Sets, Relations & Functions
Functions
star_batch_jee_advanced_2025
Grade 11
Question:
The number of real solutions to the equation $3x - 7 = [x^2 - 3x + 2]$ is ____. $[x]$ denotes greatest integer $\leq x$.
Step-by-Step Solution
Key Concept: The floor function definition $[t] \le t < [t] + 1$ translates the floor inequality into a compound linear-quadratic system.
Use the property $[t] \le t < [t] + 1$ for real $t$. Apply this to the given inequality $3x - 7 \le x^2 - 3x + 2 < (3x-7) + 1$ and solve both inequalities simultaneously to find $x \in \{\frac{7}{3}, \frac{8}{3}, \frac{9}{3}, \frac{10}{3}, \frac{11}{3}\}$.
Correct Answer: 5