Binomial Theorem
Ratio of Symmetric Terms in Binomial Expansion
nta_pyq_2023_apr
Grade 11
Question:
If the $1011$th term from the end in the binomial expansion of $\left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^{2022}$ is $1024$ times the $1011$th term from the beginning, then $32|x|$ is equal to
Step-by-Step Solution
Key Concept: The $1011$th term from the end $=T_{2022-1011+2}=T_{1013}$. Ratio $\frac{T_{1013}}{T_{1011}}=\frac{\binom{2022}{1012}(4x/5)^{1010}(-5/2x)^{1012}}{\binom{2022}{1010}(4x/5)^{1012}(-5/2x)^{1010}}$.
$\frac{(-5)^2/(2x)^2}{(4x)^2/5^2}=\frac{5^4}{2^2\cdot4^2\cdot x^4}\cdot x^2\cdot x^2=2^{10}\Rightarrow|x|=\frac{5}{16}$. $32|x|=10$.
Correct Answer: 2