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# Chapter 8 Binomial Theorem Ex-8.2 Interview Questions Answers

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Question 1 : Find the coefficient of x5 in (x + 3)8

It is known that (+1)th term, (Tr+1), in the binomialexpansion of (b)n is givenby .

Assuming that x5 occursin the (r + 1)th term of the expansion (x +3)8, we obtain Comparing the indices of x in x5 andin Tr +1, we obtain

r = 3

Thus, the coefficient of x5 is Question 2 :

Find the coefficient of a5b7 in(a – 2b)12

It is known that (+1)th term, (Tr+1), in the binomialexpansion of (b)n is givenby .

Assuming that a5b7 occursin the (r + 1)th term of the expansion (a –2b)12, we obtain Comparing the indices of a and b in a5 bandin Tr +1, we obtain

r = 7

Thus, the coefficient of a5b7 is Question 3 :

Write the general term in theexpansion of (x2 – y)6

It is known that the generalterm Tr+1 {which is the (+1)th term} in the binomial expansion of (b)n isgiven by .

Thus, the general term in theexpansion of (x2 – y6) is Question 4 :

Write the general term in theexpansion of (x2 – yx)12x ≠ 0

It is known that the generalterm Tr+1 {which is the (+1)th term} in the binomial expansion of (b)n isgiven by .

Thus, the general term in theexpansion of(x2 – yx)12 is Question 5 :

Find the 4th termin the expansion of (x – 2y)12 .

It is known that (+1)th term, (Tr+1), in the binomialexpansion of (b)n is givenby .

Thus, the 4th termin the expansion of (x – 2y)12 is Question 6 : Find the 13th term in the expansion of .

It is known that (+1)th term, (Tr+1), in the binomialexpansion of (b)n is givenby .

Thus, 13th termin the expansion of is Question 7 : Find the middle terms in the expansions of It is known that in the expansion of (a + b)n, if n is odd, then there are two middle terms, namely, term and term.
Therefore, the middle terms in the expansion of are term and term Thus, the middle terms in the expansion of are .

Question 8 : Find the middle terms in the expansions of It is known that in the expansion (a + b)n, if n is even, then the middle term is term.
Therefore, the middle term in the expansion of is term Thus, the middle term in theexpansion of is 61236 x5y5.

Question 9 :

In the expansion of (1 + a)m+ n, prove that coefficients of am and an areequal.

It is known that (+1)th term, (Tr+1), in the binomialexpansion of (b)n is givenby .

Assuming that am occursin the (r + 1)th term of the expansion (1 + a)m + n,we obtain Comparing the indices of a in am andin T+ 1, we obtain

r = m

Therefore, the coefficientof am is Assuming that an occursin the (k + 1)th term of the expansion (1 + a)m+n,we obtain Comparing the indices of a in an andin Tk + 1, we obtain

k = n

Therefore, the coefficientof an is Thus, from (1) and (2), it can beobserved that the coefficients of am and an inthe expansion of (1 + a)m + n areequal.

Question 10 :

The coefficients of the (r –1)thrth and (r + 1)th termsin the expansion of

(x + 1)n arein the ratio 1:3:5. Find n and r.

It is known that (+1)th term, (Tk+1), in the binomialexpansion of (b)n is givenby .

Therefore, (r – 1)th termin the expansion of (x + 1)n is r th termin the expansion of (x + 1)n is (r + 1)th termin the expansion of (x + 1)n is Therefore, the coefficients ofthe (r – 1)thrth, and (r +1)th terms in the expansion of (x + 1)n are respectively. Since thesecoefficients are in the ratio 1:3:5, we obtain   Multiplying (1) by 3 andsubtracting it from (2), we obtain

4– 12 = 0

r = 3

Putting the value of r in(1), we obtain

n –12 + 5 = 0

n = 7

Thus, = 7and r = 3

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