CUET UG 2025 — General Test Quantitative Reasoning

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:

• $a_1x + b_1y + c_1 = 0$a1​x+b1​y+c1​=0
• $a_2x + b_2y + c_2 = 0$a2​x+b2​y+c2​=0

Lines intersect when:

$$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$a2​a1​​=b2​b1​​

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For $kx + 3y + 1 = 0$kx+3y+1=0: $a_1 = k$a1​=k and $b_1 = 3$b1​=3

For $2x + y + 3 = 0$2x+y+3=0: $a_2 = 2$a2​=2 and $b_2 = 1$b2​=1

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Applying the intersection condition:

$$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$a2​a1​​=b2​b1​​

$$\frac{k}{2} \neq \frac{3}{1}$$2k​=13​

$$\frac{k}{2} \neq 3$$2k​=3

$$k \neq 6$$k=6

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When $k = 6$k=6, the lines are parallel and never intersect since $\frac{6}{2} = \frac{3}{1} = 3$26​=13​=3.

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

The answer is $k \neq 6$k=6.