Binomial Theorem (DP IB Analysis & Approaches (AA): HL): Exam Questions

Find the coefficient of the term in $x^3$ in the expansion of $(2-x{)}^8.$

Find the first three terms, in ascending powers of $x,$ in the expansion of ${\left(3+x\right)}^4.$

In the expansion of ${\left(a-x\right)}^4,$ the coefficient of the $x^2$ term is 96.

Given that $a>0,$ find the value of $a$.

Find the first three terms, in ascending powers of $x,$ in the expansion of ${\left(9-2x\right)}^5$.

In the expansion of ${\left(a-2x\right)}^5$, where $a\ne 0$, the coefficient of the $x^2$ term is equal to the coefficient of the $x^3$ term.

Find the value of $a$.

In the expansion of ${\left(3+px\right)}^6$, where $p\ne 0$, the coefficient of the $x^4$ term is four times the coefficient of the $x^2$ term.

Find the possible values of $p$.

Consider the expansion of ${\left(4ax-3\right)}^5.$

Write down the number of terms in this expansion.

The coefficient of the term in $x^4$ is $-61440.$

Find the value of $a$ where $a$ is a positive constant.

Consider the expansion of ${\left(x^3+\frac{4}{x}\right)}^4$.

Find the first three terms in descending powers of $x$.

Find the value of the constant term.

The coefficient of $x^7$ in the expansion of $x^3{\left(ax+3\right)}^5$ is 1215.

Find the possible values of $a.$     

Consider the binomial expansion of $\sqrt[3]{4\left(2+x\right)}$.

Find the first three terms, in ascending powers of $x$, of the expansion.

State the restriction which must be placed on $x$ for this expansion to be valid.

Use the terms found in part (a) to estimate $\sqrt[3]{12}$ . Give your answer as a fraction.

Consider the binomial expansion of $\frac{1}{\sqrt{4+x}}$.

Find the first four terms, in ascending powers of $x$, of the expansion.

State the restriction which must be placed on $x$ for this expansion to be valid.

Consider the expansion of ${\left(\frac{3}{2}x-5\right)}^6.$

Find the term in $x^3$ in the expansion.

Hence find the term in $x^4$ in the expansion of $\left(x-2\right){\left(\frac{3}{2}x-5\right)}^6.$

Consider the expansion of ${\left(\frac{x^3}{3}+\frac{k}{x}\right)}^4.$ The constant term is $-\frac{500}{3}.$ 

Find the value of $k$.

Find the coefficient of the $x^4$ term.

Consider the expansion of ${\left(\frac{ax}{2}+\frac{3}{x^2}\right)}^5.$

Find an expression, in terms of $a$, for the coefficient of the $x^{-1}$ term.

The coefficient of the $x^{-1}$ term is 90.

Find the value of $a$.

Consider the expansion of $(5x^3-x{)}^6.$

Write down the number of terms in this expansion.

Find the first three terms, in descending powers of $x$, of the expansion.

Consider the expansion $(1-3x{)}^4(1-2kx{)}^2.$

The coefficient of the $x^6$ term is 36. Find the possible values of $k$.

Find the coefficient of the $x^{16}$ term in the expansion of ${\left(2x^2-x^3\right)}^7$.

In the expansion of $2x^2{\left(3+kx\right)}^7$, the coefficient of the term in $x^5$ is 210. 

Find the value of $k$.

Consider the binomial expansion of $\frac{1}{{\left(2x^2+x-3\right)}^2}.$

Show that  $\frac{1}{{\left(2x^2+x-3\right)}^2}$  can be written in the form ${\left(2x+a\right)}^{-2}{\left(x+b\right)}^{-2}$, and find the values of $a$ and $b$.

Hence, or otherwise, find the first three terms of the expansion, in ascending powers of $x$.

Consider the identity $\frac{1-7x}{\left(x+2\right)\left(3-x\right)}=\frac{A}{x+2}+\frac{B}{3-x}$, where $A$ and $B$ are constants to be determined.

Find the values of $A$ and $B$.

Hence, or otherwise, find the binomial expansion of $\frac{1-7x}{\left(x+2\right)\left(3-x\right)}$, in ascending powers of $x$, up to and including the term in $x^2$.

State the restriction which must be placed on $x$ for the expansion found in part (b) to be valid.

