Correct Answer - Option 2 : 7
Formula used∶
Rationalization of surds, (a + b) (a – b) = a2 – b2
Calculation∶
1/(4 - √15) – 1/(√15 - √14) + 1/(√14 - √13) -1/√(13 - √12) + 1/√(12 - √11) – 1/(√11 - √10) + 1/(√10 – 3)
⇒ [1/(4 - √15) × {(4 +√15)/ (4 +√15)}] = [(4 +√15)/{42 – (√15)2}] = (4 +√15)/(16 – 15) = (4 +√15)/1 = (4 + √15)
⇒ [{1/(√15 - √14)} × {(√15 + √14)/ (√15 + √14)}] = [{(√15 + √14)}/{(√15)2 – (√14)2}] = [√15 + √14)/(15 – 14) = (√15 + √14)/1 = √15 + √14
⇒ [{1/(√14 - √13) × {(√14 + √13)/ (√14 + √13)}] = [{(√14 + √13)}/{(√14)2 – (√13)2}] = [(√14 + √13)/(14 – 13)] = [(√14 + √13)/1] = √14 + √13
⇒ [{1/(√13 -√12) × {(√13 + √12)/ (√13 + √12)}] = [{(√13 +√12)}/{(√13)2 – (√12)2}] = [(√13 + √12)/(13 – 12)] = [(√13 + √12)/1] = √13 + √12
⇒ [{1/(√12 -√11) × {(√12 + √11)/ (√12 + √11)}] = [{(√12 +√11)}/{(√12)2 – (√11)2}] = [(√12 + √11)/(12 – 11)] = [(√12 + √11)/1] = √12 + √11
⇒ [{1/(√11 -√10) × {(√11 + √10)/ (√11 + √10)}] = [{(√11 +√10)}/{(√11)2 – (√10)2}] = [(√11 + √10)/(11 – 10)] = [(√11 + √10)/1] = √11 + √10
⇒ [{1/(√10 -3) × {(√10 + 3)/ (√10 + 3)}] = [{(√10 +3)}/{(√10)2 – (3)2}] = [(√10 + 3)/(10 – 9)] = [(√10 + 3)/1] = √10 + 3
Now, put all the values in their required places in the given equation, we get
⇒ (4 + √15) – (√15 + √14) +(√14 + √13) – (√13 + √12) + (√12 + √11) – (√11 + √10) + (√10 + 3)
⇒ 4 + √15 – √15 - √14 + √14 + √13 – √13 - √12 + √12 + √11 – √11 - √10 + √10 + 3 = 7
∴ The required value is 7.
