1.9 The binomial theorem questions for IB Maths AA

Find the value of $p$.

Find the exact value of $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

Write down the number of terms in the expansion.

Find the coefficient of $x^9$.

Show that ${\displaystyle {\int }_0^1{f}_n(x)\mathrm{d}x=1}$ for all $n\in {\mathbb{Z}}^{+}$.

Consider the case $n=3$.

Find $\mathrm{E}(X_3)$.

Find the median of $X_3$, and hence state whether the median of $X_3$ is greater than or less than $\mathrm{E}(X_3)$.

By using the substitution $u=1-x$, show that $\mathrm{E}(X_n)={\displaystyle \frac{1}{n+1}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X_n)={\displaystyle \frac{n}{(n+1{)}^2(n+2)}}$.

Show that the median of $X_n$ is $m_n=1-2^{-\frac{1}{n}}$.

Show that, for $n\in {\mathbb{Z}}^{+}$, the inequality $\mathrm{E}(X_n)>m_n$ is equivalent to ${(1+{\displaystyle \frac{1}{n}})}^n>2$.

By using the binomial theorem, prove that $\mathrm{E}(X_n)>m_n$ for all integers $n\ge 2$.

Use l'Hôpital's rule to find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }n{m}_n}$.

Hence find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{m_n}{\mathrm{E}(X_n)}}$.

Write down an expression for $\mathrm{P}(X=4)$ in terms of $p$.

It is given that $\mathrm{P}(X=4)=\frac{1}{81}$. Find the value of $p$.

Show that $\mathrm{P}(X=2)=\frac{8}{27}$.

Find $\mathrm{E}(X)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

In a game, a player receives $\mathrm{\$}{3}^X$. Show that the expected amount that the player receives is $\mathrm{\$}\frac{625}{81}$.

The player pays $\mathrm{\$}c$ to play the game, where $c\in {\mathbb{Z}}^{+}$. Find the least value of $c$ for which the organiser expects to make a profit.

Now let $Y\sim \mathrm{B}(n,\frac{1}{3})$, and suppose a player receives $\mathrm{\$}{2}^Y$. Find, in terms of $n$, the expected amount the player receives.

Use technology to find the two values of $r$ for which ${\displaystyle \left(\genfrac{}{}{0px}{}{12}{r}\right)}=495$.

Find the coefficient of $x^4$ in the expansion of $(x+3{)}^{12}$.

Find the coefficient of $x^8$ in the expansion of $(x+3{)}^{12}$.

Write down the value of $R_1$, the value of $R_2$ and the value of $R_3$.

Suppose $k$ lines in general position are drawn, and then a $(k+1)$th line is added so that the $k+1$ lines are in general position. Explain why $R_{k+1}=R_k+k+1$.

Prove by mathematical induction that $R_n={\displaystyle \frac{n^2+n+2}{2}}$, for $n\in {\mathbb{Z}}^{+}$.

Now consider $n$ circles drawn in a plane so that every two circles meet at exactly two points and no three circles pass through the same point. Let $C_n$ be the number of regions into which the $n$ circles divide the plane. For example, $C_1=2$ and $C_2=4$.

Explain why $C_{k+1}=C_k+2k$, for $k\in {\mathbb{Z}}^{+}$.

Prove by mathematical induction that $C_n=n^2-n+2$, for $n\in {\mathbb{Z}}^{+}$.

Finally, $n$ points are placed on a circle and every pair of points is joined by a chord, with no three chords meeting at the same point inside the circle. Let $M_n$ be the number of regions into which the chords divide the inside of the circle. It is known that $M_1=1$, $M_2=2$, $M_3=4$, $M_4=8$ and $M_5=16$.

Suggest an expression for $M_n$ in terms of $n$.

It can be shown that $M_6=31$. Explain why this shows that your expression from part (e) is not correct.

It can be shown that $M_n={\displaystyle \left(\genfrac{}{}{0px}{}{n}{4}\right)}+{\displaystyle \left(\genfrac{}{}{0px}{}{n}{2}\right)}+1$, for $n\ge 4$.

Verify that this formula gives the correct values of $M_4$ and $M_6$.

By using the identity ${\displaystyle \left(\genfrac{}{}{0px}{}{m}{r}\right)}={\displaystyle \left(\genfrac{}{}{0px}{}{m-1}{r}\right)}+{\displaystyle \left(\genfrac{}{}{0px}{}{m-1}{r-1}\right)}$, show that, for $n\ge 5$,

Hence explain why $M_5=2^4$ and why $M_n<2^{n-1}$ for all $n\ge 6$.

Find the least value of $n$ for which $M_n>1000$.