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PSLE Foundation Mathematics

<p>Scientists are studying the growth of a newly discovered biomass that entered Earth's atmosphere together with an asteroid. A sample of $100$ grams of the biomass was collected initially. It was found that on the next day, the sample grew to $110$ grams. The scientists observed that the rate of growth of biomass was proportional to the amount of biomass present. The amount of biomass at time $t$ days is denoted by $B$ grams.</p>
<ol class="abc-list"><li data-list-item-id="item-a-4" data-list-item-path="item-a-4"><div class="list-item-paragraph">Write down a differential equation relating $B$ and $t$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b-4" data-list-item-path="item-b-4"><div class="list-item-paragraph">Solve the differential equation obtained in part <strong>(a)</strong>, expressing $B$ in terms of $t$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c-4" data-list-item-path="item-c-4"><div class="list-item-paragraph">Sketch the graph of $B$ against $t$, and explain what would happen if the biomass was left on its own.<span class="inline-marks">[2]</span></div></li></ol>
<p>To limit the growth of the biomass, the scientist collected a second sample of $100$ grams of the biomass and introduced a nutrient depletion serum to the sample the moment it was collected. The rate of growth of the biomass for this second sample is modelled by the differential equation</p>
<p style="text-align: center">\[\frac{\mathrm{d}B}{\mathrm{d}t} = \frac{mB}{3 + 2t^2}\]</p>
<p>where&nbsp;$m$ is a constant. After $1$ day, the second sample grew to $101$ grams.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d-4" data-list-item-path="item-d-4"><div class="list-item-paragraph">For this second sample, find $B$ in terms of $t$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-e-4" data-list-item-path="item-e-4"><div class="list-item-paragraph">Determine the amount of the second biomass sample after a long time, leaving your answer to $5$ decimal places.<span class="inline-marks">[1]</span></div></li></ol>

Scientists are studying the growth of a newly discovered biomass that entered Earth's atmosphere together with an asteroid. A sample of $100$ grams of the biomass was collected initially. It was found that on the next day, the sample grew to $110$ grams. The scientists observed that the rate of growth of biomass was proportional to the amount of biomass present. The amount of biomass at time $t$ days is denoted by $B$ grams.

Write down a differential equation relating $B$ and $t$.

Sketch the graph of $B$ against $t$, and explain what would happen if the biomass was left on its own.

To limit the growth of the biomass, the scientist collected a second sample of $100$ grams of the biomass and introduced a nutrient depletion serum to the sample the moment it was collected. The rate of growth of the biomass for this second sample is modelled by the differential equation

\[\frac{\mathrm{d}B}{\mathrm{d}t} = \frac{mB}{3 + 2t^2}\]

where $m$ is a constant. After $1$ day, the second sample grew to $101$ grams.

For this second sample, find $B$ in terms of $t$.

Determine the amount of the second biomass sample after a long time, leaving your answer to $5$ decimal places.

Equilibrium Solutions and Long-Term Behaviour

Solving Differential Equations by Separation of Variables

Sketching Solution Curves of Differential Equations

Exponential Growth and Decay Models

2025 NYJC P1 Q12

Differential Equations

<p>Scientists are interested in the population of a particular species. They attempt to model the population $P$ at time $t$ days using a differential equation. Initially the population is observed to be $50$ and after $10$ days the population is $100$.</p>
<p>The first model the scientists use assumes that the rate of change of the population is proportional to the population.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down a differential equation for this model and solve it for $P$ in terms of $t$.</div><p style="text-align: right;">[5]</p></li></ol>
<p>To allow for constraints on population growth, the model is refined to</p>
<p style="text-align: center">$\frac{\mathrm{d}P}{\mathrm{d}t}=\lambda P\left( 500-P \right)$</p>
<p>where $\lambda $ is a constant.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Solve this differential equation to find $P$ in terms of $t$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Using the refined model, state the population of this species in the long term. Comment on how this value suggests the refined model is an improvement on the first model.</div><p style="text-align: right;">[2]</p></li></ol>

Scientists are interested in the population of a particular species. They attempt to model the population $P$ at time $t$ days using a differential equation. Initially the population is observed to be $50$ and after $10$ days the population is $100$.

