Value of λ for which system of equations 2x - y - z = 12, x - 2y + z = - 4, x + y + λ z = 4 has no

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3 and 4 .Determinants and Matrices

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The value of $\lambda $ for which the system of equations $2x - y - z = 12,$ $x - 2y + z = - 4,$ $x + y + \lambda z = 4$ has no solution is

A

$3$

B

$-3$

C

$2$

D

$-2$

(IIT-2004)

Solution

(d) The coefficient determinant $D = \left| {\,\begin{array}{*{20}{c}}2&{ – 1}&{ – 1}\\1&{ – 2}&1\\1&1&\lambda \end{array}\,} \right|$

= $3\lambda – 6$

For no solution, the necessary condition is $D = 0$

i.e., $ – 3\lambda – 6 = 0 \Rightarrow \lambda = – 2$

It can be seen that for $\lambda = – 2,$ there is no solution for the given system or equations.

Standard 12

Mathematics

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