QJEEM24Ma8eaa: JEE Main 2024 Mathematics - Class 12 JEEM24Ma8eaa Solution: Let . If for some , then , where denotes the greatest integer…

Question

Verified Solution • 2025–26 Session moderate

Let

f(x)=\left\{\begin{array}{l}x-1, x \text { is even, } \\ 2 x, \quad x \text { is odd, }\end{array} x \in \mathbf{N}\right.

.

If for some

\mathrm{a} \in \mathbf{N}, f(f(f(\mathrm{a})))=21

, then

\lim\limits_{x \rightarrow \mathrm{a}^{-}}\left\{\frac{|x|^3}{\mathrm{a}}-\left[\frac{x}{\mathrm{a}}\right]\right\}

, where

[t]

denotes the greatest integer less than or equal to

t

, is equal to :

Accepted Answer

f(x)=\left\{\begin{array}{l}x-1, x \text { is even, } \\ 2 x, \quad x \text { is odd, }\end{array} x \in \mathbf{N}\right.

Let

a

is odd

\begin{aligned}&amp; \Rightarrow f(a)=2 a \\\\&amp; \Rightarrow f(f(a))=2 a-1 \\\\&amp; \Rightarrow f(f(f(a)))=2(2 a-1)\end{aligned}

2(2 a-1)=21

Not possible for any

a \in N

Let

a

is even

\begin{aligned}&amp; \Rightarrow f(a)=a-1 \\\\&amp; \Rightarrow f(f(a))=2(a-1) \\\\&amp; \Rightarrow f(f(f(a)))=2(a-1)-1=2 a-3 \\\\&amp; 2 a-3=21 \quad \Rightarrow a=12\end{aligned}

Now

\begin{aligned}&amp; \lim _{x \rightarrow 12^{-}}\left(\frac{|x|^3}{2}-\left[\frac{x}{12}\right]\right) \\\\&amp; =\lim _{x \rightarrow 12^{-}} \frac{|x|^3}{12}-\lim _{x \rightarrow 12^{-}}\left[\frac{x}{12}\right] \\\\&amp; =144-0=144 .\end{aligned}

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