Question

Consider the system of two linear equations as follows: 3x + 21y + p = 0; and qx + ry – 7 = 0, where p , q, and r are real numbers.
Which of the following statements DEFINITELY CONTRADICTS the fact that the lines represented by the two equations are coinciding?

• A. p and q must have opposite signs
• B. The smallest among p, q, and r is r
• C. The largest among p, q, and r is q
• D. r and q must have same signs
• E. p cannot be 0

Answer 1

The Correct Option is C

Step 1: Determine conditions for coincident lines. For lines 3x + 2y + p = 0 and qx + qy − 7 = 0 to coincide, their coefficients and constants must be proportional:

$\frac{3}{q} = \frac{2}{q} = \frac{p}{-7}$

Step 2: Evaluate each statement.

• Option 1: p and q must have opposite signs. This conflicts with the proportionality requirement, as proportional coefficients cannot have opposite signs.
• Option 2: The smallest among p, q, and r is r. This statement does not conflict with the proportionality condition.
• Option 3: The largest among p, q, and r is q. This statement does not conflict with the proportionality condition.
• Option 4: r and q must have the same signs. This aligns with the proportionality condition.
• Option 5: p cannot be 0. This statement does not conflict with the proportionality condition, as p can be non-zero.

Answer: Option 1.