JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The number of real roots of the equation $x|x| - 5|x + 2| + 6 = 0$ , is

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Explanation

Solution

Hame equation $x|x| - 5|x + 2| + 6 = 0$ ke cases banane honge:

Case 1: $x \ge 0$

Equation: $x^2 - 5(x + 2) + 6 = 0 \implies x^2 - 5x - 4 = 0$

Roots: $x = \frac{5 \pm \sqrt{25 + 16}}{2} = \frac{5 \pm \sqrt{41}}{2}$
Kyunki $x \ge 0$ , sirf $x = \frac{5 + \sqrt{41}}{2}$ sahi hai. (1 root)

Case 2: $-2 \le x < 0$

Equation: $-x^2 - 5(x + 2) + 6 = 0 \implies -x^2 - 5x - 4 = 0 \implies x^2 + 5x + 4 = 0$

$(x+4)(x+1) = 0 \implies x = -4, -1$
Is range mein sirf $x = -1$ aata hai. (1 root)

Case 3: $x < -2$

Equation: $-x^2 - 5(-(x + 2)) + 6 = 0 \implies -x^2 + 5x + 16 = 0 \implies x^2 - 5x - 16 = 0$

Roots: $x = \frac{5 \pm \sqrt{25 + 64}}{2} = \frac{5 \pm \sqrt{89}}{2}$
$x = \frac{5 - \sqrt{89}}{2} \approx \frac{5 - 9.4}{2} \approx -2.2$ (Is range mein hai). (1 root)

Total Roots: $1 + 1 + 1 = 3$

Sahi vikalp (4) hai.

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