Binomial Theorem (Cambridge (CIE) IGCSE Additional Maths): Exam Questions

Expand ${(2-x)}^5$, simplifying each coefficient.

Hence solve  $\frac{{\mathrm{e}}^{{\left(2-x\right)}^5}\times {\mathrm{e}}^{80x}}{{\mathrm{e}}^{10x^4+32}}={\mathrm{e}}^{-x^5}$.

In the expansion of ${\left(2k-\frac{x}{k}\right)}^5$ , where $k$ is a constant, the coefficient of $x^2$ is $160$.

Find the value of $k$.

(i) Find the first 3 terms in the expansion of ${(1+3x)}^6$, in ascending powers of $x$. Simplify the coefficient of each term.

(ii) When the expansion of ${(1+3x)}^6{(a+x)}^2$ is written in ascending powers of $x$, the first three terms are $4+68x+b{x}^2$, where $a$ and $b$ are constants. Find the value of $a$ and the value of $b$.

The first 3 terms in the expansion of ${(3-ax)}^5$ , in ascending powers of $x$, can be written in the form $b-81x+c{x}^2$. Find the value of each of $a,b\mathrm{and}c$.

Find the first 3 terms in the expansion of ${\left(2+\frac{k}{2}\right)}^8$ in ascending powers of $k$.

Simplify the coefficient of each term.

Hence find the first three terms in ascending powers of $p$ in the expansion of ${\left(2+\frac{p-p^2}{2}\right)}^8$.

Simplify the coefficient of each term.

Find the term independent of $x$ in the expansion of ${\left(a{x}^2-\frac{1}{x^2}\right)}^6$ where $a\ne 0$.

Give your answer in terms of $a$.

The first 3 terms in the expansion of ${\left(a+x\right)}^3{\left(1-\frac{x}{3}\right)}^5$, in ascending powers of $x$, can be written in the form $27+bx+c{x}^2$ , where $a$, $b$ and $c$ are integers. Find the values of $a$, $b$ and $c$.

The first three terms in the expansion of ${\left(a+bx\right)}^5(1+x)\mathrm{are}32-208x+c{x}^2$ . Find the value of each of the integers $a,b\mathrm{and}c.$

Given that the coefficient of $x^2$ in the expansion of $\left(1+x\right){\left(1-\frac{x}{2}\right)}^n$ is $\frac{25}{4}$, find the value of the positive integer $n$. 

In the expansion of ${\left(1+\frac{x}{2}\right)}^n$the coefficient of $x^4$ is half the coefficient of $x^6$.

Find the value of the positive constant $n$.