# If a=√3-√2/√3+√2 and b=√3+√2/√3-√2, then the value of (a²/b)+(b²/a) :

1. 970
2. 1030
3. 1025
4. 930
Anurag Mishra Professor Asked on 11th December 2015 in

Explanation:-
a=√3-√2/√3+√2 and b=√3+√2/√3-√2
a = (√3-√2) (√3-√2)/(√3+√2) (√3-√2)
={(√3)² + (√2)² – 2 x √3 x √2 }/(√3)² – (√2)²
= (3 + 2 – 2 √6)/1
= 5 – 2√6
Then, a³ = (5)³ – (2√6)³ – 3 (5) (2√6) (5 – 2√6)
= 125 – 48√6 – 3 (5) (2√6) (5 – 2√6)

And,  b = √3+√2/√3-√2
= (√3+√2) (√3+√2)/(√3-√2) (√3+√2)
= {(√3)² + (√2)² + 2 x √3 x √2 }/(√3)² – (√2)²
= (3 + 2 + 2 √6)/1
= 5 + 2 √6
Then, b³ = (5)³ + (2√6)³ + 3 (5) (2√6) (5 + 2√6)
= 125 + 48√6 + 3 (5) (2√6) (5 + 2√6)

Then, a²/b + b²/a
= (a³ + b³)/ab
Put the values of  a³ = 125 – 48√6 – 3 (5) (2√6) (5 – 2√6),  b³ = 125 + 48√6 + 3 (5) (2√6) (5 + 2√6), a = 5 – 2√6 & b = 5 + 2 √6

Then, (a³ + b³)/ab = {125 – 48√6 – 3 (5) (2√6) (5 – 2√6) +  125 + 48√6 + 3 (5) (2√6) (5 + 2√6)}/ (5 – 2√6) (5 + 2 √6)
= (125 + 125)/(5)² – (2 √6)²
= 250/(25 – 4 x 6)
= 250/1
= 250

Hence, the answer is 250.

Anurag Mishra Professor Answered on 11th December 2015.