For a real number x , let [x] denote largest integer less than or equal to x , and let {x}=x-[x] . n

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4-2.Quadratic Equations and Inequations

medium

For a real number $x$, let $[x]$ denote the largest integer less than or equal to $x$, and let $\{x\}=x-[x]$. The number of solutions $x$ to the equation $[x]\{x\}=5$ with $0 \leq x \leq 2015$ is

A

$0$

B

$3$

C

$2008$

D

$2009$

(KVPY-2015)

Solution

(d)

We have,

$\quad[x]\{x\}=5$

$x \in[0,2015]$

$\{x\}=\frac{5}{[x]}$

$\{x\} \in[0,1)$

$\frac{5}{[x]} < 1$

${[x] > 5 }$

$\therefore$ Total number of solution is $2009.$

Standard 11

Mathematics

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