Binomial Theorem (Cambridge International IGCSE Further Maths): Exam Questions

Find the first 3 terms in the expansion of $( 4 - \frac{x}{16} )^{6}$in ascending powers of $x$. Give each term in its simplest form.

Extract:

Find the first 3 terms in the expansion of (4 - x/16)^6 in ascending powers of x. Give each term in its simplest form.

(4 - x/16)^6 = 4096 - 384x + 15x^2 + ...

Hence find the term independent of x in the expansion of $( 4 - \frac{x}{16} )^{6} ( x - \frac{1}{x} )^{2}$

Find the term independent of $x$ in the binomial expansion of $( 3 x - \frac{1}{x} )^{6}$.

Find the coefficient of $x^{2}$ in the expansion of $( x - \frac{3}{x} ) ( x + \frac{2}{x} )^{5}$.

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

Extract:

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

32 - 80x + 80x² - 40x³ + 10x^4 - x^5

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

In the expansion of $( 2 k - \frac{x}{k} )^{5}$ , 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 $4+68x+bx^{2}$, 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 $b-81x+cx^{2}$. Find the value of each of $a,b\mathrm{and}c$.

Find the first 3 terms in the expansion of $( 2 + \frac{k}{2} )^{8}$ 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 $( 2 + \frac{p - p^{2}}{2} )^{8}$.

Simplify the coefficient of each term.

Find the term independent of $x$ in the expansion of $( a x^{2} - \frac{1}{x^{2}} )^{6}$ where $a\neq0$.

Give your answer in terms of $a$.

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

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

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

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

Find the value of the positive constant $n$.