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: Analyze the conditions for coinciding lines. For the lines 3x + 2y + p = 0 and qx + qy − 7 = 0 to coincide, the coefficients of x, y, and the constants must be proportional:

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

Step 2: Test each statement.

• Option 1: p and q must have opposite signs. This contradicts the proportionality condition, as proportional coefficients cannot have opposite signs.
• Option 2: The smallest among p, q, and r is r. This does not contradict the proportionality condition.
• Option 3: The largest among p, q, and r is q. This does not contradict the proportionality condition.
• Option 4: r and q must have the same signs. This is consistent with the proportionality condition.
• Option 5: p cannot be 0. This does not contradict the proportionality condition, as p can be nonzero.

Answer: Option 1.