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MTH501 ASSIGNMENT NO. 1 FALL 2022 |
MTH501 ASSIGNMENT NO. 1 FALL 2022 || 100% RIGHT SOLUTION || LINEAR ALGEBRA || BY VuTech
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ASSIGNMENT#1
Fall 2022
Total marks: 15
Due
Date: 29th November, 2022
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Question 1: Marks = 05
Find a parametric equation of the line through `\bar u
` and `\bar v` , where
SOLUTION:
Given that,
let suppose that
Now, we find the direction vector
Here
`a = 6, b = - 7` c = 11
Now parametric equations are
`x = x_0 + at`
`\Rightarrow x = - 6 + 6t`
`y = y_0 + bt`
`\Rightarrow y = 3 - 7t`
`z = z_0 + ct`
`\Rightarrow z = - 4 + 11t`
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Question 2: Marks = 10
Determine the value(s) of h and k such that the given
system
`x + 5y - 4z = 3,`
`3x + 12y - 12z = - 3,`
`2x + 7y + hz = k`
has
1. no
solution,
2. a unique
solution,
3. many
solutions.
SOLUTION:
Given\,system
`x + 5y - 4z = 3`
`3x + 12y - 12z = - 3`
`2x + 7y + hz = k`
Now, in the matrix form
let,
So,
`AX = B`
Now, in the form of Augmented matric
As
`C = [A:B]`
Now, their consistent matrices are
Now, discuss the given conditions:
i. NO
SOLUTION:
`If h = - 8 and k \ne - 6`
then `the\[R(A) < R(B)`, there will be no solution.
For that R(A)=2 and R(B)=3.
ii. UNIQUE
SOLUTION:
`If h \ne - 8 and k \ne - 6`
then the R(A) = R(B) = 3 = no. of unknown, there will be
unique solutions.
For that R(A)=3 and R(B)=3.
iii. MANY
SOLUTION:
If `h = - 8 and k = - 6` then the R(A) = R(B) =
2 < no. of unknown, there will be many solutions.
Like that R(A)=2 and R(B)=2.
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