Questions & Answers

Question

Answers

A. 302

B. 301

C. 202

D. 101

Answer
Verified

*$({a^3} + {b^3}) = (a + b)({a^2} + {b^2} - ab)$

*$({a^x} \times {a^y}) = {(a)^{x + y}}$

*$({a^x} \times {b^x}) = {(ab)^x}$

Now first of all we will try to simplify the given equation.$ \Rightarrow {(1 + x)^{101}}{(1 + {x^2} - x)^{100}}$

We will split ${(1 + x)^{101}}$into two parts such that one part should have degree 100\[ \Rightarrow (1 + x){(1 + x)^{100}}{(1 + {x^2} - x)^{100}}\] (We have applied identity$({a^x} \times {a^y}) = {(a)^{x + y}}$)\[ \Rightarrow (1 + x){[(1 + x)(1 + {x^2} - x)]^{100}}\] (We have applied identity$({a^x} \times {b^x}) = {(ab)^x}$)

Now we know$({a^3} + {b^3}) = (a + b)({a^2} + {b^2} - ab)$ so to simplify equation more we will apply this identity as mentioned below:-

$ \Rightarrow ({1^3} + {x^3}) = (1 + x)({1^2} + {x^2} - x)$

Now we will put this simplified term in the equation.

\[ \Rightarrow (1 + x){[(1 + {x^3})]^{100}}\] .................Equation (1)

We know that binomial expansion is given by ${(1 + x)^n} = \sum\limits_{r = 0}^n {^n{C_r}.{X^r} = [{C_0} + {C_1}X + {C_2}{X^2} + ...{C_n}{X^n}]} $Now we will put value of n=100 in the expansion

$ \Rightarrow {(1 + {x^3})^{100}} = [{C_0} + {C_1}{x^3} + {C_2}{x^6} + ...{C_{100}}{x^{300}}]$

Now we will put this expansion value in equation 1

\[ \Rightarrow (1 + x)[{C_0} + {C_1}{x^3} + {C_2}{x^6} + ...{C_{100}}{x^{300}}]\]

Now multiplying whole expansion with 1 and then with x\[ \Rightarrow [{C_0} + {C_1}{x^3} + {C_2}{x^6} + ...{C_{100}}{x^{300}}][x{C_0} + {C_1}{x^4} + {C_2}{x^7} + ...{C_{100}}{x^{301}}]\]

As all the terms are different so they cannot be clubbed together and hence cannot be simplified further. Therefore terms in first expansion are 101 and other terms will also contain 101 terms, so the total terms in power of x in expansion will be 202.