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  • If  and <img alt="P'=(10-3\sqrt{11})^n" src="http://e

If P=(10+3\sqrt{11})^n and P'=(10-3\sqrt{11})^n then which one of the following is not true?

Option: 1

PP'=1


Option: 2

P+P' is an even integer


Option: 3

P'=1-f where, f is the fractional part of P


Option: 4

f+P'\in (0,1)


Answers (1)

best_answer

Finding nature of an integral part of the expression.

If the given expansion is in the form of \mathrm{N}=(\mathrm{a}+\sqrt{\mathrm{b}})^{\mathrm{n}} \quad(\mathrm{n} \in \mathrm{N})

Working rule:

Step 1: \mathrm{Choose\;\;N}^{\prime}=(\mathrm{a}-\sqrt{\mathrm{b}})^{\mathrm{n}} \text { or }(\sqrt{\mathrm{b}}-\mathrm{a})^{\mathrm{n}} \text { according as a }>\sqrt{\mathrm{b}} \text { or } \sqrt{\mathrm{b}}>\mathrm{a}

Step 2: Use N + N’ or N - N’ such that result is an integer

I.e. \mathrm{(a+\sqrt{b})^n+(a-\sqrt{b})^n\;\;or\;\;(a+\sqrt{b})^n-(a-\sqrt{b})^n\;\;is\;\;an\;integer}

Step 3: Now use N = I + f, where I is an integral part of N and f is a fractional part of N (0 < f < 1)

 

Now  

i) P P^{\prime}=(10+3 \sqrt{11})^{n}(10-3 \sqrt{11})^{n}=(10^2-(3 \sqrt{11})^2)^{n}=1^n=1
 

ii) P+P^{\prime}=2 \left[^{n}C_0\;(10)^{n}+^{n}C_2\;(10)^{n-2}(3 \sqrt{3})^{2} +^{n}C_2\;(10)^{n-4}(3 \sqrt{3})^{4}\cdots\cdots \right]

hence, it is an even integer

iii) 

Let P=[P]+f , where [P] is the integer just less than P 

Now, from (ii) 

P+P'=\text{ even integer } 

P+P'=[P]+f+P'= \text{Integer}

[P] is an integer hence f+P' should be an integer 

But P^{\prime}\in(0,1) and f\in(0,1)

hence f+P'\in (0,2)

or f+P'=1

hence, option D is not true

Posted by

Ajit Kumar Dubey

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