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

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

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

Write down the quadratic expression in the form $(px-q)(x-r).$

Find the coefficient of the $x^8$ term in the expansion of $(5x^2-15x+10{)}^5$. Give your answer in the form $a\times {10}^k,$ where $1\le a<10,$ $k\in \mathrm{\mathbb{Z}}$.

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.

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

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

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.

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 expansion of ${\left(4x-2\right)}^4.$

Find the term in $x^4$ in the expansion.

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

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 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 interval of convergence for the complete expansion.

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  ​​$\sqrt{\frac{x}{2\left(x+3\right)}}$.

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

State the interval of convergence for the complete expansion.

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

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 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 in $x$ is $-\frac{2}{27}$,  show that $n=3^{n-2}$.

For the value of $n$ found in part (a), find the coefficient of $x^2$. 

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 $x$.

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

Find the value of $n$.

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

Find the value of $n$.

Consider the expansion ${\left(5+x\right)}^5$.

Write down and simplify the expansion in descending powers of$x$.

Hence, find the exact value of ${\left(5.1\right)}^5$.

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

Write down and simplify the expansion in descending powers of $x$.

Hence find the exact value of ${\left(1.8\right)}^3$.

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 $b$.

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

Find $a$, giving your answer to 3 significant figures.

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 $k$.

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

Find the value of $p.$

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.

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

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

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

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

State the interval of convergence for the expansion found in part (b).

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

Given that the coefficient in $x^2$ 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 $\sqrt{2^n{\left(1-x\right)}^{n+1}}$, in ascending powers of $x$.