QJEEM20Ma1244: JEE Main 2020 Mathematics - Class 12 JEEM20Ma1244 Solution: Let {x} and [x] denote the fractional part of x and the…

Question

Verified Solution • 2025–26 Session moderate

Let {x} and [x] denote the fractional part of x and
the greatest integer

\le

x respectively of a real
number x. If

\int_0^n {\left\{ x \right\}dx} ,\int_0^n {\left[ x \right]dx}

and 10(n² – n),

\left( {n \in N,n &gt; 1} \right)

are three consecutive terms of a G.P., then n is equal to_____.

Accepted Answer

\int\limits_0^n {\left\{ x \right\}} dx = n\int\limits_0^1 x dx = n\left( {{{{x^2}} \over 2}} \right)_0^1 = {n \over 2}

[As period of {x} = 1]

\int\limits_0^n {\left[ x \right]} dx = \int\limits_0^1 0 dx + \int\limits_1^2 1 dx + ... + \int\limits_{n - 1}^n {\left( {n - 1} \right)} dx

= 1 + 2 + 3 + ....+ (n - 1)

=

{{n\left( {n - 1} \right)} \over 2}

As

{n \over 2}

,

{{n\left( {n - 1} \right)} \over 2}

, 10(n² – n) are in GP.

\therefore

{\left[ {{{n\left( {n - 1} \right)} \over 2}} \right]^2} = {n \over 2} \times 10\left( {{n^2} - n} \right)

\Rightarrow

n² = 21n

\Rightarrow

n = 21

Step-by-Step Verified Solution