JEE Main 2017 · 9 Apr · Shift 1 · Mathematics Q61

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MathematicsSingle correct+4 / -1Question 61 of 90

The function $f:\mathbb N\to\mathbb N$ defined by $f(x)=x-5\left[\dfrac x5\right]$ , where $\mathbb N$ is the set of natural numbers and [x] denotes the greatest integer less than or equal to $x$ , is :

one-one and onto.

one-one but not onto.

onto but not one-one.

neither one-one nor onto.

Answer and worked solution

Official answer: Bonus: full marks to everyone

$f(x)=x-5\left[\frac x5\right]$ is the remainder of $x$ on division by 5, so $f(5)=f(10)=\dots=0$ . Since $0\notin\mathbb N$ , $f$ does not map $\mathbb N$ into $\mathbb N$ , and the question is ill-posed.

(Read with a codomain that includes 0, $f$ takes only the values 0 to 4, so it would be neither one-one nor onto.) The question was given marks to all.

Topics:Types of functions

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JEE Main 2017 · 9 Apr · Shift 1, Mathematics question 61