Consider the quadratic expression $5x^2-15x+10$.

Write down the quadratic expression in the form $p\left(x-q\right)\left(x-r\right)$, where $q<r$.

Find the coefficient of the $x^8$ term in the expansion of ${\left(5x^2-15x+10\right)}^5$.

The coefficient of $x^7$ in the expansion of ${\left(\frac{x}{3}\right)}^5{\left(ax+5\right)}^2$ is $\frac{1}{3}.$  

Find the possible values of $a.$

The sum of the coefficients of the expansion is $\frac{196}{243}.$

Determine which value of $a$ found in part (a) is correct.

In the expansion of ${\left(\frac{1}{2}x+1\right)}^n,$ the coefficient of the $x^2$ term is $8n$, where $n\in {\mathrm{\mathbb{Z}}}^{+}.$

Find $n$.

Consider the binomial expansion of $\frac{1+x^2}{1-x^2}$.

Find the first four terms, in ascending powers of $x$, of the expansion.

State the restriction which must be placed on $x$ for this expansion to be valid.

By substituting an appropriate value into the expression found in part (a), find an approximation for the value of $\frac{101}{99}$.

Consider the binomial expansion of $\sqrt{\frac{x}{2\left(x+3\right)}}$.

Find the first three terms, in ascending powers of $x$, of the expansion.

State the restriction which must be placed on $x$ for this expansion to be valid.

Using the expansion found in part (a), find an approximation for the value of $\frac{1}{\sqrt{8}}$, giving your answer as an exact value in as simple a form as possible.

Consider the binomial expansion of $\left(1+2x\right){\left(1-ax\right)}^{\frac{2}{3}}$, where $a\in \mathrm{\mathbb{N}}$.

Given that the coefficient of the term in $x^2$ is $-5$, find

(i) the value of $a$

(ii) the coefficient of the term in $x$

(iii) the first three non-zero terms of the expansion, in ascending powers of $x$.

Consider the binomial expansion of $\frac{1}{{\left(2x+3\right)}^n}$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

For the case where the coefficient of $x$ is $-\frac{2}{27}$, show that $n=3^{n-2}$.

Using your answer to part (a), find the value of $n$. Hence find the coefficient of $x^2$.

Given that ${\left(2+nx\right)}^2{\left(1-2x\right)}^n=4-24x+\dots$

Find the value of $n$.

Given that ${\left(1+nx\right)}^2{\left(1+\frac{2x}{3}\right)}^n=1+40x+...$

Find the value of $n.$

Consider the expansion of ${\left(2x^3-\frac{k}{x^2}\right)}^{12},$ where $k>0.$ The coefficient of the term in $x^6$ is equal to the coefficient of the term in $x^{16}.$

Find $k.$

Consider the expansion of ${\left(\frac{x^3}{a}+3x^5\right)}^9$, where $a>0$. The coefficient of the $x^{39}$ term is five times the coefficient of the $x^{31}$ term.

Find $a$.

Find the coefficient of the term in $x^3$ in the expansion of ${\left(-2x^2+7x-3\right)}^{-1}.$

Consider the binomial expansion of $\sqrt{2^n{\left(1-x\right)}^{n+1}}$, where $n\in \mathrm{\mathbb{Z}}$.

Given that the coefficient of $x^2$ is $\frac{3}{16}$, show that

$2^{\frac{n+2}{2}}\left(n^2-1\right)=3$

Given also that the constant term is $\frac{1}{2}$, find

(i) the value of $n$

(ii) the first three terms of the expansion of $\sqrt{2^n{\left(1-x\right)}^{n+1}}$, in ascending powers of $x$.

Given that ${\left(1-2x\right)}^2{\left(1+yx\right)}^3=1+zx+40x^2+\dots +k{y}^3{x}^5.$

Determine the value of $k$.

Find the possible values of $y$ and $z.$

Given that ${\left(1-2ax\right)}^3{\left(1+3x\right)}^3=1+bx-27x^2+\dots +k{a}^3{x}^6.$

Determine the value of $k$.

Find the possible values of $a$ and $b.$

The coefficient of the $x^5$ term in the expansion of $(1+2x{)}^4(1-px{)}^3$ is $-120.$

Find the value of $p.$