The values of $λ a n d μ$ such that the system of equations $x + y + z = 6, 3 x + 5 y + 5 z = 26, x + 2 y + λ z = μ$ has no solution, are:

Option 1 -

$λ = 2, μ \neq 10$

Option 2 -

$λ = 3, μ \neq 10$

Option 3 -

$λ = 3, μ = 5$

Option 4 -

$λ \neq 2, μ = 10$

2 Views | Posted a month ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 1

Detailed Solution:

System of equations can be written as

$(\begin{matrix} 1 & 1 & 1 \\ 3 & 5 & 5 \\ 1 & 2 & λ \end{matrix}) (\begin{array}{l} x \\ y \\ z \end{array}) = (\begin{array}{l} 6 \\ 26 \\ μ \end{array})$

$R'_{3} = R_{3} - R_{1}, R'_{2} = R_{2} - 5 R_{1} \Rightarrow$

$(\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 0 \\ 0 & 1 & λ - 1 \end{matrix}) (\begin{array}{l} x \\ y \\ z \end{array}) = (\begin{array}{l} 6 \\ - 4 \\ μ - 6 \end{array})$

Again $R''_{3} = R'_{3} - R'_{1} \Rightarrow (\begin{matrix} 0 & 1 & 1 \\ - 2 & 0 & 0 \\ 0 & 0 & λ - 2 \end{matrix}) (\begin{array}{l} x \\ y \\ z \end{array}) = (\begin{array}{l} 4 \\ - 4 \\ μ - 10 \end{array})$

The system will have no solution for

$λ = 2 a n d μ \neq 10 .$

System of equations can be written as <math> <mrow> <mrow> <mo>(</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>3</mn> </mtd> <mtd columnalign="center"> <mn>5</mn> </mtd> <mtd columnalign="center"> <mn>5</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>2</mn> </mtd> <mtd columnalign="center"> <mi>λ</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mi>x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mn>6</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mn>6</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>μ</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </math>                           <math> <mrow> <mi>R</mi> <msub> <mrow> <mo>'</mo> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>R</mi> <msub> <mrow> <mo>'</mo> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>5</mn> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>⇒</mo> </mrow> </math>                 <math> <mrow> <mrow> <mo>(</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mo>−</mo> <mn>2</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mi>λ</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mi>x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mn>6</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>μ</mi> <mo>−</mo> <mn>6</mn> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </math>                              Again <math> <mrow> <mi>R</mi> <mo>'</mo> <msub> <mrow> <mo>'</mo> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>R</mi> <msub> <mrow> <mo>'</mo> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mi>R</mi> <msub> <mrow> <mo>'</mo> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>⇒</mo> <mrow> <mo>(</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mo>−</mo> <mn>2</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mi>λ</mi> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mi>x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mn>4</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>μ</mi> <mo>−</mo> <mn>1</mn> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </math>  The system will have no solution for <math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>2</mn> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mi>μ</mi> <mo>≠</mo> <mn>1</mn> <mn>0</mn> <mo>.</mo> </mrow> </math>

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
687k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

The values of $λ a n d μ$ such that the system of equations $x + y + z = 6, 3 x + 5 y + 5 z = 26, x + 2 y + λ z = μ$ has no solution, are:

1 Answer

Learn more about...

Most viewed information

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

The values of λ a n d μ such that the system of equations x + y + z = 6 , 3 x + 5 y + 5 z = 2 6 , x + 2 y + λ z = μ has no solution, are:

1 Answer

Learn more about...

Most viewed information

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

The values of $λ a n d μ$ such that the system of equations $x + y + z = 6, 3 x + 5 y + 5 z = 26, x + 2 y + λ z = μ$ has no solution, are: