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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 356
Question 356 of 472
Medium
MCQ
An ordered pair (\[\alpha \], \[\beta \]) for which the system of linear equations (1 + \[\alpha \]) x + \[\beta \]y + z = 2 \[\alpha \]x + (1 + \[\beta \])y + z = 3 \[\alpha \]x + \[\beta \]y + 2z = 2 has a unique solution, is :
A
(--3, 1)
B
(1, --3)
C
(--4, 2)
D
(2, 4)
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Official solution
For unique solution \[\Delta \] \[ \ne \] 0 \[ \Rightarrow \] \[\left| {
(
1
+
α
β
1
α
1
+
β
1
α
β
2
)
\begin{pmatrix} {1 + \alpha } & \beta & 1 \\ \alpha & {1 + \beta } & 1 \\ \alpha & \beta & 2 \\ \end{pmatrix}
1
+
α
α
α
β
1
+
β
β
1
1
2
} \right| \ne 0\] \[\left| {
(
1
−
1
0
0
1
−
1
α
β
2
)
\begin{pmatrix} 1 & { - 1} & 0 \\ 0 & 1 & { - 1} \\ \alpha & \beta & 2 \\ \end{pmatrix}
1
0
α
−
1
1
β
0
−
1
2
} \right| \ne 0 \Rightarrow \alpha + \beta \ne - 2\]
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