The Binomial Theorem

[Maximum mark: 4]

Expand $(2x + 1)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 4]

Expand $(2x - 3)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 5]

Consider the expansion of $\left(\dfrac{x^2}{2} + \dfrac{a}{x}\right)^6$. The constant term is $960$.

Find the possible values of $a$.

[Maximum mark: 5]

In the expansion of $(2x + 1)^n$, the coefficient of the term in $x^2$ is $40n$, where $n \in \mathbb{Z}^+$. Find $n$.

[Maximum mark: 7]

Consider the expansion of $\bigg(2x^6+\dfrac{x^2}{q}\bigg)^{10}$,  $q \neq 0$. The coefficient of the term

in $x^{40}$ is twelve times the coefficient of the term in $x^{36}$. Find $q$.