The pair of linear equations \(kx + 3y + 1 = 0\) and \(2x + y + 3 = 0\) intersect each other, if
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(k\ne6\)
Two lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) intersect at one point when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\), that is, when their slopes are different. If the two ratios are equal, the lines are parallel (or the same line) and do not cut each other.
Here \(a_1=k,\ b_1=3\) and \(a_2=2,\ b_2=1\)
Condition: \(\displaystyle \frac{k}{2}\ne\frac{3}{1}\)
So \(k\ne2\times3\), that is, \(k\ne6\)
Why the other options are incorrect
Option (1) \(k\ne-6\) — Incorrect. It still allows \(k=6\), the one value that makes the lines parallel.
Option (2) \(k=6\) — Incorrect. With \(k=6\), \(\frac{6}{2}=\frac{3}{1}\ne\frac{1}{3}\), so the lines are parallel and never meet.
Option (4) \(k\ne-5\) — Incorrect. It also allows \(k=6\), for which the lines do not intersect.