| Assertion (A): The system of equations \[\text{3x}-\text{y}-\text{5}=0,\]\[\text{6x}-\text{2y}-\text{k}=0\] has no solution if \[\text{k}=\text{1}0\]. |
| Reason (R): The pair of equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\]has no solution if \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}.\] |
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] Given, system of equations is |
| \[3x-y-5=0\] |
| and \[6x-2y-k=0\] |
| Here, \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{3}{6}=\frac{1}{2},\] |
| \[\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{-1}{-2}=\frac{1}{2}\] |
| and \[\frac{{{c}_{1}}}{{{c}_{2}}}=\frac{-5}{-k}=\frac{5}{k}\] |
| For no solution, \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\] |
| \[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2}=\frac{1}{2}\ne \frac{5}{k}\] |
| \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{5}{k}\ne \frac{1}{2}\] |
| \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k\ne 10\] |
| Hence, for every value of k except 10, given system has no solution. |
| \[\therefore \] Assertion: False; Reason: True. |
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