   ### FROM THE FUNCTIONS

Here is another simple but a really good concept building problem from the topic Functions and Graphs
The problem must be approached systematically to arrive at the result. After solving the question you will build some confidence to solve the problems related specially to the Greatest Integer Functions.

Problem :
The number of solutions of | [x] – 3x | = 6, where [x] is the greatest integer < or = x, is
1.   2
2.   4
3.   3
4.   No solution
Solution:
First of all we must know what is [x].

[x] is the greatest integer function i.e. if the value of x is 2.5 then the greatest integer less than 2.5 is 2. If the value of x is -2.5 then the greatest integer less than -2.5 is -3 and Not -2.

In this problem the nature of x is not specified, so we have to consider the whole range of real numbers.
First we will consider that x is an integer and later as a non-integer.
When x is an integer, which means
[x] = x,
hence the equation | [x] – 3x | = 6 reduces to | x – 3x | = 6 which can be easily solved as below:

x – 3x = 6    &   -(x – 3x) = 6
-2x = 6        &   -(-2x) = 6
x = -3          &      x = 3

When x is not an integer, we can write x as x = n+ k. Where n is the integer and k is the fraction part such that 0 < k < 1.
so the equation | [x] – 3x | = 6 will now reduce to | [n+k] – 3(n+k) | = 6 i.e.
| n – 3n - 3k | = 6
| – 2n - 3k | = 6
What is important to note is that for the result to be an integer as can bee seen on the R.H.S. the factor 3k on L.H.S. must be an integer which is possible only when 3k=1 i.e. when k = 1/3

With k = 1/3, the equation becomes | -2n - 1 | = 6 which can be solved very easily as
-2n - 1 = 6       &      -( -2n - 1 ) = 6
-2n = 7            &        2n = 5
n = -7/2          &        n = 5/2

Hence we see that four solutions are possible to the question. Hence the option (b)
This was a very simple problem and hardly takes two minutes.

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IIT JEE EXAM 2011- EXAM DATE 10 April 2011, Last date to Apply = 15 December 2010