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Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.?
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Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (w...
The fractional part function {x} returns the fractional part of a number x, which is defined as the part of x that is to the right of the decimal point.
If x = [√2 + 1]^n, where n is an odd natural number, then the value of x will be a positive integer greater than 1. This is because the square root of 2 is an irrational number, so √2 + 1 is an irrational number that is greater than 1. When an irrational number is raised to an odd power, the result will always be an irrational number that is greater than 1.
If x is a positive integer greater than 1, then {x} = 0, because the fractional part of a positive integer is always zero. Therefore, if x = [√2 + 1]^n, where n is an odd natural number, then {x} = 0.
If x is a positive integer greater than 1, then 1 - x^2 will always be a negative number. This is because (x - 1)^2 = x^2 - 2x + 1, which is a difference of squares that can be factored as (x + 1)(x - 1). If x is a positive integer greater than 1, then both x + 1 and x - 1 will be positive, so (x + 1)(x - 1) will be positive. Therefore, 1 - x^2 will be negative.
If x is a positive integer greater than 1, then the value of (1 - {x}^2) /{x} will be 0 / 0, which is an indeterminate form. This is because both the numerator and the denominator are zero, so the fraction is undefined.
Therefore, the correct answer is:
A. Irrational
Community Answer
Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (w...
Understanding the Expression
Let n be an odd natural number. We define x as follows:
- x = (√2 + 1)^n
This expression can be simplified as it relates to the properties of irrational and rational numbers.
Analyzing the Fractional Part Function
The fractional part function, denoted as {x}, is defined as:
- {x} = x - ⌊x⌋
Where ⌊x⌋ is the greatest integer less than or equal to x.
Since (√2 + 1)^n is irrational for any natural number n, its floor, ⌊x⌋, is also an integer. Thus, {x} remains irrational.
Examining the Expression (1 - {x}^2) / {x}
Now we consider:
- (1 - {x}^2) / {x}
Substituting the property of {x}, we know:
- {x}^2 is also irrational because the square of an irrational number is irrational.
Thus, we have:
- 1 - {x}^2 is rational (since 1 is rational and subtracting an irrational from a rational results in irrational).
Now, dividing by {x} (irrational) gives us:
- (1 - {x}^2) / {x} = irrational / irrational
Conclusion
Since we have an irrational number divided by another irrational number, the result can be:
- A. Irrational
This means the final expression is irrational.
In conclusion, the answer to the problem is that the expression (1 - {x}^2) / {x} is irrational when n is an odd natural number.
Let n be an odd natural number. We define x as follows:
- x = (√2 + 1)^n
This expression can be simplified as it relates to the properties of irrational and rational numbers.
Analyzing the Fractional Part Function
The fractional part function, denoted as {x}, is defined as:
- {x} = x - ⌊x⌋
Where ⌊x⌋ is the greatest integer less than or equal to x.
Since (√2 + 1)^n is irrational for any natural number n, its floor, ⌊x⌋, is also an integer. Thus, {x} remains irrational.
Examining the Expression (1 - {x}^2) / {x}
Now we consider:
- (1 - {x}^2) / {x}
Substituting the property of {x}, we know:
- {x}^2 is also irrational because the square of an irrational number is irrational.
Thus, we have:
- 1 - {x}^2 is rational (since 1 is rational and subtracting an irrational from a rational results in irrational).
Now, dividing by {x} (irrational) gives us:
- (1 - {x}^2) / {x} = irrational / irrational
Conclusion
Since we have an irrational number divided by another irrational number, the result can be:
- A. Irrational
This means the final expression is irrational.
In conclusion, the answer to the problem is that the expression (1 - {x}^2) / {x} is irrational when n is an odd natural number.
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Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.?
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Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.?.
Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let n be an odd natural number. If x= [√2 +1]^n then (1-{x}^2) /{x} (where {x} denotes fractional part function). A. Irrational B. Rational but not an integer C. Even integer D. Odd integer.?.
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