If system of linear equations (a – 1) x + z = a, x + (b – 1) y = b and y + (c – 1) z = $γ$ where a, b, c $\in$ I and $γ \in R,$ does not have a unique solution, then maximum possible value of $| a + b + c |$ is

Option 1 -

Option 2 -

Option 3 -

Option 4 -

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Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 4

Detailed Solution:

$(a - 1) x + 0 y + z = α$ ……(1)

$x + (b - 1) y + 0 z = β$ ……(2)

$0 x + y + (c - 1) z = γ$ ……(3)

$| \begin{matrix} (a - 1) & 0 & 1 \\ 1 & (b - 1) & 0 \\ 0 & 1 & (c - 1) \end{matrix} | = 0$ For no unique solution D = 0

$(a - 1) (b - 1) (c - 1) + 1 = 0$

$∴ a = 2; b = 2; c = 0$

Hence, |a + b + c| = 4

<math> <mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mi>α</mi> </mrow> </math>        ……(1) <math> <mrow> <mi>x</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>y</mi> <mo>+</mo> <mn>0</mn> <mi>z</mi> <mo>=</mo> <mi>β</mi> </mrow> </math>                       ……(2) <math> <mrow> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>z</mi> <mo>=</mo> <mi>γ</mi> </mrow> </math>                       ……(3) <math> <mrow> <mrow> <mo>|</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> <mtd columnalign="center"> <mn>0</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </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"> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math>           For no unique solution D = 0 <math> <mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mrow> </math>             <math> <mrow> <mo>∴</mo> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mo>;</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mo>;</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mrow> </math>            Hence, |a + b + c| = 4

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If system of linear equations (a – 1) x + z = a, x + (b – 1) y = b and y + (c – 1) z = $γ$ where a, b, c $\in$ I and $γ \in R,$ does not have a unique solution, then maximum possible value of $| a + b + c |$ is

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If system of linear equations (a – 1) x + z = a, x + (b – 1) y = b and y + (c – 1) z = γ where a, b, c ∈ I and γ ∈ R , does not have a unique solution, then maximum possible value of | a + b + c | is

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If system of linear equations (a – 1) x + z = a, x + (b – 1) y = b and y + (c – 1) z = $γ$ where a, b, c $\in$ I and $γ \in R,$ does not have a unique solution, then maximum possible value of $| a + b + c |$ is