| Assertion (A): The system of equations \[x+2y-5=0\] and \[\text{2x}-\text{6y}+\text{9}=0\] has infinitely many solutions. |
| Reason (R): The system of equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\]and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] has infinitely many solutions when \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\]. |
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: D
Solution :
| [d] The given system of equations are |
| \[x+2y-5=0\] and \[2x-6y+9=0\] |
| Here, \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{1}{2},\] \[\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{2}{-6}=\frac{-1}{3},\] \[\frac{{{c}_{1}}}{{{c}_{1}}}=\frac{-5}{9}\] |
| Since, \[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\] |
| \[\therefore \] The given system of equations has a unique, solution. |
| So, Assertion: False; Reason: True. |
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