Sets, Relations & Functions
Greatest Integer Function / Equations
nta_pyq_2023_jan
Grade 11
Question:
The equation $x^2 - 4x + [x] + 3 = x[x]$, where $[x]$ denotes the greatest integer function, has:
exactly two solutions in (-∞, ∞)
no solution
a unique solution in (-∞, 1)
a unique solution in (-∞, ∞)
Step-by-Step Solution
Key Concept: Rearrange to factor and analyze $\{x\} = x - [x]$; the fractional part cannot equal 3.
Rearranging: $(x-1)(x-3) = [x](x-1)$, so $x=1$ or $x-3=[x]$. The latter gives $\{x\}=3$, which is impossible. Only solution is $x=1 \in (-\infty,\infty)$.
Correct Answer: 4