The first model the scientists use assumes that the rate of change of the population is proportional to the population.

To allow for constraints on population growth, the model is refined to

$\frac{\mathrm{d}P}{\mathrm{d}t}=\lambda P\left( 500-P \right)$

Assumptions and Limitations of Differential Equation Models

Logistic Growth Models

N2022 P1 Q12

<p>A patient is injected with $150$ mg of a drug for treatment of a disease. Every $8$ hours, $15\%$ of the drug passes out of his bloodstream. To compensate the loss, a further $x$ mg dose is given every $8$ hours.</p>
<ol class="i-list"><li data-list-item-id="item-1760248150400-0epltp4ef" data-list-item-path="item-1760248150400-0epltp4ef"><div class="list-item-paragraph">Show that the amount of drug (mg) in the patient’s bloodstream immediately after the nth dose is given by $\left( 150-\frac{20}{3}x \right){{\left( 0.85 \right)}^{n-1}}+\frac{20}{3}x$.<span class="inline-marks">[4]</span></div></li></ol>
<p>The safety limit of the amount of drug in a person’s bloodstream is $165$ mg.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760248166593-pabd8waju" data-list-item-path="item-1760248166593-pabd8waju"><div class="list-item-paragraph">In the case where&nbsp;$x=25$, find the maximum number of doses that can be administered to the patient for the amount of drug to not exceed the safety limit.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another patient is initially injected with $165$mg of the drug. The amount of drug present at time $t$ hours in the bloodstream of the patient is $V$mg. The drug is removed naturally from his bloodstream at a rate $kV$mg/hour for some positive constant $k$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760248184363-9pwsv6491" data-list-item-path="item-1760248184363-9pwsv6491"><div class="list-item-paragraph">Write down and solve a differential equation, giving $V$ in terms of $t$ and $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760248186138-ah07mlf6k" data-list-item-path="item-1760248186138-ah07mlf6k"><div class="list-item-paragraph"><p>The drug is effective for treatment only if there is at least 85 mg in the body. Given that&nbsp;𝑉 =100 after 24 hours, find the time taken for the amount of drug to drop from 165 mg to the minimum effective level, giving your answer to the nearest hour.</p><span class="inline-marks">[2]</span></div></li></ol>

A patient is injected with $150$ mg of a drug for treatment of a disease. Every $8$ hours, $15\%$ of the drug passes out of his bloodstream. To compensate the loss, a further $x$ mg dose is given every $8$ hours.

The safety limit of the amount of drug in a person’s bloodstream is $165$ mg.

Another patient is initially injected with $165$mg of the drug. The amount of drug present at time $t$ hours in the bloodstream of the patient is $V$mg. The drug is removed naturally from his bloodstream at a rate $kV$mg/hour for some positive constant $k$.

