1.10 Counting principles and the extended binomial theorem questions for IB Maths AA

Show that the probability that the two counters are the same colour, given that bag 1 was chosen, is $\frac{2}{5}$.

Find the probability that the two counters are the same colour, given that bag 2 was chosen.

Given that the two counters are the same colour, find the probability that Leah chose bag 2.

Find the first three terms of the Maclaurin series for $(1-u{)}^{-\frac{1}{2}}$.

Hence show that $E\approx {\displaystyle \frac{1}{2}}m{v}^2+{\displaystyle \frac{3m{v}^4}{8c^2}}$.

A particle moves at speed $v=0.4c$.

Find the percentage error when the approximation in part (b) is used to find the kinetic energy of the particle.

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 $M_n>1000$.

Show that $\mathrm{P}(X=0)=\frac{1}{12}$.

Find $\mathrm{P}(X=1)$ and $\mathrm{P}(X=2)$.

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

Given that both balls are red, find the probability that they were taken from bag C.

Given that at least one of the balls is red, find the probability that they were taken from bag A.

A player pays $c$ dollars to play the game and receives 3 dollars for each red ball taken. Find the value of $c$ for which the game is fair.

Show that $f(x)=2{(1+{\displaystyle \frac{x}{8}})}^{\frac{1}{3}}$.

Hence find the first three terms of the Maclaurin series for $f(x)$.

Use your answer to part (b) to find an approximate value for $\sqrt[3]{9}$, and find the percentage error in this approximation.

Find the value of $I_0$, the value of $I_1$ and the value of $I_2$.

By writing ${\cos }^{n}x={\cos }^{n-1}x\cos x$ and using integration by parts, show that $I_n={\displaystyle \frac{n-1}{n}}{I}_{n-2}$, for $n\ge 2$.

Find the value of $I_4$ and the value of $I_5$.

Prove by mathematical induction that $I_{2n}={\displaystyle \frac{(2n)!\pi }{2^{2n+1}(n!{)}^2}}$, for $n\in \mathbb{N}$.

Prove by mathematical induction that $n{I}_n{I}_{n-1}={\displaystyle \frac{\pi }{2}}$, for $n\in {\mathbb{Z}}^{+}$.

Explain why $I_n\le I_{n-1}$, for $n\in {\mathbb{Z}}^{+}$.

Hence show that $\sqrt{{\displaystyle \frac{\pi }{2(n+1)}}}\le I_n\le \sqrt{{\displaystyle \frac{\pi }{2n}}}$, for $n\in {\mathbb{Z}}^{+}$.

Use your GDC to find the value of $I_{20}$, and verify that it satisfies the inequalities in part (g).