JEE Main 2025 Apr 7 Shift 2, Mathematics Q4: Some Important Properties Of Functions

JEE Main2025Apr 7, Shift 2Mathematics

If the range of the function $f (x) = \frac{5 - x}{x ^{2} - 3 x + 2}$ , $x \neq = 1, 2$ , is $(- \infty, α] \cup [β, \infty)$ , then $α^{2} + β^{2}$ is equal to :

A
190
B
192
C
188
D
194

Solution

Setting $y = \frac{5 - x}{x ^{2} - 3 x + 2}$ and cross-multiplying gives $y x^{2} + (1 - 3 y) x + (2 y - 5) = 0$ .

If $y = 0$ : the equation reduces to $x - 5 = 0$ , giving $x = 5$ , a valid value (not 1 or 2), so $y = 0$ is attained.

If $y \neq = 0$ : treating this as a quadratic in $x$ , real $x$ requires the discriminant to be non-negative: $(1 - 3 y)^{2} - 4 y (2 y - 5) \geq 0 \Rightarrow y^{2} + 14 y + 1 \geq 0 \Rightarrow (y + 7)^{2} \geq 48$ .

This gives $y \leq - 7 - 43$ or $y \geq - 7 + 43$ (and $y = 0$ falls inside the second branch since $-7+4\sqrt{3}\approx -0.07 < 0$, so no separate range is added).

Comparing with $(- \infty, α] \cup [β, \infty)$ : $α = - 7 - 43$ and $β = - 7 + 43$ .

$α^{2} + β^{2} = (43 + 7)^{2} + (43 - 7)^{2} = 2 (48) + 2 (49) = 194$ .

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JEE Main 2025 Apr 7 Shift 2, Mathematics Q4: Some Important Properties Of Functions

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