The pair of linear equations $kx + 3y+ 1 = 0$ and $2x + y + 3 = 0$ intersect each other, if

$k ≠ 6$

The correct answer is Option (3) → $k ≠ 6$

Step 1: Condition for intersection

Two lines:

$a_1x + b_1y + c_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2 = 0$

• They intersect if they are not parallel:

$\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$

Step 2: Apply to given lines

• Line 1: $a_1 = k, b_1 = 3, c_1 = 1$
• Line 2: $a_2 = 2, b_2 = 1, c_2 = 3$

Parallel condition:

$\frac{k}{2} = \frac{3}{1} ⇒k = 6$

So, lines are parallel if k = 6.

Hence, lines intersect if:

$k \ne 6$