 Value of 1st order divided difference f[2 , 5] is

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@zareen said in MTH603 Mid Term Past and Current Solved Paper Discussion:

If n x n matrices A and B are similar, then they have the same eigenvalues (with the same multiplicities).

True
False

Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is non-zero since P is invertible) is an eigenvector for B with eigenvalue λ.

Assignment No: 01
Question #1: Find the root of the equation, Perform three iteration of the equation,
ln (x −1) + sinx =0 by using Newton Raphson method.

Ans: Let f(x) = ln(x+1) + sinx = 0 and f(x) = 1/(x-1) + cosx

F (1.5) = ln(0.5) + (1.5) = - 0.0667

F(2) = ln(1) + sin(2) = 0.035

Since f (1.5) f (2) < 0 so roots lies in interval [1.5, 2]

Let x0 = 1.75 . x0 can be taken in the interval any real number [ 1.5 , 2 ], we let mid point

of this interval .
As we know Newton Raphson method is

Xn+1 = xn – f ( xn ) / f(xn)
First iteration
X1 = x0 –f(x0) / f(x0) = 1.75 - f(1.75) / f(1.75)
= 1.75 – (-0.2571 / 2.3329) = 1.8602
Second iteration:
X2 = x1 - f(x) / f(x) = 1.8602 –[ f(1.8602) / f(1.8602)]
= 1.8602 - ( -0.1181 / 2.1620 ) = 1.9148
Third iteration:
X3 = x2- f(x2) / f(x2) = 1.9148 –f(1.9148) / f(1.9148)
= 1.9148 – [-0.0556/2.0926]
= 1.9414
Question #2: Solve the system of linear equations with the help of Gaussian elimination method.
x + y + z = 6;2x − y + z = 3;x + z = 4

ANS: In Gaussian elimination method we convert the augmented matrix into reduce

Echelon form therefore,

Augmented matrix is

R2- 2R1 , R3 – R1

-1R2 , -1R3
R23
R3-3R2

X + Y+ Z = 6 ;………………….(1)
Y = 2,
Z = 3
Put into eq (1),
we get X = 1 ,