Class XI Chapter 8 – Binomial Theorem Maths ______________________________________________________________________________ 9 a 7 Thus, the required value of a is 9/7. Question 3: Find the coefficient of Solution 3: 5 x 6 7 in the product 1 2x 1 x using binomial theorem. 6 7 Using Binomial Theorem, the expressions, 1 2x and 1 x , can be expanded as 6 6 6 6 6 6 6 1 2x C C 2x C 2x C 2x C 2x C 2x C 2x 6 2 3 4 5 6 0 1 2 3 4 5 6 2 3 4 5 6 1 6 2x 15 2x 20 2x 15 2x 6 2x 2x **11**2x 60x 160x 240x 192x 64x 2 3 4 5 6 7 7 7 7 7 7 7 7 1 x C C x C x C x C x C x C x C x 7 2 3 4 5 6 7 0 1 2 3 4 5 6 7 1 7x 21x 35x 35x 21x 7x x 1 2x 1 x 2 3 4 5 6 7 6 7 **11**2x 60x 160x 240x 192x 64x 1 7x 21x 35x 35x 21x 7x x 2 3 4 5 6 2 3 4 5 6 7 The complete multiplication of the two brackets is not required to be carried out. Only those terms, which involve The terms containing 5 x 5 x , are required. are x 5 x x 4 x 2 x 3 x 3 x 2 x 4 x x 5 1 21 12 35 60 35 160 21 240 7 192 1 5 171x Thus, the coefficient of 5 x in the given product is 171. Question 4: If a and b are distinct integers, prove that a b n integer. [Hint: write a a b b n Solution 4: In order to prove that a n n a b k a b and expand] b is a factor of a n is a factor of b , where k is some natural number It can be written that, a a b b n a n b , it has to be proved that n , whenever n is a positive Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 8 – Binomial Theorem Maths ______________________________________________________________________________ n n a a b b a b b n n .... n n 1 n n 1 n n C0 a b C1 a b b Cn 1 a b b Cnb n n ... n 1 n n 1 n 1 n1 a b C a b b C a b b b n n n 1 n n 2 n n 1 1 .... n1 a b a b a b C a b b C b n n a b k a b Where, k a b C a b b C b This shows that a n n 1 n n 2 n n 1 1 .... n1 b is a factor of a n b n , where n is a natural number is a positive integer. Question 5: 6 6 3 2 3 2 Evaluate Solution 5: 6 6 Firstly, the expression a b a b done as is simplified by using Binomial Theorem. This can be 6 6 6 6 5 6 4 2 6 3 3 6 2 4 6 1 5 6 6 a b C a C a b C a b C a b C a b C a b C b 0 1 2 3 4 5 6 6 5 4 2 3 3 2 4 5 6 a 6a b 15a b 20a b 15a b 6ab b 6 6 6 6 5 6 4 2 6 3 3 6 2 4 6 1 5 6 6 a b C a C a b C a b C a b C a b C a b C b 0 1 2 3 4 5 6 6 5 4 2 3 3 2 4 5 6 a 6a b 15a b 20a b 15a b 6ab b 6 6 5 3 3 5 a b a b 2 6a b 20a b 6ab Putting a 3 and b 2 , we obtain 3 2 3 2 2 6 3 2 20 3 2 6 3 2 6 6 5 3 3 5 254 6 120 6 24 6 2 198 6 396 6 Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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