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How do I find the number of integer solutions of the equation which is [x] + [2x] + [3x] = {x} + {2x} +2x^2 ,where [.], {.} are greatest integer and fractional part function??
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How do I find the number of integer solutions of the equation which is...
Introduction:
The given equation is [x] [2x] [3x] = {x} {2x} 2x^2, where [.] and {.} represents greatest integer and fractional part function respectively. We need to find the number of integer solutions of this equation.

Solution:
Let's consider each term of the equation separately.

[x]:
As [x] is an integer, the equation becomes [x] [2x] [3x] = 0 {2x} 2x^2 for 0 ≤ x < />

[2x]:
When 1 ≤ x < 3/2,="" [2x]="2," and="" when="" 3/2="" ≤="" x="" />< 2,="" [2x]="3." so,="" the="" equation="" becomes="" [x]="" [2x]="" [3x]="0" {2x}="" 2x^2="" for="" 1="" ≤="" x="" />< />

[3x]:
When 2 ≤ x < 3,="" [3x]="6." so,="" the="" equation="" becomes="" [x]="" [2x]="" [3x]="0" {2x}="" 2x^2="" for="" 2="" ≤="" x="" />< />

Combining the intervals:
If we combine the intervals obtained from the above three cases, we get the range of x for which the equation holds true. It is 0 ≤ x < 1="" and="" 1="" ≤="" x="" />< 2="" and="" 2="" ≤="" x="" />< />

Checking for integer solutions:
Now, we need to check how many integer solutions are there in each of these intervals.

For 0 ≤ x < 1,="" the="" equation="" becomes="" [x]="" [2x]="" [3x]="0" {2x}="" 2x^2.="" as="" [x]="" [2x]="" [3x]="0," the="" equation="" reduces="" to="" {2x}="" 2x^2="0." this="" implies="" that="" {2x}="0" and="" 2x^2="0." therefore,="" x="" />

For 1 ≤ x < 2,="" the="" equation="" becomes="" [x]="" [2x]="" [3x]="2" {2x}="" 2x^2.="" as="" [x]="" [2x]="" [3x]="2," the="" equation="" reduces="" to="" {2x}="" 2x^2="1." this="" implies="" that="" {2x}="1/2" and="" 2x^2="1/2." therefore,="" x="" />

For 2 ≤ x < 3,="" the="" equation="" becomes="" [x]="" [2x]="" [3x]="6" {2x}="" 2x^2.="" as="" [x]="" [2x]="" [3x]="6," the="" equation="" reduces="" to="" {2x}="" 2x^2="1.5." this="" implies="" that="" {2x}="3/4" and="" 2x^2="3/4." therefore,="" x="" />

Conclusion:
Thus, there are three integer solutions of the given equation, which are x = 0, x = 1/
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How do I find the number of integer solutions of the equation which is [x] + [2x] + [3x] = {x} + {2x} +2x^2 ,where [.], {.} are greatest integer and fractional part function??
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How do I find the number of integer solutions of the equation which is [x] + [2x] + [3x] = {x} + {2x} +2x^2 ,where [.], {.} are greatest integer and fractional part function?? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about How do I find the number of integer solutions of the equation which is [x] + [2x] + [3x] = {x} + {2x} +2x^2 ,where [.], {.} are greatest integer and fractional part function?? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How do I find the number of integer solutions of the equation which is [x] + [2x] + [3x] = {x} + {2x} +2x^2 ,where [.], {.} are greatest integer and fractional part function??.
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