The existence of the unique solution of the system for equations:
$
x + y + z = \lambda \\
5x - y + \mu z = 10 \\
2x + 3y - z = 6 \\
$
depends on
A. $\mu $ only
B. $\lambda $ only
C. $\lambda $ and $\mu $ both
D. neither $\lambda $ nor $\mu $
Last updated date: 26th Mar 2023
•
Total views: 311.1k
•
Views today: 8.89k
Answer
311.1k+ views
Hint: Express the given system of equations in matrix form and find the determinant of the coefficients of x,y and z.
We will write given equations in the matrix form as $A.X = B$
Where $A = \left( {\begin{array}{*{20}{c}}
1&1&1 \\
5&{ - 1}&\mu \\
2&3&{ - 1}
\end{array}} \right)$ , \[X = \left( {\begin{array}{*{20}{c}}
x \\
y \\
z
\end{array}} \right)\] and \[B = \left( {\begin{array}{*{20}{c}}
\lambda \\
{10} \\
6
\end{array}} \right)\]
Now, we will find determinant of A i.e. $\left| A \right|$
\[
\left| A \right| = \left( {\begin{array}{*{20}{c}}
1&1&1 \\
5&{ - 1}&\mu \\
2&3&{ - 1}
\end{array}} \right) \\
\left| A \right| = 1\left( {1 - 3\mu } \right) - 1\left( { - 5 - 2\mu } \right) + 1\left( {15 + 2} \right) \\
\left| A \right| = 1 - 3\mu + 5 + 2\mu + 17 \\
\left| A \right| = 23 - \mu \\
\]
From the above equation, we can see that the uniqueness of the system depends only on $\mu $.
$\therefore $Correct option is A.
Note: In a practical case, a system of linear equations will have a unique solution if the lines
representing the equations intersect each other at only one unique point i.e. the lines are
neither parallel nor coincident.
We will write given equations in the matrix form as $A.X = B$
Where $A = \left( {\begin{array}{*{20}{c}}
1&1&1 \\
5&{ - 1}&\mu \\
2&3&{ - 1}
\end{array}} \right)$ , \[X = \left( {\begin{array}{*{20}{c}}
x \\
y \\
z
\end{array}} \right)\] and \[B = \left( {\begin{array}{*{20}{c}}
\lambda \\
{10} \\
6
\end{array}} \right)\]
Now, we will find determinant of A i.e. $\left| A \right|$
\[
\left| A \right| = \left( {\begin{array}{*{20}{c}}
1&1&1 \\
5&{ - 1}&\mu \\
2&3&{ - 1}
\end{array}} \right) \\
\left| A \right| = 1\left( {1 - 3\mu } \right) - 1\left( { - 5 - 2\mu } \right) + 1\left( {15 + 2} \right) \\
\left| A \right| = 1 - 3\mu + 5 + 2\mu + 17 \\
\left| A \right| = 23 - \mu \\
\]
From the above equation, we can see that the uniqueness of the system depends only on $\mu $.
$\therefore $Correct option is A.
Note: In a practical case, a system of linear equations will have a unique solution if the lines
representing the equations intersect each other at only one unique point i.e. the lines are
neither parallel nor coincident.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Ray optics is valid when characteristic dimensions class 12 physics CBSE

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

Alfred Wallace worked in A Galapagos Island B Australian class 12 biology CBSE

Imagine an atom made up of a proton and a hypothetical class 12 chemistry CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

How do you define least count for Vernier Calipers class 12 physics CBSE

Why is the cell called the structural and functional class 12 biology CBSE

A 30 solution of H2O2 is marketed as 100 volume hydrogen class 11 chemistry JEE_Main
