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

To determine which statement DEFINITELY CONTRADICTS the fact that the lines represented by the given equations are coinciding, let's start by understanding the conditions for two linear equations to represent coinciding lines.

Understanding Coinciding Lines

For two lines represented by the equations:

• Equation 1: \(3x + 21y + p = 0\)
• Equation 2: \(qx + ry - 7 = 0\)

To coincide, these lines must be equivalent, meaning their coefficients must be proportional. The conditions are:

• \(\frac{3}{q} = \frac{21}{r} = \frac{p}{-7}\)
• Furthermore, \(q \neq 0\) and \(r \neq 0\).

Analysis of the Statements

Let's analyze each given statement:

1. p and q must have opposite signs: This does not directly contradict the proportionality condition as \(\frac{p}{-7}\) could be negative while \(\frac{3}{q}\) is positive.
2. The smallest among p, q, and r is r: This is possible as the actual values of \(p, q,\) and \(r\) are not restricted beyond their proportionality.
3. The largest among p, q, and r is q: This would contradict the condition if \(q\) needs to be in proportion to both \(3\) and \(r\). If \(q\) is the largest, it may disrupt the required proportion, making this statement a viable contradiction.
4. r and q must have the same signs: This can occur if \(\frac{3}{q}\) and \(\frac{21}{r}\) both have the same sign, hence aligning with the coinciding line requirement.
5. p cannot be 0: \(p\) being zero does not contradict proportionality, it simply changes the type of line.

Conclusion

Out of all the options, the statement "The largest among p, q, and r is q" contradicts the proportionality condition necessary for the lines to coincide, potentially disrupting the required relationships between coefficients. Thus, this statement definitely contradicts the fact that the given lines are coinciding.

Answer 2

To determine which statement definitely contradicts the fact that the lines represented by the two equations are coinciding, we need to understand the condition for two lines to be coincident. Two lines are coincident if they represent the same line, i.e., their equations need to be proportional.

Let's write the equations of the lines properly:

• \(3x + 21y + p = 0\)
• \(qx + ry - 7 = 0\)

For the lines represented by these equations to be coincident, their coefficients must be proportional:

• \(\frac{3}{q} = \frac{21}{r} = \frac{p}{-7}\)

Analyzing this, we can comment on the conditions:

• p and q must have opposite signs: This statement does not necessarily contradict the condition for the lines to coincide. It depends on the values chosen for other coefficients.
• The smallest among p, q, and r is r: This statement alone does not necessarily contradict since all terms could be proportional despite which is smallest.
• The largest among p, q, and r is q: If q is largest, it suggests that other coefficients, for example, 3 (which corresponds proportionally) must be smaller than q, which could contradict the requirement for proportionality depending on the values of p and r.
• r and q must have same signs: This could hold true if all terms uphold the proportions.
• p cannot be 0: If p is 0, with correct q and r maintaining the proportion, the lines could still coincide.

Therefore, the statement "The largest among p, q, and r is q" definitely contradicts the condition of proportional coefficients when attempting to have coincident lines, since it could disrupt the proportional balance needed for coincidence.