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Joint Entrance Examination
Mathematics
Matrices and Determinants
Question 373
Question 373 of 472
Medium
MCQ
The system of linear equations \[\lambda \]x + 2y + 2z = 5 2\[\lambda \]x + 3y + 5z = 8 4x + \[\lambda \]y + 6z = 10 has
A
a unique solution when \[\lambda \] = --8
B
no solution when \[\lambda \] = 2
C
infinitely many solutions when \[\lambda \] = 2
D
no solution when \[\lambda \] = 8
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Official solution
\[\Delta \] = \[\left| {
(
λ
2
2
2
λ
3
5
4
λ
6
)
\begin{pmatrix} \lambda & 2 & 2 \\ {2\lambda } & 3 & 5 \\ 4 & \lambda & 6 \\ \end{pmatrix}
λ
2
λ
4
2
3
λ
2
5
6
} \right|\] = \[\lambda \] ( 18 -- 5\[\lambda \]) -- 2(12\[\lambda \] -- 20) + 2(2\[\lambda \]2 -- 12) = 18\[\lambda \] -- 5\[\lambda \]2 -- 24\[\lambda \] + 40 + 4\[\lambda \]2 -- 24 = -- \[\lambda \]2 -- 6\[\lambda \] + 16 = -- (\[\lambda \] + 8)(\[\lambda \] -- 2) For no solutions \[\lambda \] = 0 \[ \Rightarrow \] \[\lambda \] = -- 8, \[\lambda \] = 2 when \[\lambda \] = 2 \[\Delta \]x = \[\left| {
(
5
2
2
8
3
5
10
2
6
)
\begin{pmatrix} 5 & 2 & 2 \\ 8 & 3 & 5 \\ {10} & 2 & 6 \\ \end{pmatrix}
5
8
10
2
3
2
2
5
6
} \right|\] = 5 (18 -- 10) -- 2 (48 -- 50) + 2 (16 -- 30) = 40 + 4 -- 28 \[ \ne \] 0 So no solution for \[\lambda \] = 2
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