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BINOMIAL THEOREM — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSBINOMIAL THEOREM1 Mark
MCQ
If $(1-\text{x}^{2})^{\text{n}}=\sum_{\text{r}=0}^{\text{n}}\text{a}_{\text{r}}\text{x}^{\text{r}}(1-\text{x})^{2\text{n}-\text{r}},$ then $a_r$​ is equal to:
  • A
    ${ }^n C_r$
  • ✓
    ${ }^n \mathrm{C}_{\mathrm{r}} 3^{\mathrm{r}}$
  • C
    ${ }^{2 n} \mathrm{C}_{\mathrm{r}}$
  • D
    ${ }^n \mathrm{C}_{\mathrm{r}} 2^{\mathrm{r}}$
✓

Answer

Correct option: B.
${ }^n \mathrm{C}_{\mathrm{r}} 3^{\mathrm{r}}$
  1. ${ }^n \mathrm{C}_{\mathrm{r}} 3^{\mathrm{r}}$
Solution:
$(1-\text{x})^{\text{n}}(1+\text{x})^{\text{n}}=\sum_{\text{r}=0}^{\text{n}}\text{a}_{\text{r}}\text{x}^{\text{r}}(1-\text{x})^{\text{n}}(1-\text{x})^{\text{n}-\text{r}}$
$\Rightarrow(1-\text{x}+2\text{x})^{\text{n}}=\sum_{\text{r}=0}^{\text{n}}\text{a}_{\text{r}}\text{x}^{\text{r}}(1-\text{x})^{\text{n}-\text{r}}$
$\Rightarrow\sum_{\text{r}=0}^{\text{n}}{^\text{n}}\text{C}_{\text{r}}(1-\text{x})^{\text{n}-\text{r}}(2\text{x})^{\text{r}}$
$=\sum_{\text{r}=0}^{\text{n}}\text{a}_{\text{r}}\text{x}^{\text{r}}(1-\text{x})^{\text{n}-\text{r}}$
Comparing general term, we get $\text{a}^{\text{r}}={^\text{n}}\text{C}_{\text{r}}2^{\text{r}}$

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