CUET UG 2025 — Mathematics Algebra

Given points: $(2, -3)$(2,−3), $(\lambda, -1)$(λ,−1), and $(0, 4)$(0,4)

For three points to be collinear, the slope between any two pairs of points must be equal.

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Using points $(2, -3)$(2,−3) and $(0, 4)$(0,4):

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$m=x2​−x1​y2​−y1​​

$$m = \frac{4 - (-3)}{0 - 2}$$m=0−24−(−3)​

$$m = \frac{7}{-2}$$m=−27​

$$m = -\frac{7}{2}$$m=−27​

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Using points $(2, -3)$(2,−3) and $(\lambda, -1)$(λ,−1):

$$m = \frac{-1 - (-3)}{\lambda - 2}$$m=λ−2−1−(−3)​

$$m = \frac{2}{\lambda - 2}$$m=λ−22​

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Since the points are collinear, the slopes must be equal:

$$\frac{2}{\lambda - 2} = -\frac{7}{2}$$λ−22​=−27​

$$2 \times 2 = -7 \times (\lambda - 2)$$2×2=−7×(λ−2)

$$4 = -7\lambda + 14$$4=−7λ+14

$$7\lambda = 14 - 4$$7λ=14−4

$$7\lambda = 10$$7λ=10

$$\lambda = \frac{10}{7}$$λ=710​

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Therefore, $\lambda = \frac{10}{7}$λ=710​