Question

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Answer 1

Answer:

correct option is B

Step-by-step explanation:

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:x = \sqrt{5} + 1$

and

$\rm :\longmapsto\:y = \dfrac{1}{2 \sqrt{5} - 3}$

Consider,

$\rm :\longmapsto\:y = \dfrac{1}{2 \sqrt{5} - 3}$

On rationalizing the denominator, we get

$\rm :\longmapsto\:y = \dfrac{1}{2 \sqrt{5} - 3} \times \dfrac{2 \sqrt{5} + 3}{2 \sqrt{5} + 3}$

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

$\red{\bigg \{ \because \:(x + y)(x - y) = {x}^{2} - {y}^{2} \bigg \}}$

$\rm :\longmapsto\:y =\dfrac{2 \sqrt{5} + 3 }{20 - 9 }$

$\rm :\longmapsto\:y =\dfrac{2 \sqrt{5} + 3 }{11}$

Now, Consider

$\red{\rm :\longmapsto\:x + y}$

On substituting the values of x and y, we get

$\red{\rm : = \: \sqrt{5} + 1 + \dfrac{2 \sqrt{5} + 3 }{11}}$

On taking LCM, we get

$\red{\rm = \: \dfrac{11 \sqrt{5} + 11 + 2 \sqrt{5} + 3 }{11}}$

On rearranging the terms, we get

$\red{\rm = \: \dfrac{11 \sqrt{5} + 2 \sqrt{5} + 3 + 11 }{11}}$

$\red{\rm = \: \dfrac{13 \sqrt{5} + 14 }{11}}$

$\red{\bf\implies \:\boxed{ \tt{ \: x + y = \dfrac{13 \sqrt{5} + 14 }{11}} \: }}$

Hence, option (B) is correct.

More Identities to know :-

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)