A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

The number of cells in an experiment can be modelled by a function $N(t)=50000\left(1.6^{\frac{t}{2}}\right)$,

where $t$ is the number of hours that have passed from the start of the experiment.

Find the number of cells present at the start of the experiment.

Find the time taken for the number of cells to be ten times the initial number of cells. Give your answer to the nearest hour.

It is given that the number of cells at time $t=t_2$ is double the number of cells at the time $t=t_1$. Show that the difference between the two timings $(t_2-t_1)$ is approximately 2.95 hours, corrected to 3 significant figures.

Applications of Exponential and Logarithmic Functions

2025 NASS PRELIMS P1 Q9

Exponential and Logarithmic Functions

A drink cools according to T=22+58e-0.18t, where T is in degrees Celsius and t is in minutes. Find when T=35.

A drink cools according to $T=22+58e^{-0.18t}$, where $T$ is in degrees Celsius and $t$ is in minutes. Find when $T=35$.

Cooling model

<p>Let&nbsp;$f(x) = A\mathrm{e}^{kx} + 3$. Part of the graph of $f$ is shown below.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="262" style="width:300px;height:262px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The $y$-intercept is at $ (0, 13) $.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$A = 10$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$f(15) = 3.49$ (correct to $3$ significant figures), find the value of $k$.<span class="inline-marks">[3]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Using your value of $k$, find $f'(x)$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Hence, explain why $f$ is a decreasing function.</div></li><li><div class="list-item-paragraph">Write down the equation of the horizontal asymptote of the graph $f$.</div></li></ol></li></ol>
<p>Let&nbsp;$g(x) = -x^2 + 12x - 24$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the area enclosed by the graphs of $f$ and $g$.<span class="inline-marks">[6]</span></div></li></ol>

Let $f(x) = A\mathrm{e}^{kx} + 3$. Part of the graph of $f$ is shown below.

Increasing and Decreasing Functions

Asymptotes and Discontinuities

Derivatives of Exponential Functions

Area between Two Curves

Techniques of Differentiation

Graphing Techniques

Integration

Applications of Differentiation

Derivatives of Exponential, Logarithmic and Trigonometric Functions

<p>The decay of a certain radioactive isotope can be modelled by the exponential equation $N=N_0e^{-at}$ after t weeks, where&nbsp;$N$ represents the amount of radioactive isotope, $N_0$ and $a$ are constants. A sample of this radioactive isotope has a mass of $100.9 g$ initially.</p>
<ol class="abc-list"><li data-list-item-id="item-1773466642173-v29a5921z" data-list-item-path="item-1773466642173-v29a5921z"><div class="list-item-paragraph">After $2$ weeks, it is found that the amount of this sample left is $84.6 g$.<br>Calculate the value of $a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1773466642985-kiyhgmmaf" data-list-item-path="item-1773466642985-kiyhgmmaf"><div class="list-item-paragraph">What percentage of this sample has decayed after $5$ weeks?<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1773466646814-6i1uq7q7s" data-list-item-path="item-1773466646814-6i1uq7q7s"><div class="list-item-paragraph">Find the number of weeks when the amount of radioactive isotope decayed first exceeds $60 g$. Give your answer to the nearest week.<span class="inline-marks">[3]</span></div></li></ol>

The decay of a certain radioactive isotope can be modelled by the exponential equation $N=N_0e^{-at}$ after $t$ weeks, where $N$ represents the amount of radioactive isotope, $N_0$ and $a$ are constants. A sample of this radioactive isotope has a mass of 100.9 g initially. After 2 weeks, it is found that the amount of this sample left is 84.6 g.

What percentage of this sample has decayed after 5 weeks?

Find the number of weeks when the amount of radioactive isotope decayed first exceeds 60 g. Give your answer correct to the nearest week.

