Question

Assertion (A): The homogeneous system of equations \[3x-4y=0\], \[2x-3y=0\] has a unique solution x = 0 and y = 0.

Reason (R): The homogeneous system of equations \[a_{1}x+b_{1}y=0,\ a_{2}x+b_{2}y=0\] has a unique solution x = 0 and y = 0 when \[\frac{a_{1}}{a_{2}}\ne\frac{b_{1}}{b_{2}}.\]

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 coefficient ratios are \[\frac{3}{2}\] and \[\frac{-4}{-3}=\frac{4}{3},\] which are unequal. Therefore, the homogeneous system has the unique solution \[x=0,\ y=0.\]

Step 2 – Reason: R is true. It states that a homogeneous system has the unique zero solution when the coefficient ratios are unequal.

Step 3 – Link: The given ratios are unequal, so R correctly explains A.