| Directions: Each of these questions contains two statements: Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below. |
| Assertion [A] When k = - 4, then linear equations\[x+\left( k+1 \right)=5\],\[\left( k+1 \right)x+9y=8k-1\]have infinitely many solutions. |
| Reason [R] \[{{a}_{1}}x+{{b}_{1}}y={{c}_{1}}\]and \[{{a}_{2}}x+{{b}_{2}}y={{c}_{2}}\]have infinitely many solutions, if\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\]. |
A) A is true, R is true; R is a correct explanation for A.
B) A is true, R is true; R is not a correct explanation for A.
C) A is true; R is False.
D) A is false; R is true.
Correct Answer: D
Solution :
| The given system of equations will have infinitely many solutions, if |
| \[\frac{1}{k+1}=\frac{k+1}{9}=\frac{5}{8k-1}\] |
| \[\Rightarrow \,\,\,\frac{1}{k+1}=\frac{k+1}{9}\] and \[\frac{k+1}{9}=\frac{5}{8k-1}\] |
| \[\Rightarrow \,\,{{\left( k+1 \right)}^{2}}=9\] and \[\left( k+1 \right)\left( 8k-1 \right)=45\] |
| Now, \[{{\left( k+1 \right)}^{2}}=9\] |
| \[\Rightarrow \,\,k+1=\pm 3\Rightarrow k=2,\,-4\] |
| We observe that k = 2 satisfies the equation \[\left( k+1 \right)\left( 8k-1 \right)=45\] but \[k=-4\]does not satisfy. |
| Hence, the given system of equations will have infinitely many solutions, if k = 2. |
| Assertion : False; Reason : True |
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