2025 MSHS PRELIMS P2 Q3

<p>In $2020$, there were an estimated $7$ million insects of a particular species in a certain country. In $2021$, the numbers were estimated to have fallen to $5$ million. Scientists believe that the number of these insects, $N$ million, can be modelled by the formula $N = 3 + a\mathrm{e}^{-kt}$, where&nbsp;$t$ is the time in years after $2020$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Find the exact values of $a$ and of $k$.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">What is the approximate size of the population for large values of $t$?</span><span class="inline-marks">[1]</span></div></li></ol>

In 2020, there were an estimated 7 million insects of a particular species in a certain country. In 2021, the numbers were estimated to have fallen to 5 million. Scientists believe that the number of these insects, $N$ million, can be modelled by the formula $N = 3 + ae^{-kt}$, where $t$ is the time in years after 2020.

Find the exact values of $a$ and of $k$.

What is the approximate size of the population for large values of $t$?

(A Math) 2025 CCHY PRELIMS P2 Q1

<p>In an experiment, $5$ million cells were stored in a container. It was observed that&nbsp;$40$% of the cells are dying every minute. The number of cells remaining, $N$, after $t$ minutes, is given by $N = Ae^{kt}$, where&nbsp;$A$ and $k$ are constants.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Explain why $k = \ln \frac{3}{5}$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Find the time taken when the number of cells decreases to $2000$.</span><span class="inline-marks">[3]</span></div></li></ol>

In an experiment, 5 million cells were stored in a container. It was observed that 40% of the cells are dying every minute. The number of cells remaining, $N$, after $t$ minutes, is given by $N = Ae^{kt}$, where $A$ and $k$ are constants.

Find the time taken when the number of cells decreases to 2000.

2025 SCGS PRELIMS P2 Q1

<p>The temperature, $T^\circ\mathrm{C}$, of glass being heated in a kiln is given by the formula $T = 2030 - 2000e^{-kt}$, where&nbsp;$t$ is the time in hours since the glass was placed in the kiln. The temperature of the glass after $90$ minutes is $1000^\circ\mathrm{C}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the temperature of the glass before it was heated.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The glass can be moulded into different shapes when its temperature reaches at least $1300^\circ\mathrm{C}$. What is the earliest time that the glass can be removed from the kiln and moulded?<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The instruction manual of the kiln gives the range of the temperature of the kiln as&nbsp;$T &lt; y^\circ\mathrm{C}$. State the value of $y$.<span class="inline-marks">[1]</span></div></li></ol>

The temperature, $T^\circ\mathrm{C}$, of glass being heated in a kiln is given by the formula $T = 2030 - 2000e^{-kt}$, where $t$ is the time in hours since the glass was placed in the kiln. The temperature of the glass after $90$ minutes is $1000^\circ\mathrm{C}$.

Find the temperature of the glass before it was heated.

The glass can be moulded into different shapes when its temperature reaches at least $1300^\circ\mathrm{C}$. What is the earliest time that the glass can be removed from the kiln and moulded?

The instruction manual of the kiln gives the range of the temperature of the kiln as $T < y^\circ\mathrm{C}$. State the value of $y$.

IP3 Class Test (May 2025) Q9 (2023 ACSBR AM Prelim P1 Q3)

<p>Dan invested ${\$\,x}$ in a bank at the start of 2023. The amount of money, ${\$\,P}$, at the end of ${t}$ years is denoted by ${P = 6800(1.015)^t}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">State the value of $x$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the year in which the amount of money in the bank first reach $\$8000$.<span class="inline-marks">[2]</span></div></li></ol>

Dan invested ${\$\,x}$ in a bank at the start of 2023. The amount of money, ${\$\,P}$, at the end of ${t}$ years is denoted by ${P = 6800(1.015)^t}$.

2023 Ahmad Ibrahim Sec AM S3 EOY Q1

<p>An object is heated to a temperature of $T{}^\circ\mathrm{C}$ before it is allowed to cool. The temperature of the object when it is cooling for $t$ minutes can be modelled by $T = 23 + 15e^{-\frac{3}{5}t}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $t$ when $T = 30$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why the temperature of the object will never go below $23{}^\circ\mathrm{C}$.</span><span class="inline-marks">[1]</span></div></li></ol>

An object is heated to a temperature of $T{}^\circ\mathrm{C}$ before it is allowed to cool. The temperature of the object when it is cooling for $t$ minutes can be modelled by $T = 23 + 15e^{-\frac{3}{5}t}$.

