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GATE 2025 ME – Question 4

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

The ceiling function of a real number $x$, denoted by $ce(x)$, is defined as the smallest integer that is greater than or equal to $x$. Similarly, the floor function, denoted by $fl(x)$, is defined as the largest integer that is smaller than or equal to $x$. Which one of the following statements is NOT correct for all possible values of $x$?

  1. $ce(x) \geq x$
  2. $fl(x) \leq x$
  3. $ce(x) \geq fl(x)$
  4. $fl(x) < ce(x)$

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Show answer and explanation

Correct answer: (D) $fl(x) < ce(x)$

Explanation

Options A, B and C always hold. Option D fails when $x$ is an integer, because then the floor and the ceiling are both equal to $x$, so $fl(x) = ce(x)$ and the strict inequality is false.