CUET UG 2025 — Mathematics Algebra

An inconsistent system means the equations have NO solution — the two lines are parallel and never meet.

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For the system

$$2x + 5y = 7$$2x+5y=7

$$6x + \lambda y = 28$$6x+λy=28

the condition for inconsistency is:

$$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$$a2​a1​​=b2​b1​​=c2​c1​​

This means the lines have the same slope (parallel) but different intercepts (never intersect).

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Here, $a_1 = 2,\; a_2 = 6,\; b_1 = 5,\; b_2 = \lambda,\; c_1 = 7,\; c_2 = 28$a1​=2,a2​=6,b1​=5,b2​=λ,c1​=7,c2​=28

Applying the first part of the condition:

$$\dfrac{2}{6} = \dfrac{5}{\lambda}$$62​=λ5​

$$\dfrac{1}{3} = \dfrac{5}{\lambda}$$31​=λ5​

Cross-multiplying:

$$\lambda = 5 \times 3 = 15$$λ=5×3=15

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Now we verify the "not equal to" part to confirm the system is truly inconsistent and not dependent:

$$\dfrac{a_1}{a_2} = \dfrac{1}{3}$$a2​a1​​=31​

$$\dfrac{c_1}{c_2} = \dfrac{7}{28} = \dfrac{1}{4}$$c2​c1​​=287​=41​

$\dfrac{1}{3} \neq \dfrac{1}{4}$31​=41​ ✅

This confirms the system is inconsistent.

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$$\lambda = 15$$λ=15