Explain why the temperature of the object will never go below $23{}^\circ\mathrm{C}$.

2023 ACSBR AM S3 EOY Q1

<p>In an experiment, $200$ grams of a particular radioactive substance is stored in a container. The amount of substance present after $t$ days is given by $N = 200e^{kt}$, where $k$ is a constant. It is found that the substance is halved after $20$ days.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the amount of substance remaining after $1$ week.<span class="inline-marks">[4]</span></div></li></ol>
<p>The radioactive substance is considered to be safe to be discarded using the proper procedures when the amount is reduced to $15$ grams.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">John claimed that it is safe to discard the radioactive substance after $74$ days. Is his claim justified? Show the calculations and the reasons on which you based your answer.<span class="inline-marks">[2]</span></div></li></ol>

In an experiment, $200$ grams of a particular radioactive substance is stored in a container. The amount of substance present after $t$ days is given by $N = 200e^{kt}$, where $k$ is a constant. It is found that the substance is halved after $20$ days.

The radioactive substance is considered to be safe to be discarded using the proper procedures when the amount is reduced to $15$ grams.

2021 MSS AM S4 Q6

<p>The value of an antique is estimated to increase by $k$% per year. Given that its value at the beginning of $2017$ was $\$20\ 000$. The value $V$, after $t$ years, can be modelled by $V = A(1.11)^t$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the value of $A$ and of $k$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Calculate the year when its estimated value will first reach $\$190\ 000$.<span class="inline-marks">[2]</span></div></li></ol>

The value of an antique is estimated to increase by $k$% per year. Given that its value at the beginning of $2017$ was $\$20\ 000$. The value $V$, after $t$ years, can be modelled by $V = A(1.11)^t$.

2022 SGSS AM S4 P1 Q2 (b)

<p>Clarification: $P$ denotes the fraction remaining, and $100P$ is the percentage remaining. Use $P=0.50$ for 50% and $P=0.34$ for 34%.</p>
<p>It was found that the fraction of the initial carbon-14, $P$, contained in the bones of an animal $n$ years after it has died is given by $P = 2^{-kn}$, where&nbsp;$k$ is a positive constant. The percentage of carbon-14 contained in the bones after the animal has been dead for 7200 years was 50%. How long, to the nearest year, was the animal dead if the percentage of carbon-14 found in the bones was only 34%?<span class="inline-marks">[4]</span></p>

Clarification: $P$ denotes the fraction remaining, and $100P$ is the percentage remaining. Use $P=0.50$ for 50% and $P=0.34$ for 34%.

It was found that the fraction of the initial carbon-14, $P$, contained in the bones of an animal $n$ years after it has died is given by $P = 2^{-kn}$, where $k$ is a positive constant. The percentage of carbon-14 contained in the bones after the animal has been dead for 7200 years was 50%. How long, to the nearest year, was the animal dead if the percentage of carbon-14 found in the bones was only 34%?

2022 FMSS AM S4 P1 Q2

<p>The value, $V$, of a piece of rare jewel at the beginning of $2022$ was $\$32\ 000$. This value increased continuously such that, after a period of $15$ years, the value of the jewel increased exponentially to $\$160\ 000$.<br><br></p>
<p>Given that $V=A(1.25)^{kt}$, where $A$ and $k$ are constants and $t$ is the time in years from the beginning of $2022$, find<br><br></p>
<ol class="abc-list"><li><div class="list-item-paragraph">the value of $A$ and $k$,<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">the year in which the value of the stone first reached $\$200\ 000$.<span class="inline-marks">[3]</span></div></li></ol>

The value, $V$, of a piece of rare jewel at the beginning of $2022$ was $\$32\ 000$. This value increased continuously such that, after a period of $15$ years, the value of the jewel increased exponentially to $\$160\ 000$.



