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CGPET 2018 Solved Paper

Section: Mathematics

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Question : 121 of 150

Marks: +1, -0

The system of linear equations

x+y+z=2

3x-y+2z=6

3x+y+z=-18

has

no solution
zero solution as the only solution
a unique solution
an infinite number of solutions

Solution:

Given, system of linear equations is

x+y+z=2

,

3x-y+2z=6

and

3x+y+z=-18

We can write system of equations in this form as

\left[ \begin{array}{ccc} 1 & 1 & 1 \\ 3 & -1 & 2 \\ 3 & 1 & 1 \end{array} \right] \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]

= \left[ \begin{array}{c} 2 \\ 6 \\ -18 \end{array} \right]

Now ,

\left[ \begin{array}{ccc} 1 & 1 & 1 \\ 3 & -1 & 2 \\ 3 & 1 & 1 \end{array} \right]

=1(-1-2)-1(3-6)+1(3+3)

=-3+3+6=6

Since,

\left[ \begin{array}{ccc} 1 & 1 & 1 \\ 3 & -1 & 2 \\ 3 & 1 & 1 \end{array} \right] \neq 0

Hence, a unique solution is exist.

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