Q43 / 75 · JEE Main 2025 (7 Apr Shift 2) · Mathematics
The number of real roots of the equation $\mathrm{x}|\mathrm{x}-2|+3|\mathrm{x}-3|+1=0$ is :
- A4
- B2
- Correct answerC1
- D3
Solution
$\begin{aligned}<br/>& \text { (I) } x \lt 2 \\
& -x^2+2 x-3 x+9+1=0 \\
& \Rightarrow x^2+x-10=0 \\
& \Rightarrow x=\frac{-1+\sqrt{41}}{2}, \frac{-1-\sqrt{41}}{2} \\
& \quad \times \quad \sqrt{ }<br/>\end{aligned}$
$\begin{aligned}<br/>& \text { (II) } 2 \leq x \lt 3 \\ & \Rightarrow x^2-2 x-3 x+9+1=0 \\ & \Rightarrow x^2-5 x+10=0 \\ & \text { D } \lt 0 \\ & \text { (III) } x \geq 3 \\ & x^2-2 x+3 x-9+2=0 \\ & \Rightarrow x^2+x-8=0 \\ & x=\frac{-1+\sqrt{32}}{2}, \frac{-1-\sqrt{32}}{2} \\ & \quad \times \quad \times<br/>\end{aligned}$
1 real roots
$\begin{aligned}<br/>& \text { (II) } 2 \leq x \lt 3 \\ & \Rightarrow x^2-2 x-3 x+9+1=0 \\ & \Rightarrow x^2-5 x+10=0 \\ & \text { D } \lt 0 \\ & \text { (III) } x \geq 3 \\ & x^2-2 x+3 x-9+2=0 \\ & \Rightarrow x^2+x-8=0 \\ & x=\frac{-1+\sqrt{32}}{2}, \frac{-1-\sqrt{32}}{2} \\ & \quad \times \quad \times<br/>\end{aligned}$
1 real roots
Hence, the correct answer is option: 'C'.