Given that $V=A(1.25)^{kt}$, where $A$ and $k$ are constants and $t$ is the time in years from the beginning of $2022$, find



2022 CHS AM S4 P1 Q13

<p>Carbon-14 has a half-life of 5730 years, which means that it takes 5730 years for the original mass of Carbon-14 to be reduced by 50%. To determine the age of a plant or animal fossil, scientists determine the amount of Carbon-14 in the specimen as the Carbon-14 undergoes radioactive decay.</p>
<p>The amount of Carbon-14 in a piece of fossilised bone is given by $M = M_0e^{-kt}$, where $k$ is a constant, $M$ is the mass of Carbon-14 in the specimen, $M_0$ is the initial mass of Carbon-14 and $t$ is measured in years.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Determine the value of $k$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine the percentage of Carbon-14 that has decayed in the specimen if the fossil is estimated to be about 900 years old.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Express the rate of change of percentage of Carbon-14 in terms of $t$.<span class="inline-marks">[2]</span></div></li></ol>

Carbon-14 has a half-life of 5730 years, which means that it takes 5730 years for the original mass of Carbon-14 to be reduced by 50%. To determine the age of a plant or animal fossil, scientists determine the amount of Carbon-14 in the specimen as the Carbon-14 undergoes radioactive decay.

The amount of Carbon-14 in a piece of fossilised bone is given by $M = M_0e^{-kt}$, where $k$ is a constant, $M$ is the mass of Carbon-14 in the specimen, $M_0$ is the initial mass of Carbon-14 and $t$ is measured in years.

Derivative as an Instantaneous Rate of Change

2022 AHS AM S4 P1 Q4

Differentiation

<p>At the beginning of $1990$, the population, $P$, of a certain species of insects after a period of $n$ years is given by $P = 40000e^{-0.05n}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the population of the insects</div><ol class="i-list"><li><div class="list-item-paragraph">at the beginning of $1990$.</div></li><li><div class="list-item-paragraph">at the beginning of $2015$, correct to $3$ significant figures.</div></li></ol></li><li><div class="list-item-paragraph">Sketch the graph of $P$ against $n$.</div></li></ol>

At the beginning of $1990$, the population, $P$, of a certain species of insects after a period of $n$ years is given by $P = 40000e^{-0.05n}$.

Sketching Exponential Functions and Solving Equations Graphically

<p>The stated population 1226 is rounded to the nearest animal. Treat the given time of the maximum as exact and the supplied count as approximate.</p>
<p>The population of a gold-mining town, $N$, initially grew over time, reaching a maximum of $1226$ after $3\frac{1}{3}$ years. However, as the gold supply was depleted, the population began to decline. </p>
<p>The population at any time $t$ (in years) is modelled by the equation:$N=1000t{{\mathrm{e}}^{-at}}$, where&nbsp;$a$ is a constant. The diagram illustrates the behaviour of the population over time.</p>
<ol class="i-list"><li data-list-item-id="item-1766126722510-eufrg6ije" data-list-item-path="item-1766126722510-eufrg6ije"><div class="list-item-paragraph">Calculate the population of the town at $t=0$.</div></li><li data-list-item-id="item-1766126723784-5wxiutucb" data-list-item-path="item-1766126723784-5wxiutucb"><div class="list-item-paragraph">Find the population of the town when&nbsp;$t=2$.</div></li><li data-list-item-id="item-1766126724295-os4f14rbc" data-list-item-path="item-1766126724295-os4f14rbc"><div class="list-item-paragraph">Describe the behaviour of the population as&nbsp;$t$ increases indefinitely.</div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1766126780025-6kwsnf4gzgp.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjYxMjY3ODAwMjUtNmt3c25mNGd6Z3AucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcxNzMwNSwiZXhwIjoxNzkyNzE3NjA1fQ.ExjCynvXjDp_LyHsT37ZwyRYrPG7Ev_Nh6NJe1_YS48" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The stated population 1226 is rounded to the nearest animal. Treat the given time of the maximum as exact and the supplied count as approximate.

