| Assertion (A): If the system of equations \[\text{2x}+\text{3y}=\text{7}\] and \[2ax+(a+b)y=28\] has infinitely many solutions, then\[\text{2a}-\text{b}=0\]. |
| Reason (R): The system of equations \[\text{3x}-\text{5y}=\text{9}\] and \[\text{6x}-\text{l0y}=\text{8}\] has a unique solution. |
A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
B) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
C) Assertion (A) is true but reason (R) is false.
D) Assertion (A) is false but reason (R) is true.
Correct Answer: C
Solution :
| [c] Assertion: given system of equations has infinitely many solutions if, |
| \[\frac{2}{2a}=\frac{3}{a+b}\] |
| \[=\frac{-7}{-28}\] |
| i.e., \[\frac{1}{4}\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{a}=\frac{3}{a+b}=\frac{1}{4}\] |
| \[3a=a+b\] |
| \[2a-b=0\] |
| Also clearly, \[a=4\] and \[a+b=12\] |
| \[b=8\] |
| \[2a-b=8-8=0\] |
| Assertion is true but reason is false, |
| \[\frac{3}{6}=\frac{-5}{-10}\] \[[3(-10)=(-5)\,\,(6)=-30]\] |
| For unique solution if, |
| \[{{a}_{1}}x+{{b}_{2}}y+{{c}_{2}}=0,\] |
| Then \[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\] |
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