| Assertion (A): If the pair of equations \[x+2y+7=0,\] \[3x+ky+21=0\] represents coincident lines, then the value of k is 6. |
| Reason (R): The pair of linear equations are coincident Lines if they have no 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] The given equations are of the form |
| \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\]and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0,\] |
| where\[{{a}_{1}}=1,\,{{b}_{1}}=2,\,\,{{c}_{1}}=7\]and |
| \[{{a}_{2}}=3,\,{{b}_{2}}=k,\,\,{{c}_{2}}=21\] |
| The given equations will represent coincident lines, if they have infinitely many solutions. |
| The condition for which is |
| \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\,\,\Rightarrow \,\frac{1}{3}=\frac{2}{k}=\frac{7}{21}\,\,\,\Rightarrow \,\,\,k=6\] |
| Hence, the given system of equations will represent coincident lines, if \[k=6\]. |
| \[\therefore \] Assertion: True; Reason: False. |
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