CUET UG 2025 — General Test Quantitative Reasoning

Two straight lines are given:
Line 1: $mx + 2y + 3 = 0$mx+2y+3=0
Line 2: $3x + 6y + 2 = 0$3x+6y+2=0

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Two lines can either intersect, be parallel, or be coincident.

For lines $a_1x + b_1y + c_1 = 0$a1​x+b1​y+c1​=0 and $a_2x + b_2y + c_2 = 0$a2​x+b2​y+c2​=0:

Lines are parallel when: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}$a2​a1​​=b2​b1​​

Lines intersect when: $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$a2​a1​​=b2​b1​​

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From the given equations:

Line 1: $a_1 = m$a1​=m, $b_1 = 2$b1​=2, $c_1 = 3$c1​=3

Line 2: $a_2 = 3$a2​=3, $b_2 = 6$b2​=6, $c_2 = 2$c2​=2

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The ratios are:

$$\dfrac{a_1}{a_2} = \dfrac{m}{3}$$a2​a1​​=3m​

$$\dfrac{b_1}{b_2} = \dfrac{2}{6} = \dfrac{1}{3}$$b2​b1​​=62​=31​

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For the lines to intersect:

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

$$\dfrac{m}{3} \neq \dfrac{1}{3}$$3m​=31​

$$m \neq 1$$m=1

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Therefore, the lines intersect each other when $m \neq 1$m=1.