JEE Mathematics 2023 Question with Solution

Given: The coefficients of $x^7$ in $\left( ax^2 + \frac{1}{2} bx \right)^{11}$ and $x^7$ in $\left( ax - \frac{1}{3} bx^2 \right)$ are equal.

Find: The correct relation between $a$ and $b$.

From the solution, the general form of the coefficient of $x^7$ in $\left( ax^2 + \frac{1}{2} bx \right)^{11}$ is obtained using the binomial expansion:

Thus, the coefficient of $x^7$ is given by:

Similarly, for $\left( ax - \frac{1}{3} bx^2 \right)$, the coefficient of $x^7$ is stated in the solution as:

Equating the two coefficients, the solution gives:

Thus,

Therefore, the correct option is C.

Given: A coefficient comparison problem involving binomial expansion.

Find: The required relation between $a$ and $b$.

When dealing with binomial expansions, carefully apply the binomial theorem to find the desired term. Do not forget to adjust the powers of the terms according to the required power of $x$.