Binomial Expansion (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Expand $(2+x{)}^4$.

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

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

Use your answer to part (a) to estimate $(3.1{)}^4$.

In the expansion of $(a-x{)}^4$, the coefficient of $x^2$ 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 $(9-2x{)}^5$.

Use your answer to part (a) to estimate $(8.9{)}^5$.

In the expansion of $(a-2x{)}^5$, the coefficient of $x^2$ is equal to the coefficient of $x^3$. Find the value of $a$.

In the expansion of $(3+px{)}^6$, the coefficient of $x^4$ is four times the coefficient of $x^2$. Find the possible values of $p$.

Find the first three terms, in ascending powers of $x$, in the expansion of $(3+2x{)}^8$.

Given that $x$ is small, so that $x^3$ and higher powers of $x$ can be ignored, show that

$(1+x)(3+2x{)}^8\approx 6561+41553x+116640x^2$

In the expansion of $(p+qx{)}^5$, the coefficients of $x^2$ and $x^3$ are equal. Find $p$ in terms of $q$.

In the expansion of $(a+bx{)}^4$, the coefficient of $x^2$ is equal to the coefficient of $x^3$. Show that $\frac{a}{b}=\frac{2}{3}$.

Given that $a$ and $b$ are integers, and that $10<b<15$, find the values of $a$ and $b$.

Expand $(3-2x{)}^5$.

Fully expand $(4-x{)}^4$.

Fully expand ${\left(2-\frac{1}{3}x\right)}^4$.

Find the coefficient of $x^4$ in the expansion of $(3+2x{)}^9$.

Find the first three terms, in ascending powers of $x$, in the expansion of $(5-2x{)}^4$.

Use your answer to part (a) to estimate $(4.5{)}^4$.

In the expansion of $(4-px{)}^6$, the coefficient of $x^4$ is 19 440. Given that $p$ is a positive integer, find the value of $p$.

In the expansion of $(3a-2x{)}^6$, the coefficient of $x^3$ is equal to the coefficient of $x^4$. Find the value of $a$.

Find the first three terms, in ascending powers of $x$, in the expansion of $(2-3x{)}^7$.

Given that $x$ is small, so that $x^3$ and higher powers of $x$ can be ignored, show that

$(1-2x)(2-3x{)}^7\approx 128-1600x+8736x^2$

In the expansion of $(p+qx{)}^8$, the coefficients of $x^2$ and $x^6$ are equal. Find $p$ in terms of $q$.

In the expansion of $(1+x{)}^n$, the coefficient of $x^3$ is 84. Find the value of $n$.

In the expansion of $(a+bx{)}^4$, the coefficient of $x^3$ is 216.

In the expansion of $(a+bx{)}^6$, the coefficient of $x^4$ is 4860.

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

Use the first three terms, in ascending powers of $x$, in the expansion of $(3-5x{)}^4$ to find an approximation for $(2.6{)}^4$.

Using your calculator, find the percentage error in the approximation from part (a) to the exact value of $(2.6{)}^4$.

In the expansion of ${\left(3a+\frac{1}{2}x\right)}^6$, the coefficient of $x^3$ is equal to the coefficient of $x^5$. Find the values of $a$, giving your answers in the form $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are integers to be found.

Find the coefficient of $x^4$ in the expansion of $(4-3x{)}^7$.

Given that ${}^{n}C_3=35$, find the value of $n$.

In the expansion of ${\left(m-\frac{1}{4}x\right)}^5$, the coefficient of $x^3$ is $-10$. Find the possible values of $m$.

Find the first three terms, in ascending powers of $x$, in the expansion of $(4-3x{)}^9$.

Given that $x$ is small, so that $x^3$ and higher powers of $x$ can be ignored, show that

$(3-2x^2)(4-3x{)}^9\approx 786432-5308416x+15400960x^2$

In the expansion of $(p+qx{)}^9$, the coefficient of $x^3$ is double that of the $x^5$ term. Find $p$ in terms of $q$.

In the expansion of $(1-3x{)}^n$, the coefficient of $x^3$ is $-3240$. Find the value of $n$.

In the expansion of $(a+bx{)}^8$, the coefficient of $x^5$ is $-870 912$. In the expansion of $(a+bx{)}^{12}$, the coefficient of $x^3$ is $-1 557 135 360$. Find the possible values of $a$ and $b$.