10th Class Mathematics Pair of Linear Equations in Two Variables Question Bank Case Based (MCQs) - Pair of Linear Equations in Two Variables

Find the value(s) of k for which the pair of linear equations given by \[\text{2x}+\text{5y}=\text{2};\] \[(k+2)x+(2k+1)y=2(k-1)\] will have infinitely many solutions.

A) \[6\]

B) \[-4\]

C) \[4\]

D) \[-6\]

Correct Answer: C

Solution :

For infinitely many solutions,

\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\Rightarrow \frac{2}{k+2}=\frac{3}{2k+1}=\frac{2}{2(k-1)}\]

\[\Rightarrow \,\,\,\,\frac{2}{k+2}=\frac{3}{2k+1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{2}{k+2}=\frac{2}{2(k-1)}\]

\[\Rightarrow \,\,\,\,4k+2=3k+6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,2k-2=k+2\]

\[\Rightarrow \,\,\,\,k=4\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,k=4\]

\[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k=4\]

So, option [c] is correct.