Practice IB Mathematics Applications & Interpretation (AI) Topic SL 1.8—use of Technology to Solve Systems of Linear Equations and Polynomial Equations with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 1.8—use of Technology to Solve Systems of Linear Equations and Polynomial Equations and mirrors Paper 1, 2, 3 style where relevant.
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The path of a soccer ball kicked from the ground at the edge of a sports field can be modelled by a quadratic function where is the height of the ball in metres above the ground and is the horizontal distance in metres from the point where the ball was kicked.
A coach uses high-speed cameras to record three points on the ball's path.
| (m) | 3 | 7 | 12 |
|---|---|---|---|
| (m) | 3.55 | 5.55 | 5.80 |
A defender stands m from the point where the ball was kicked. With an outstretched arm, the defender can reach a maximum height of m.
Write down a system of three linear equations in , , and .
Use technology to solve the system and find the values of , , and .
Write down the equation of the model.
Find the horizontal distance the ball travels before it lands on the ground.
Determine, with reasoning, whether the defender is able to intercept the ball as it passes the defender's position.
A small business produces ceramic pots. The daily production cost, , in dollars, when pots are produced is modelled by Each pot is sold for $, so the daily revenue, , in dollars, when pots are sold is
Calculate the daily production cost when pots are produced.
Hence calculate the daily profit when pots are produced and sold.
Find the two values of , within the domain, at which the business breaks even.
Find the maximum daily profit and the value of at which it occurs.
Determine the range of values of for which the business makes a profit.
Suggest one reason why this model would not be appropriate for .
A flare's height is modelled by , where is in metres and is in seconds. In the first two parts, use technology where appropriate.
Find the time when the flare hits the ground, giving your answer correct to three significant figures.
A signal reading is modelled by . Write down an equation whose solutions are the -values for which the signal reading is , and solve it.
The cost of a subscription is modelled by , where is in dollars and is in years. Find the value of when the cost reaches .
Hence determine the first whole year for which the cost exceeds .
The amount of substrate in a chemical reactor, , in kilograms, is modelled by where is the time, in hours, since the start of the reaction.
The chemist records the times at which .
Write down the initial amount of substrate in the reactor.
Find the three values of at which .
Hence determine the total number of distinct times during the -hour period at which .
A bakery sells three types of bread: sourdough, multigrain, and rye. The table below shows the number of loaves sold of each type to three customers on a particular day, and the total amount each customer paid.
| Customer | Sourdough | Multigrain | Rye | Total ($$$) |
|---|---|---|---|---|
| Alex | 3 | 2 | 1 | 38.00 |
| Brenda | 2 | 4 | 3 | 61.00 |
| Cyrus | 1 | 1 | 4 | 44.80 |
Let , , and be the prices, in dollars, of one loaf of sourdough, multigrain, and rye respectively.
Write down a system of three linear equations in , , and to represent this information.
Use technology to solve your system and find the price of each type of bread.
Daria buys sourdough loaves, multigrain loaves, and rye loaves.
Calculate the total amount Daria pays.