Q172 / 270 · NDA 2022 (Phase 2) · Mathematics

Paragraph: Consider the binomial expansion of $(p+q x)^9$ :Question: What is the value of $q$ if the coefficients of $x^3$ and $x^6$ are equal ?

  • Correct answer
    A
    $p$
  • B
    $9 p$
  • C
    $\frac{1}{p}$
  • D
    $p^2$

Solution

Given expansion is $(p+q x)^9$ $T^{\mathrm{r}+1}={ }^9 \operatorname{Cr} p^{9-\mathrm{r}}(q x)^{\mathrm{r}}={ }^9 \operatorname{Cr} p p^{\mathrm{q}-\mathrm{r}} q^{\mathrm{r}} x^{\mathrm{r}}$ $\because \quad$ coefficient of $x^3=$ coefficient $x^6$ $\begin{aligned} & { }^9 C_3 p^6 q^3={ }^9 C_6 p^3 q^6 \\ & \Rightarrow p^3=q^3 \\ & \Rightarrow p=q \end{aligned}$

Hence, the correct answer is option: 'A'.

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