Question

Options are:
(A) 1

(B) 2√2 - 1

(C) $\dfrac{\sqrt {5}}{2}$

(D) $\dfrac{2}{\sqrt {\sqrt {5} + 1}}$

[ Give full explanation. ]​

Answer 1

Required Answer:-

Given:

$\rm \mapsto N = \dfrac{ \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } }{ \sqrt{ \sqrt{5} + 1 } } - \sqrt{3 - 2 \sqrt{2} }$

To find:

• The value of N.

Solution:

Let us assume that,

$\rm \bigstar \: x = \dfrac{ \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } }{ \sqrt{ \sqrt{5} + 1 } }$

$\rm \bigstar \: y = \sqrt{3 - 2 \sqrt{2} }$

So,

$\rm \implies N = x + y$

Solve for x.

We have,

$\rm x = \dfrac{ \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } }{ \sqrt{ \sqrt{5} + 1 } }$

$\rm = \dfrac{( \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } )( \sqrt{ \sqrt{5} - 1} )}{ (\sqrt{ \sqrt{5} + 1 } )( \sqrt{ \sqrt{5} - 1 } )}$

$\rm = \dfrac{( \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } )( \sqrt{ \sqrt{5} - 1} )}{ \sqrt{ (\sqrt{5} + 1 )( \sqrt{5} - 1)}}$

$\rm = \dfrac{( \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } )( \sqrt{ \sqrt{5} - 1} )}{ \sqrt{5 - 1}}$

$\rm = \dfrac{( \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } )( \sqrt{ \sqrt{5} - 1} )}{ \sqrt{4}}$

$\rm = \dfrac{( \sqrt{ \sqrt{5} + 2 } + \sqrt{ \sqrt{5} - 2 } )( \sqrt{ \sqrt{5} - 1} )}{2}$

$\rm = \dfrac{( \sqrt{ \sqrt{5} + 2 } )( \sqrt{ \sqrt{5} - 1} )+ (\sqrt{ \sqrt{5} - 2 } )( \sqrt{ \sqrt{5} - 1} )}{2}$

$\rm = \dfrac{\sqrt{ (\sqrt{5} + 2) ( \sqrt{5} - 1)} + \sqrt{( \sqrt{5} - 2)( \sqrt{5} - 1) } }{2}$

$\rm = \dfrac{\sqrt{ 5 - \sqrt{5} + 2 \sqrt{5} - 2} + \sqrt{5 - \sqrt{5} - 2 \sqrt{5} + 2 } }{2}$

$\rm = \dfrac{\sqrt{ 3 + \sqrt{5}} + \sqrt{7 - 3\sqrt{5} } }{2}$

$\rm = \dfrac{\sqrt{ \dfrac{ 3 + \sqrt{5}}{2} \times 2} + \sqrt{ \dfrac{7 - 3\sqrt{5} }{2}} \times 2 }{2}$

$\rm = \dfrac{\sqrt{ \dfrac{ 6 + 2\sqrt{5}}{2}} + \sqrt{ \dfrac{14 - 6\sqrt{5} }{2}} }{2}$

$\rm = \dfrac{ \dfrac{1}{ \sqrt{2} } \times \bigg( \sqrt{6 + 2\sqrt{5}} + \sqrt{14 - 6\sqrt{5}} \bigg)}{2}$

$\rm = \dfrac{ \sqrt{6 + 2\sqrt{5}} + \sqrt{14 - 6\sqrt{5}} }{2 \sqrt{2} }$

$\rm = \dfrac{ \sqrt{1 + 5+ 2\sqrt{5}} + \sqrt{9 + 5 - 6\sqrt{5}} }{2 \sqrt{2} }$

$\rm = \dfrac{ \sqrt{ {(1)}^{2} + {( \sqrt{5} )}^{2} + 2 \times 1 \times \sqrt{5}} + \sqrt{{(3)}^{2} + {( \sqrt{5} )}^{2} - 2 \times 3 \times \sqrt{5}} }{2 \sqrt{2} }$

$\rm = \dfrac{ \sqrt{ {( 1 + \sqrt{5})}^{2} } + \sqrt{{( 3 - \sqrt{5} )}^{2}} }{2 \sqrt{2} }$

$\rm = \dfrac{ {( 1 + \sqrt{5} )}+ ( 3 - \sqrt{5} ) }{2 \sqrt{2} }$

$\rm = \dfrac{4}{2 \sqrt{2} }$

$\rm = \dfrac{2}{ \sqrt{2} }$

$\rm = \sqrt{2}$

So,

$\rm \implies x = \sqrt{2}$

Now, solve for y.

We have,

$\rm y = \sqrt{3 - 2 \sqrt{2} }$

$\rm = \sqrt{1 + 2 - 2 \sqrt{2} }$

$\rm = \sqrt{ {(1)}^{2} + {( \sqrt{2}) }^{2} - 2 \times 1 \times \sqrt{2} }$

$\rm = \sqrt{ {(1 - \sqrt{2} )}^{2} }$

$\rm = 1 - \sqrt{2}$

So,

$\rm \implies y = 1 - \sqrt{2}$

Therefore,

$\rm \implies N = x + y$

$\rm \implies N = \sqrt{2} + 1 - \sqrt{2}$

$\rm \implies N =1$

Hence, the value of N is 1.

So, Option A is the correct answer for this question.

Answer:

• The value of N is 1(Option A)

Answer 2

$\tt\huge\purple{hello!!!}$

HERE IS UR ANSWER

$_____________________________$

option a) is correct

the value of N is 1.