System of equations λ x + y + z = 0, - x + λ y + z = 0, - x

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3 and 4 .Determinants and Matrices

medium

The system of equations $\lambda x + y + z = 0,$ $ - x + \lambda y + z = 0,$ $ - x - y + \lambda z = 0$, will have a non zero solution if real values of $\lambda $ are given by

$0$

$1$

$3$

$\sqrt 3 $

(IIT-1984)

Solution

(a) Accordingly, $\left| {\,\begin{array}{*{20}{c}}\lambda &1&1\\{ – 1}&\lambda &1\\{ – 1}&{ – 1}&\lambda \end{array}\,} \right| = 0 \Rightarrow {\lambda ^3} + 3\lambda = 0$

Therefore $\lambda = 0$, since $\lambda = i\sqrt 3 $ does not exist.

Standard 12

Mathematics

How many values of $k $ , systeam of linear equations $\left( {k + 1} \right)x + 8y = 4k\;,\;kx + \left( {k + 3} \right)y$$ = 3k – 1$ has no solutions.

medium

(IIT-2002)

Let $k_1$, $k_2$ be the maximum and minimum values of $k$ for which the system of equations given by

$x + ky = 1$ ; $kx + y = 2$; $x + y = k$ are consistent then $k_1^2 + k_2^2$ is equal to

normal

Find equation of line joining $(1,2)$ and $(3,6)$ using determinates

easy

Let $\lambda \in R .$ The system of linear equations

$2 x_{1}-4 x_{2}+\lambda x_{3}=1$

$x_{1}-6 x_{2}+x_{3}=2$

$\lambda x_{1}-10 x_{2}+4 x_{3}=3$ is inconsistent for

hard

(JEE MAIN-2020)

If $\left| {\,\begin{array}{*{20}{c}}{x + 1}&3&5\\2&{x + 2}&5\\2&3&{x + 4}\end{array}\,} \right| = 0$, then $ x =$

medium

The system of equations $\lambda x + y + z = 0,$ $ - x + \lambda y + z = 0,$ $ - x - y + \lambda z = 0$, will have a non zero solution if real values of $\lambda $ are given by

Solution

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Let $k_1$, $k_2$ be the maximum and minimum values of $k$ for which the system of equations given by

$x + ky = 1$ ; $kx + y = 2$; $x + y = k$ are consistent then $k_1^2 + k_2^2$ is equal to

Let $\lambda \in R .$ The system of linear equations

$2 x_{1}-4 x_{2}+\lambda x_{3}=1$

$x_{1}-6 x_{2}+x_{3}=2$

$\lambda x_{1}-10 x_{2}+4 x_{3}=3$ is inconsistent for