# If X= √ 3 − √ 2 √ 3 + √ 2 and Y = √ 3 + √ 2 √ 3 − √ 2 , Then X2 + Y +Y2 = - Mathematics

MCQ

If x= $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$ and y = $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ , then x2 + y +y2 =

• 101

• 99

• 98

• 102

#### Solution

Given that  x= (sqrt3 - sqrt2) /(sqrt3 + sqrt2) and y = (sqrt3 + sqrt2) /(sqrt3 - sqrt2).

We need to find  x^2 +xy +y^2

Now we will rationalize x. We know that rationalization factor for   sqrt3+sqrt2 is sqrt3-sqrt2   sqrt3+sqrt2. We will multiply numerator and denominator of the given expression x= (sqrt3 - sqrt2) /(sqrt3 + sqrt2) by sqrt3 - sqrt2, to get

x =  x= (sqrt3 - sqrt2) /(sqrt3 + sqrt2) xx (sqrt3 - sqrt2) /(sqrt3 - sqrt2)

= ((sqrt3)^2 +(sqrt2)^2 - 2 xx sqrt3 xx sqrt2)/((sqrt3)^2 - (sqrt2)^2)

= (3+2-2sqrt6)/(3-2)

= 5-2sqrt6

Similarly, we can rationalize y. We know that rationalization factor for sqrt3 - sqrt2 is  sqrt3 +sqrt2. We will multiply numerator and denominator of the given expression (sqrt3 +sqrt2) /(sqrt3 - sqrt2)by sqrt3 + sqrt2, to get

y = (sqrt3 + sqrt2) /(sqrt3 - sqrt2) xx (sqrt3 + sqrt2) /(sqrt3 +sqrt2)

= ((sqrt3)^2 +(sqrt2)^2 +2 xx sqrt3 xx sqrt2)/((sqrt3)^2 - (sqrt2)^2)

= (3+2-2sqrt6)/(3-2)

= 5-2sqrt6

Therefore,

x^2 + xy + y ^ 2  = ( 5 - 2 sqrt6 )^2+ (5-2sqrt6) (5 + 2 sqrt6)+ (5+2sqrt6)^2

= 5^2 +(2sqrt6 )^2 - 2 xx 5 xx 2 sqrt6 +5^2 - (2sqrt6 )^2+ 5^2 + (2sqrt6)^2 + 2xx 5 xx 2sqrt6

= 25 +24 -20sqrt6 +25 - 24 +25 +24 +20sqrt6

= 49 + 1+ 49

= 99`

Concept: Laws of Exponents for Real Numbers
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#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 3 Rationalisation
Exercise 3.4 | Q 14 | Page 17

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