Binomial Theorem (DP IB Analysis & Approaches (AA): SL): 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 $\left(x-2\right){\left(\frac{3}{2}x-5\right)}^6.$

Given that $(2+nx{)}^2(1-2x{)}^n=4-24x+....$

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 $(5+x{)}^5.$

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 $(2-x{)}^3.$

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

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

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

Determine the value of $k.$

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

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

Determine the value of $k.$

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

In the expansion of $2x^2(3+kx{)}^7,$ 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 $p.$