1.8 Equations and systems using technology questions for IB Maths AI

Write down the gradient of $L_1$ in terms of $k$.

Find the value of $k$.

Find the coordinates of the point where the two cables cross.

An inspection point is located where $L_2$ crosses the $y$-axis. Find the distance between the inspection point and the point where the two cables cross.

Write down the distance between the centre of the Earth and the centre of the Moon in metres, in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Calculate the gravitational force between the Earth and the Moon. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

A weather satellite of mass 1850 kg orbits the Earth. The gravitational force between the Earth and the satellite is $4.6\times {10}^3$ N.

Find the distance, $r$, between the centre of the Earth and the satellite.

The radius of the Earth is $6.37\times {10}^6$ m.

Find the height of the satellite above the surface of the Earth, in km.

The speed of the satellite, $v$ m s⁻¹, is given by $v=\sqrt{{\displaystyle \frac{GM}{r}}}$, where $M$ is the mass of the Earth.

Calculate the speed of the satellite.

The satellite travels in a circle around the centre of the Earth.

Find the time, in hours, that the satellite takes to complete one orbit.

Write down the matrix $\mathit{L}$ such that $\left(\begin{array}{c}{J}_{n+1}\\ A_{n+1}\end{array}\right)=\mathit{L}\left(\begin{array}{c}{J}_n\\ A_n\end{array}\right)$.

(i) Find $J_1$ and $A_1$.

(ii) Find the total number of females in the reserve after 6 years, according to this model.

Show that the eigenvalues of $\mathit{L}$ are $1.1$ and $-0.3$.

Find an eigenvector of $\mathit{L}$ for each eigenvalue.

Show that $\left(\begin{array}{c}310\\ 130\end{array}\right)=50\left(\begin{array}{c}5\\ 3\end{array}\right)+20\left(\begin{array}{c}3\\ -1\end{array}\right)$.

Hence show that the total number of females after $n$ years is $T_n=400(1.1{)}^n+40(-0.3{)}^n$.

Find the first year in which the total number of females is predicted to exceed 3000.

Describe the long-term behaviour of the beaver population according to this model. Justify your answer.

The reserve cannot support an ever-growing population, so the ecologists plan to relocate some adult females each year. This changes only the proportion of adult females that remain as adults in the reserve, from 0.6 to $a$.

The population is stable in the long term when $\mathit{L}$ has an eigenvalue equal to 1.

Find the value of $a$ for which the population is stable.

The natural survival rate of adults remains 60 %, and the relocated females are chosen from the adults who would otherwise have remained.

Find the percentage of these adult females that must be relocated each year.

Explain what the model predicts if the ecologists relocate slightly more than this percentage every year.

State two limitations of this model.

Write down this information as a matrix equation of the form $\mathit{A}\mathit{x}=\mathit{b}$, where $\mathit{x}=\left(\begin{array}{c}a\\ c\\ s\end{array}\right)$.

Use an inverse matrix to find the price of each type of ticket.

On Monday, 40 adult, 70 child and 50 senior tickets were sold. Find the ticket income on Monday.

Use the point $(2,3.9)$ to write down an equation in $a$, $b$ and $c$.

Write down two more equations in $a$, $b$ and $c$.

Find the equation of the path of the ball.

Write down the height of the ball when Nadia hits it.

Find the maximum height reached by the ball.

The net is 9 m horizontally from Nadia and its top is 2.43 m above the floor. Determine whether the ball passes over the net.

The far end of the court is 18 m horizontally from Nadia. Determine whether the ball lands inside the court.

State one limitation of this model.

Write down the transition matrix $\mathit{T}$ such that ${\mathit{s}}_{n+1}=\mathit{T}{\mathit{s}}_n$, where ${\mathit{s}}_n=\left(\begin{array}{c}{a}_n\\ b_n\\ c_n\end{array}\right)$ gives the proportions of customers with Aero, Brio and Cellux after $n$ months.

Ji-ho is currently an Aero customer.

Find the probability that he is with Brio at the end of each of the next two months and with Cellux at the end of the third month.

At the start of January, 50 % of customers are with Aero, 30 % with Brio and 20 % with Cellux.

Find the percentage of customers with each network six months later.

Find the exact long-term proportion of customers with each network.

Interpret your answer to part (d)(i) in context.

Brio launches a loyalty scheme. Brio now keeps a proportion $p$ of its customers each month. Of the customers who leave Brio, one quarter move to Aero and three quarters move to Cellux. The other columns of $\mathit{T}$ do not change.

Write down the new second column of $\mathit{T}$ in terms of $p$.

Brio wants to have the largest long-term share of the three networks.

Find the minimum value of $p$ needed, giving your answer as a percentage correct to the nearest whole number.

Several years later Aero closes, and its customers join the other two networks. Daniela now models monthly switching between Brio and Cellux with the transition matrix

where the first row and column refer to Brio.

Find the eigenvalues of $\mathit{T}$.

Find an eigenvector for each eigenvalue.

Hence write down matrices $\mathit{P}$ and $\mathit{D}$ such that $\mathit{T}=\mathit{P}\mathit{D}{\mathit{P}}^{-1}$, where $\mathit{D}$ is a diagonal matrix.

Immediately after Aero closes, 45 % of customers are with Brio.

Show that the proportion of customers with Brio after $n$ months is $\frac{7}{19}+\frac{31}{380}(0.81{)}^n$.

Find the number of months after Aero closes until Brio's share first falls below 40 %.

State one reason why Daniela's Markov chain model may not be realistic.