Question

Assertion :The number of distinct (dissimilar) term in ${(x-a)}^{100}+{(x+a)}^{100}$ is 202 Reason: The number of term in ${(1+x)}^{n}is(n+1).$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion,
• C. Assertion is true but Reason is false,
• D. Assertion is false but Reason is true.

Answer 1

The correct option is D Assertion is false but Reason is true.

Considering an expansion
$(1+x{)}^n$ total number of terms will be n+1.
Now consider $(a-x{)}^{100}$ and $(a+x{)}^{100}$
$=[a^{100}-{}^{100}{C}_1{a}^{99}x+{}^{100}{C}_2{a}^{98}{x}^2+....x^{100}]+[a^{100}+{}^{100}{C}_1{a}^{99}x+{}^{100}{C}_2{a}^{98}{x}^2+....x^{100}]$
Therefore all the even terms will cancel out and we will be left with
$2[T_1+T_3+T_5...+T_{2n+1}...T_{101}]$
Noe $1,3,5,...101$ forms an A.P
$a_n=a+(n-1)d$
$101=1+(n-1)\left(2\right)$
$50=n-1$
$n=51$
Hence there will be in total 51 terms.