QJEEAdv23Mac1aa: JEE Advanced 2023 Mathematics - Class 11 JEEAdv23Mac1aa Solution: Let and be two nonzero real numbers. If the coefficient of in…

Question

Verified Solution • 2025–26 Session moderate

Let

a

and

b

be two nonzero real numbers. If the coefficient of

x^5

in the expansion of

\left(a x^2+\frac{70}{27 b x}\right)^4

is equal to the coefficient of

x^{-5}

in the expansion of

\left(a x-\frac{1}{b x^2}\right)^7

, then the value of

2 b

is :

Accepted Answer

\begin{aligned}&amp; \mathrm{T}_{\mathrm{r}+1}={ }^4 \mathrm{C}_{\mathrm{r}}\left(\mathrm{a} \cdot \mathrm{x}^2\right)^{4-\mathrm{r}} \cdot\left(\frac{70}{27 \mathrm{bx}}\right)^{\mathrm{r}} \\\\&amp; ={ }^4 \mathrm{C}_{\mathrm{r}} \cdot \mathrm{a}^{4-\mathrm{r}} \cdot \frac{70^{\mathrm{r}}}{(27 b)^{\mathrm{r}}} \cdot x^{8-3 \mathrm{r}}\end{aligned}

\begin{aligned}&amp; \text { here } 8-3 r=5 \\\\&amp; 8-5=3 r \Rightarrow r=1 \\\\&amp; \therefore \text { coeff. }=4 \cdot a^3 \cdot \frac{70}{27 b}\end{aligned}

\begin{aligned}&amp; \mathrm{T}_{\mathrm{r}+1}={ }^7 \mathrm{C}_{\mathrm{r}}(\mathrm{ax})^{7-\mathrm{r}}\left(\frac{-1}{\mathrm{bx}^2}\right)^{\mathrm{r}} \\\\&amp; ={ }^7 \mathrm{C}_{\mathrm{r}} \cdot a^{7-\mathrm{r}}\left(\frac{-1}{\mathrm{~b}}\right)^{\mathrm{r}} \cdot \mathrm{x}^{7-3 \mathrm{r}} \\\\&amp; 7-3 \mathrm{r}=-5 \Rightarrow 12=3 \mathrm{r} \Rightarrow \mathrm{r}=4\end{aligned}

\begin{aligned}&amp; \text { now } \frac{35 \mathrm{a}^3}{\mathrm{~b}^4}=\frac{280 \mathrm{a}^3}{27 \mathrm{~b}} \\\\&amp; \mathrm{~b}^3=\frac{35 \times 27}{280}=\mathrm{b}=\frac{3}{2} \Rightarrow 2 \mathrm{~b}=3\end{aligned}

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