Binomial Expansion (Cambridge International A-Level Mathematics): Exam Questions

Evaluate

(i) $4 !$

(ii) {}^{5}C_{2}

(iii) {}^{6}C_{3}

Show that, for all values of $k$,

{}^{k}C_{1} = k

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

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

Expand $( 2 x - 3 )^{6}$.

In the expansion of $( p + x )^{12}$, the coefficient of $x^{5}$ is 12 976 128. Find the value of $p$.

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

Use your answer to part (a) to estimate the value of $( 5 . 04 )^{5}$.

In the expansion of $( p + x )^{4}$, where $p$ is a non-zero constant, the coefficient of $x^{2}$ is twice the coefficient of $x$. Find the value of $p$.

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 $( 9 - 2 x )^{5}$.

Use your answer to part (a) to estimate $( 8 . 9 )^{5}$.

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

In the expansion of $( 3 + p x )^{6}$, 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 $( 3 + 2 x )^{8}$.

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

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

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

In the expansion of $( a + b x )^{4}$, 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 $( 3 - 2 x )^{5}$.

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

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

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

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

Use your answer to part (a) to estimate $( 4 . 5 )^{4}$.

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

In the expansion of $( 3 a - 2 x )^{6}$, 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 $( 2 - 3 x )^{7}$.

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

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

In the expansion of $( p + q x )^{8}$, 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 $( a + b x )^{4}$, the coefficient of $x^{3}$ is 216.

In the expansion of $( a + b x )^{6}$, 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 $( 3 - 5 x )^{4}$ 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 + \dfrac{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 $( 4 - 3 x )^{7}$.

Given that {}^{n}C_{3} = 35, find the value of $n$.

In the expansion of \left(m - \dfrac{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 $( 4 - 3 x )^{9}$.

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

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

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

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

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