The Binomial Theorem

[Maximum mark: 5]

The third term in the expansion of $(x+p)^8$ is $252x^6$. Find the possible values of $p$.

[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: 6]

In the expansion of $px^2(5 + px)^8$, the coefficient of the term in $x^6$ is $3402$. Find the value of $p$.

[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: 5]

In the expansion of $x(2x + 1)^n$, the coefficient of the term in $x^3$ is $20n$, where $n \in \mathbb{Z}^+$. Find $n$.

[Maximum mark: 8]

Use the extension of the binomial theorem for $n \in \mathbb{Q}$ to show that $\sqrt{\dfrac{1+x}{1-x}} \approx 1 + x + \dfrac{x^2}{2}$, $|x| < 1$.

[Maximum mark: 7]

1. Write down the quadratic expression $3x^2 + 5x - 2$ in the form $(ax-b)(x+c)$.[2]
   

2. Hence, or otherwise, find the coefficient of the term in $x^9$ in the expansion
   of $(3x^2+5x-2)^5$. [5]