A) Infinitely many
B) 3
C) 1
D) 2
Correct Answer: C
Solution :
A non-homogeneous system of linear equations has a unique non-trivial solution if and only if its determinant is non-zero. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions. \[\left[ \begin{matrix} (k+2) & 10 \\ k & (k+3) \\ \end{matrix} \right]\] \[\left[ \frac{x}{y} \right]=\left[ \frac{k}{k-1} \right]\] Now it is of the Form \[Ax=B\] Now to for the system to have no Solution, determinant of A must be 0, as follows \[\Rightarrow \]\[|A|=(k+2)(k+3)-k\times 10=0\Rightarrow {{k}^{2}}-5k+6=\]\[(k-2)(k-3)=0\] Therefore for k = 2, 3 system will have no solution. For k = 2, we get infinitely many solutions, after substituting the value of k = 2 in the equations. Thus \[k=3\] Thus, the number of solutions is 1.
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