Problems involving inequalities

Finding Explicit Terms of Linear Recurrences

2022 ASRJC J2 MYCT P1 Q11

Sequence and Series

<p>Glucose in the blood stream is reduced at a rate proportional to the amount of glucose present in the blood stream.</p>
<p>Let $G$ milligrams (mg) be the amount of glucose in $1$ decilitre (dL) of blood stream at time $t$ minutes and let $\lambda $ denote the positive constant of proportionality.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Write down a differential equation relating $G$, $\lambda $ and $t$. Solve this differential equation to find an expression of $G$ in terms of $\lambda $ and $t$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Through extensive research, doctors recommended that a healthy glucose level in the blood stream of an adult should be between $70$ mg/dL to $100$ mg/dL.</p>
<p>An adult patient, Neo, has $80$ mg/dL of glucose in his blood stream at time $t=0$ minutes. Due to a medical condition, he is not able to extract glucose from the food he eats. As he is not able to replenish the glucose in his blood stream, he is thus losing his glucose in the blood stream at a rate (in mg/dL per minute) corresponding to $\lambda =0.005$ per minute.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the approximate time it would take for Neo’s glucose level in the blood stream to fall below the healthy range if there is no intake of glucose.</div><p style="text-align: right;">[2]</p></li></ol>
<p>In order to maintain Neo’s glucose level in the blood stream in the healthy range, glucose is injected into his blood stream intravenously at a constant rate of $\mu $ mg/dL per minute.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Write down a differential equation relating $G$, $\mu $ and $t$ to model Neo’s situation. Solve this differential equation to find an expression of $G$ in terms of $\mu $ and $t$.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Explain whether it is recommended to keep injecting glucose at a rate of $0.7$ mg/dL per minute into Neo’s blood stream.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the maximum range of values of $\mu $ so that Neo’s glucose level in the blood stream will always be in the healthy range.</div><p style="text-align: right;">[1]</p></li></ol>

Glucose in the blood stream is reduced at a rate proportional to the amount of glucose present in the blood stream.

Let $G$ milligrams (mg) be the amount of glucose in $1$ decilitre (dL) of blood stream at time $t$ minutes and let $\lambda $ denote the positive constant of proportionality.

Through extensive research, doctors recommended that a healthy glucose level in the blood stream of an adult should be between $70$ mg/dL to $100$ mg/dL.

An adult patient, Neo, has $80$ mg/dL of glucose in his blood stream at time $t=0$ minutes. Due to a medical condition, he is not able to extract glucose from the food he eats. As he is not able to replenish the glucose in his blood stream, he is thus losing his glucose in the blood stream at a rate (in mg/dL per minute) corresponding to $\lambda =0.005$ per minute.

In order to maintain Neo’s glucose level in the blood stream in the healthy range, glucose is injected into his blood stream intravenously at a constant rate of $\mu $ mg/dL per minute.

Modelling with Multiple Rates of Change

2022 R1 P1 Q10

<p>Scientists are investigating the effect of a disease on the number of sheep on a small island. They discover that every year the death rate of the sheep is greater than the birth rate of the sheep. The difference every year between the death rate and the birth rate for the population of sheep on the island is $3%$. The number of sheep on the island is $P$ at a time $t$ years after the scientists begin observations.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Write down a differential equation relating $P$ and $t$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Solve this differential equation to find an expression for $P$ in terms of $t$. Explain what happens to the number of sheep if this situation continues over many years.</div><p style="text-align: right;">[4]</p></li></ol>
<p>The scientists import sheep at a constant uniform rate of $n$ sheep per year. (The difference every year between the death rate and the birth rate remains at $3%$.)</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Write down a differential equation to model the new situation.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Solve the differential equation to find an expression for $P$ in terms of $t$ and $n$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Given that the number of sheep settles down to $500$ after many years, find $n$.</div><p style="text-align: right;">[2]</p></li></ol>

Scientists are investigating the effect of a disease on the number of sheep on a small island. They discover that every year the death rate of the sheep is greater than the birth rate of the sheep. The difference every year between the death rate and the birth rate for the population of sheep on the island is $3%$. The number of sheep on the island is $P$ at a time $t$ years after the scientists begin observations.

The scientists import sheep at a constant uniform rate of $n$ sheep per year. (The difference every year between the death rate and the birth rate remains at $3%$.)

N2020 P1 Q10

<p>A hospital patient is receiving a certain drug through a drop at a constant rate of $50$ mg per hour. The rate of loss of the drug from the patient’s body is proportional to $x$, where&nbsp;$x$ (in mg) is the amount of drug in the patient at time $t$ (in hour). Form a differential equation connecting $x$ and $t$ and show that</p>
<p style="text-align: center">$x=\frac{1}{k}\left( 50-A{{e}^{-kt}} \right)$<span class="inline-marks">[3]</span></p>
<p>Where&nbsp;$A$ and $k$ are constants.</p>
<p>Given that&nbsp;$k=\frac{1}{20}$.</p>
<ol class="i-list"><li data-list-item-id="item-1759600785251-vx7a0df9k" data-list-item-path="item-1759600785251-vx7a0df9k"><div class="list-item-paragraph"><p>Find the time needed for the drug in the patient to reach $200$ mg, if initially there is no trace of this drug in the patient’s body.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759600785859-cq4ina19g" data-list-item-path="item-1759600785859-cq4ina19g"><div class="list-item-paragraph"><p>When there is $80$ mg of the drug in the patient’s body, the drip is disconnected. Assuming that the rate of loss remains the same, find the time taken or the amount of drug in the patient to fall from $80$ mg to $20$ mg.</p><span class="inline-marks">[3]</span></div></li></ol>

