The pair of linear equations kx + 3y + 1 = 0 and 2x + y + 3 = 0 intersect each other, if Option 1: k ≠ -6 Option 2: k = 6 Option 3: k ≠ 6 Option 4: k ≠ -5

Answer: k ≠ 6. Solution: Two lines can intersect, be parallel, or be coincident (the same line). Lines intersect when they are neither parallel nor coincident. For two lines in the form: - a_1x + b_1y + c_1 = 0 - a_2x + b_2y + c_2 = 0 Lines intersect when: a_1/a_2 ≠ b_1/b_2 For kx + 3y + 1 = 0: a_1 = k and b_1 = 3 For 2x + y + 3 = 0: a_2 = 2 and b_2 = 1 Applying the intersection condition: a_1/a_2 ≠ b_1/b_2 k/2 ≠ 3/1 k/2 ≠ 3 k ≠ 6 When k = 6, the lines are parallel and never intersect since 6/2 = 3/1 = 3. Therefore, the lines intersect for all values of k except 6. The answer is k ≠ 6.

Quantitative Reasoning

Linear Equation

Medium

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

$k \neq -6$

$k = 6$

$k \neq 6$

$k \neq -5$

✅ Correct Option: 3

Two lines can intersect, be parallel, or be coincident (the same line). Lines intersect when they are neither parallel nor coincident.

For two lines in the form:

$a_1x + b_1y + c_1 = 0$
$a_2x + b_2y + c_2 = 0$

Lines intersect when:

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

For $kx + 3y + 1 = 0$ : $a_1 = k$ and $b_1 = 3$

For $2x + y + 3 = 0$ : $a_2 = 2$ and $b_2 = 1$

Applying the intersection condition:

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

$\frac{k}{2} \neq \frac{3}{1}$

$\frac{k}{2} \neq 3$

$k \neq 6$

When $k = 6$ , the lines are parallel and never intersect since $\frac{6}{2} = \frac{3}{1} = 3$ .

Therefore, the lines intersect for all values of $k$ except $6$ .

The answer is $k \neq 6$ .

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Solution

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CUET General Test Past Year Questions (Topic-Wise)

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