JEE 2023 — Mathematics PYQ

Binomial expansion ke general term formula ka use karke hum ise solve karenge.

1. General Term Nikalna

Expansion $(a+b{)}^n$ ke liye general term $T_{r+1}$ hota hai:

Yahan hamare paas hai:

• $a=\frac{4x}{5}$

• $b=\frac{5}{2x^2}=\frac{5}{2}{x}^{-2}$

• $n=9$

Isliye, general term hoga:

2. $x$ ki Power ko Separate Karna

3. $r$ ki Value Find Karna

Hume $x^{-6}$ ka coefficient chahiye, isliye $x$ ki power ko $-6$ ke barabar rakhenge:

4. Coefficient Calculate Karna

Ab $r=5$ ko formula mein rakhte hain:

Ise simplify karte hain:

• $\left(\genfrac{}{}{0ex}{}{9}{5}\right)=\frac{9\times 8\times 7\times 6}{4\times 3\times 2\times 1}=126$

• ${\left(\frac{4}{5}\right)}^4=\frac{256}{625}$

• ${\left(\frac{5}{2}\right)}^5=\frac{3125}{32}$

Final Answer:

The coefficient of $x^{-6}$ is 5040.