Question

Assertion (A): The system of equations \[3x-6y=0\] and \[x-2y-6=0\] has infinitely many solutions.

Reason (R): The system of equations \[a_{1}x+b_{1}y+c_{1}=0,\ a_{2}x+b_{2}y+c_{2}=0\] has infinitely many solutions, if \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}.\]

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Assertion (A) is false and Reason (R) is true.

Explanation:

Step 1 – Assertion: A is false. The ratios are \[\frac{a_{1}}{a_{2}}=\frac{3}{1}=3,\quad \frac{b_{1}}{b_{2}}=\frac{-6}{-2}=3,\quad \frac{c_{1}}{c_{2}}=\frac{0}{-6}=0.\] The first two ratios are equal but differ from the third, so the equations have no solution.

Step 2 – Reason: R is true. It correctly states that infinitely many solutions require all three coefficient ratios to be equal.