| 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] If the system of equations \[2x+3y=7\] and \[2ax+\left( a+b \right)y=28\]has infinitely many solutions, then\[2a-b=0\]. |
| Reason [R] The system of equations |
| \[3x-5y=9\]and \[6x-10y=8\] has a unique solution. |
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: C
Solution :
| Assertion : Given system of equations has infinitely many solutions if, |
| \[\frac{2}{2a}=\frac{3}{a+b}=\frac{-7}{-28}\] |
| \[\frac{1}{a}=\frac{3}{a+b}=\frac{1}{4}\] |
| \[3a=a+b\Rightarrow \,\,2a-b=0\] |
| Also, clearly a = 4, and \[a+b=12\] |
| \[b=8\] |
| \[2a-b=8-8=0\] |
| For unique solution |
| \[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\] or |
| \[\frac{3}{6}=\frac{-5}{-10}\left[ 3\left( -10 \right)=\left( -5 \right)\left( 6 \right)=-30 \right]\] |
| Assertion is true But reason is false. |
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