Spring 2023_MTH603_1 Solution

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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:

While solving a system of linear equations by Gauss Jordon Method, after all the elementary row operations if there lefts also zeros on the main diagonal then which of the is true about the system?

The question asked that while solving a system of linear equations where ghost Children method, after all the elementary operations, if their lives are all widows on the main diagonal, then which of the following history in the system. So first we have to know about the Gaussian elimination method. So Gaussian elimination is the name of the matter. We used to perform the three types of metrics. Cooperation on an undocumented metrics coming from a linear system of equations in order to find the solutions for such a system. This technique is also cultural reduction and it conjures up two stages forward elimination and backs institutions. The forward elimination estates refers to the road except needed to simplify the metrics in questions into the chloroform such states has the proposed to demonstrate if the system of equations for trade in the metrics have a unique possible solutions infinitely many solutions or just no solutions at all, he found that the system has no solutions, then there is no reason to continue the reduction the magic through the next states. So according to the given a statement, the correct officer is an officer and a. That each system me how infinitely many solutions. Thank you This after applying all the elementary row operations on the system. If the main diagonal is still conjures of zeros, that means that the system may have infinitely many solar cells. Thank you

@zaasmi said in MTH603 Quiz 3 Solution and Discussion:

Euler’s Method numerically computes the approximate ________ of a function.

Euler’s method is a numerical tool for approximating values for solutions of differential equations.

A system is inconsistent if it has no solution. In a system of two equations in two variables, the equations are dependent if one equation is a multiple of the other. Dependent systems have an infinite number of solutions – every point is a solution.

@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 ,