A hospital patient is receiving a certain drug through a drop at a constant rate of $50$ mg per hour. The rate of loss of the drug from the patient’s body is proportional to $x$, where $x$ (in mg) is the amount of drug in the patient at time $t$ (in hour). Form a differential equation connecting $x$ and $t$ and show that

$x=\frac{1}{k}\left( 50-A{{e}^{-kt}} \right)$

HCI P1 Q7

<p>A tank has a capacity of 100 litres. Initially, the tank contains 10 litres of water thoroughly mixed with 300 grams of salt. Salt water with a concentration of 5 g/litre is poured into the tank at a constant rate of 2 litres per minute, while the mixture flows out at a constant rate of 1 litre per minute. </p>
<p>Let&nbsp;$S$ denote the amount of dissolved salt in the tank (in grams) at time $t$ minutes after salt water is poured into the tank.</p>
<ol class="i-list"><li data-list-item-id="item-1759145628578-sxiuu5hpw" data-list-item-path="item-1759145628578-sxiuu5hpw"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}S}{\mathrm{d}t}=10-\frac{S}{t+10}$, stating your assumption(s) clearly.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759145629261-n2k09ecec" data-list-item-path="item-1759145629261-n2k09ecec"><div class="list-item-paragraph">By substituting $Q=\left( t+10 \right)S$, solve the differential equation in (i), and find $S$ in terms of $t$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759145630889-5xy7sgk7c" data-list-item-path="item-1759145630889-5xy7sgk7c"><div class="list-item-paragraph">Hence, find the concentration of salt in the tank at the point just before it overflows.<span class="inline-marks">[2]</span></div></li></ol>
<p>Once the volume of salt solution in the tank reaches 100 litres, the pouring stops, and the tank is allowed to drain off. The salt solution drains from the tank at a rate proportional to the volume of solution in the tank. Let&nbsp;$V$ denote the volume of solution in the tank at time $T$ after the tank starts to drain off.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1759145654830-s1565fpw8" data-list-item-path="item-1759145654830-s1565fpw8"><div class="list-item-paragraph">If the tanks takes 10 minutes to drain half of its contents, find $V$ in terms of  $T$.<span class="inline-marks">[4]</span></div></li></ol>

A tank has a capacity of 100 litres. Initially, the tank contains 10 litres of water thoroughly mixed with 300 grams of salt. Salt water with a concentration of 5 g/litre is poured into the tank at a constant rate of 2 litres per minute, while the mixture flows out at a constant rate of 1 litre per minute. 

Let $S$ denote the amount of dissolved salt in the tank (in grams) at time $t$ minutes after salt water is poured into the tank.

Show that $\frac{\mathrm{d}S}{\mathrm{d}t}=10-\frac{S}{t+10}$, stating your assumption(s) clearly.

By substituting $Q=\left( t+10 \right)S$, solve the differential equation in (i), and find $S$ in terms of $t$.

Hence, find the concentration of salt in the tank at the point just before it overflows.

Once the volume of salt solution in the tank reaches 100 litres, the pouring stops, and the tank is allowed to drain off. The salt solution drains from the tank at a rate proportional to the volume of solution in the tank. Let $V$ denote the volume of solution in the tank at time $T$ after the tank starts to drain off.

If the tanks takes 10 minutes to drain half of its contents, find $V$ in terms of  $T$.

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