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3 and 4 .Determinants and Matrices
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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
A
$\frac{{7 - \sqrt {13} }}{2}$
B
$5$
C
$\frac{{9 - \sqrt {13} }}{2}$
D
$7$
Solution
For the system to be consistent $\Delta=0$
$\Rightarrow\left|\begin{array}{lll}{1} & {k} & {1} \\ {k} & {1} & {2} \\ {1} & {1} & {k}\end{array}\right|=0$
$\Rightarrow(k-1)\left(k^{2}+k-3\right)=0$
$\Rightarrow \mathrm{k}=1$ or $\mathrm{k}^{2}+\mathrm{k}-3=0$
But if $\mathrm{k}=1$ equations will have no solution
$\Rightarrow \mathrm{k}^{2}+\mathrm{k}=3$
Further if $\mathrm{k}^{2}+\mathrm{k}=3$ then none of the pair of lines are parallel
$\Rightarrow \mathrm{k}_{1}^{2}+\mathrm{k}_{2}^{2}=7$
Standard 12
Mathematics
