Question

Assertion (A): The homogeneous system of equations \[2x+3y=0\], \[4x+6y=0\] has an infinite number of solution.

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

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

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

Explanation:

Step 1 – Assertion: A is true. The two equations are proportional: the second equation is twice the first. Thus, they represent the same line and have infinitely many solutions.

Step 2 – Reason: R is true. It states that a homogeneous pair has infinitely many solutions when the ratios of the coefficients of x and y are equal.

Step 3 – Link: Here \[\frac{2}{4}=\frac{3}{6}=\frac{1}{2},\] so R correctly explains A.