Question

Assertion (A) : Number of the disimilar terms in the sum of expansion $(x+a{)}^{102}+(x-a{)}^{102}$ is $206$

Reason (R) : Number of terms in the expansion of $(x+b{)}^n$ is n + 1

• A. Both A and R are individually true and R is the correct explanation of A.
• B. Both A and R are individually true and R is not correct explanation of A.
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is D A is false but R is true

$(x+a{)}^{102}+(x-a{)}^{102}$
Since a is constant let $a=1$
Therefore the above question can be re-written as
$(1+x{)}^{102}+(1-x{)}^{102}$
$=[1+{}^{102}{C}_1{x}^1+{}^{102}{C}_2{x}^2...+{}^{102}{C}_{102}{x}^{102}]+[1-{}^{102}{C}_1{x}^1+{}^{102}{C}_2{x}^2-{}^{102}{C}_3{x}^3...+{}^{102}{C}_{102}{x}^{102}]$
$=2[1+{}^{102}{C}_2{x}^2+{}^{102}{C}_2{x}^4...+{}^{102}{C}_{102}{x}^{102}]$ ...(i)
Hence number of dissimilar terms in Eq(i) will be $\frac{102}{2}+1$
$=52$ terms.
Hence A is false.
Reason is true.
Hence option D is the correct option.