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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The set of all $a \in \mathbb{R}$ for which the equation $x|x - 1| + |x + 2| + a = 0$ has exactly one real root is:

Choose the correct answer:

A.
$(-\infty, -3)$
B.
$(-\infty, \infty)$
(Correct Answer)
C.
$(-6, \infty)$
D.
$(-6, -3)$

Explanation

Is sawal ko solve karne ke liye, aap equation ko aise likh sakte hain:

$f(x) = x|x - 1| + |x + 2| = -a$

Intervals break karein: $x < -2$ , $-2 \le x < 1$ , aur $x \ge 1$ .

Graph draw karein: $f(x)$ ka graph banakar dekhein ki horizontal line $y = -a$ graph ko sirf ek jagah kab cut karti hai.

Local Maxima/Minima: Check karein ki function monotonic hai ya nahi. Is case mein, $f(x)$ hamesha badhne wala (strictly increasing) function nikalta hai, isliye $a$ ki kisi bhi real value ke liye hamesha ek hi root hoga.

Correct Option: (2) $(-\infty, \infty)$

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