Chapter 8 Binomial Theorem Ex 8.2 Solutions
Question - 11 : - Prove that the coefficientof xn in the expansion of (1 + x)2n istwice the coefficient of xn in the expansion of (1+ x)2n–1 .
Answer - 11 : -
It is known that (r +1)th term, (Tr+1), in the binomialexpansion of (a + b)n is givenby 
.
Assuming that xn occursin the (r + 1)th term of the expansion of (1+ x)2n, we obtain
Comparing the indices of x in xn andin Tr + 1, we obtain
r = n
Therefore, the coefficient of xn inthe expansion of (1 + x)2n is
Assuming that xn occursin the (k +1)th term of the expansion (1 + x)2n –1, we obtain
Comparing the indices of x in xn and Tk +1, we obtain
k = n
Therefore, the coefficient of xn inthe expansion of (1 + x)2n –1 is
From (1) and (2), it is observedthat
Therefore, the coefficientof xn in the expansion of (1 + x)2n istwice the coefficient of xn in the expansion of (1+ x)2n–1.
Hence, proved.
Question - 12 : - Find a positive value of m forwhich the coefficient of x2 in the expansion
(1 + x)m is6.
Answer - 12 : -
It is known that (r +1)th term, (Tr+1), in the binomialexpansion of (a + b)n is givenby .
Assuming that x2 occursin the (r + 1)th term of the expansion (1 +x)m,we obtain
Comparing the indices of x in x2 andin Tr + 1, we obtain
r = 2
Therefore, the coefficientof x2 is 
.
It is given that the coefficientof x2 in the expansion (1 + x)m is6.
Thus, the positive value of m,for which the coefficient of x2 in the expansion
(1 + x)m is6, is 4.