| Directions: Assertion-Reason type questions. Each of these questions contains two statements: Statement I (Assertion) and Statement II (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice in the codes [a], [b], [c] and [d] in the given below: |
| Statement I: The term independent of an the expansion of \[{{\left( x+\frac{1}{x}+2 \right)}^{21}}\]is \[^{42}{{C}_{21}}\]. |
| Statement II: In a binomial expansion middle term is independent of x |
A) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
B) Statement I is true, Statement II is false.
C) Statement I is false. Statement II is true.
D) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
Correct Answer: B
Solution :
Now, \[{{\left( x+\frac{1}{x}+2 \right)}^{21}}={{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{42}}\] \[\therefore \]\[{{T}_{r+1}}{{=}^{42}}{{C}_{r}}{{(\sqrt{x})}^{42-r}}.{{\left( \frac{1}{\sqrt{x}} \right)}^{r}}\]\[{{=}^{42}}{{C}_{r}}.{{(x)}^{21-r}}\] For independent of \[x,21-r=0\Rightarrow r=21\] \[\therefore \]\[{{T}_{21+1}}{{=}^{42}}{{C}_{21}}\] Hence, Statement I is true. In a binomial expansion \[{{(x+a)}^{n}}\] (say) middle term is independent of x which is possible only when x . a = 1 Hence, it is not necessary that middle term is independent of x.
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