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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 389
Question 389 of 472
Medium
MCQ
The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9 x - y + 4z = 8
A
does not have any solution
B
has a solution (\[\alpha\], \[\beta\], \[\gamma\]) satisfying \[\alpha\] + \[\beta\]2 + \[\gamma\]3 = 12
C
has a unique solution
D
has infinitely many solutions
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Official solution
\[\Delta = \left| {
(
2
3
2
3
2
2
1
−
1
4
)
\begin{pmatrix} 2 & 3 & 2 \\ 3 & 2 & 2 \\ 1 & { - 1} & 4 \\ \end{pmatrix}
2
3
1
3
2
−
1
2
2
4
} \right| = - 20 \ne 0\] \[ \therefore \] unique solution \[{\Delta _x} = \left| {
(
9
3
2
9
2
2
8
−
1
4
)
\begin{pmatrix} 9 & 3 & 2 \\ 9 & 2 & 2 \\ 8 & { - 1} & 4 \\ \end{pmatrix}
9
9
8
3
2
−
1
2
2
4
} \right| = 0\] \[{\Delta _y} = \left| {
(
2
9
2
3
9
2
1
8
4
)
\begin{pmatrix} 2 & 9 & 2 \\ 3 & 9 & 2 \\ 1 & 8 & 4 \\ \end{pmatrix}
2
3
1
9
9
8
2
2
4
} \right| = - 20\] \[{\Delta _z} = \left| {
(
2
3
9
3
2
9
1
−
1
8
)
\begin{pmatrix} 2 & 3 & 9 \\ 3 & 2 & 9 \\ 1 & { - 1} & 8 \\ \end{pmatrix}
2
3
1
3
2
−
1
9
9
8
} \right| = - 40\] \[ \therefore \] \[x = {{{\Delta _x}} \over \Delta } = 0\] \[y = {{{\Delta _y}} \over \Delta } = 1\] \[z = {{{\Delta _z}} \over \Delta } = 2\] Unique solution : (0, 1, 2)
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