Find a + b 4 - a - b 4 . Hence, evaluate 3 + 2 4 - 3 - 2 4 .

To find: a + b 4 - a - b 4 We know that, Binomial expansion formula is given by: ( a + b ) n = n C 0 a n + n C 1 a n - 1 b + n C 2 a n - 2 b 2 + … + n C n - 1 a b n - 1 + n C n b n By using binomial expansion we get, ( a + b ) 4 = C 0 4 a 4 + C 1 4 a 3 b + C 2 4 a 3 b 2 + C 3 4 a b 3 + C 4 4 b 4 and ( a - b ) 4 = C 0 4 a 4 - C 1 4 a 3 b + C 2 4 a 3 b 2 - C 3 4 a b 3 + C 4 4 b 4 ∴ a + b 4 + a - b 4 = C 0 4 a 4 + C 1 4 a 3 b + C 2 4 a 3 b 2 + C 3 4 a b 3 + C 4 4 b 4 - C 0 4 a 4 - C 1 4 a 3 b + C 2 4 a 3 b 2 - C 3 4 a b 3 + C 4 4 b 4 = 2 C 1 4 a 3 b + C 3 4 a b 3 = 2 4 a 3 b + 4 a b 3 = 8 a b a 2 + b 2 … . ( i ) In equation ( i ) we putting a = 3 , b = 2 . 3 + 2 4 - 3 - 2 4 = 8 3 2 3 2 + 2 2 = 8 6 3 + 2 = 40 6

Find x + 1 6 + x - 1 6 . Hence or otherwise evaluate 2 + 1 6 + 2 - 1 6 .

To find: x + 1 6 + x - 1 6 We know that, Binomial expansion formula is given by: ( a + b ) n = n C 0 a n + n C 1 a n - 1 b + n C 2 a n - 2 b 2 + … + n C n - 1 a b n - 1 + n C n b n By using binomial expansion, ( x + 1 ) 6 = C 0 6 x 6 + C 1 6 x 5 + C 2 6 x 4 + C 3 6 x 3 + C 4 6 x 2 + C 5 6 x + C 6 6 x 0 and ( x - 1 ) 6 = C 0 6 x 6 - C 1 6 x 5 + C 2 6 x 4 - C 3 6 x 3 + C 4 6 x 2 - C 5 6 x + C 6 6 x 0 ∴ ( x + 1 ) 6 + ( x - 1 ) 6 = 2 C 0 6 x 6 + C 2 6 x 4 + C 4 6 x 2 + C 6 6 = 2 x 6 + 15 x 4 + 15 x 2 + 1 … . ( i ) In equation ( i ) we put x = 2 , 2 + 1 6 + 2 - 1 6 = 2 2 6 + 15 2 4 + 15 2 2 + 1 = 2 8 + 60 + 30 + 1 = 2 99 = 198

Prove that ∑ r = 0 n 3 r C r n = 4 n

To prove: ∑ r = 0 n 3 r C r n = 4 n We know that, according to binomial theorem, ∑ r = 0 n C r n a n - r b r = ( a + b ) n … … ( i ) As, ∑ r = 0 n C r n a n - r b r = n C 0 a n + n C 1 a n - 1 b + n C 2 a n - 2 b 2 + … + n C n - 1 a b n - 1 + n C n b n If b = 3 and a = 1 the equation ( i ) becomes, ∑ r = 0 n C r n ( 1 ) n - r ( 3 ) r = ( 1 + 3 ) n ⇒ ∑ r = 0 n 3 r . C r n = 4 n Hence proved

E M B I B E

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.1

EXERCISE 8.1

Author:Jharkhand Board

Jharkhand Board Mathematics Solutions for EXERCISE 8.1

Simple step-by-step solutions to EXERCISE 8.1 questions of Binomial Theorem from MATHEMATICS Textbook for Class XI. Also get 3D topic explainers, cheat sheets, and unlimited doubts solving on EMBIBE.