The population of a gold-mining town, $N$, initially grew over time, reaching a maximum of $1226$ after $3\frac{1}{3}$ years. However, as the gold supply was depleted, the population began to decline. 

The population at any time $t$ (in years) is modelled by the equation:$N=1000t{{\mathrm{e}}^{-at}}$, where $a$ is a constant. The diagram illustrates the behaviour of the population over time.

<p>The population model is an approximation: at $t=0$, it gives&nbsp;$1640/2900\approx0.566$, which rounds to one animal. Keep the supplied model constants.</p>
<p>On a farm with $1640$ animals, one animal was infected with a contagious disease. The spread of the disease among the animals is modelled by the equation $N=\frac{1640}{1+2899{{\mathrm{e}}^{-0.65t}}}$, where&nbsp;$N$ is the total number of infected animals after $t$ days. The farm management plans to quarantine the animals for a week if more than $40\%$ of the animals are infected. Estimate, correct to the nearest whole number:</p>
<ol class="abc-list"><li data-list-item-id="item-1766126608938-ixwryhdhh" data-list-item-path="item-1766126608938-ixwryhdhh"><div class="list-item-paragraph">the number of animals infected after $3$ days,</div></li><li data-list-item-id="item-1766126610340-bb3bc0k5z" data-list-item-path="item-1766126610340-bb3bc0k5z"><div class="list-item-paragraph">the number of days it will take for more than $40\%$ of the animals to be infected.<br><br></div></li></ol>

The population model is an approximation: at $t=0$, it gives $1640/2900\approx0.566$, which rounds to one animal. Keep the supplied model constants.

On a farm with $1640$ animals, one animal was infected with a contagious disease. The spread of the disease among the animals is modelled by the equation $N=\frac{1640}{1+2899{{\mathrm{e}}^{-0.65t}}}$, where $N$ is the total number of infected animals after $t$ days. The farm management plans to quarantine the animals for a week if more than $40\%$ of the animals are infected. Estimate, correct to the nearest whole number:

<p>Loudness is a subjective measure of how strong a sound feels, as it can vary between individuals. A more objective way to measure sound strength is the sound intensity level, $L$, given by the formula $L=10\lg \frac{I}{{{I}_{0}}}$,</p>
<p>where&nbsp;$I$ represents the sound intensity in W/m$^{2}$, and ${{I}_{0}}$ is the minimum sound intensity detectable by the human ear (also measured in W/m$^{2}$). The unit of $L$ is decibels (dB).</p>
<ol class="i-list"><li><div class="list-item-paragraph">What is the lowest value of $L$ (in dB) that the human ear can detect?</div></li><li><div class="list-item-paragraph">If&nbsp;$L=90$ dB when&nbsp;$I={{10}^{-3}}$ W/m$^{2}$, find the minimum sound intensity detectable by the human ear.</div></li><li><div class="list-item-paragraph">Find the sound intensity level for a refrigerator with a sound intensity of $4\times {{10}^{-8}}$ W/m$^{2}$.</div></li><li><div class="list-item-paragraph">Given that the sound intensity level of a machine is $120$ dB, find the corresponding sound intensity $I$.</div></li></ol>

Loudness is a subjective measure of how strong a sound feels, as it can vary between individuals. A more objective way to measure sound strength is the sound intensity level, $L$, given by the formula $L=10\lg \frac{I}{{{I}_{0}}}$,

where $I$ represents the sound intensity in W/m$^{2}$, and ${{I}_{0}}$ is the minimum sound intensity detectable by the human ear (also measured in W/m$^{2}$). The unit of $L$ is decibels (dB).

<p>In a garden with $10\,000$ plants, a disease begins spreading after one plant is infected by virus. Assuming all plants are susceptible to the disease, the number of plants $N$ infected $t$ days after the disease begins spreading is modelled by the equation $N=\frac{10\,000}{1+9999{{\mathrm{e}}^{-0.2t}}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the expected number of plants infected after $7$ days.</div></li><li><div class="list-item-paragraph">Estimate the number of days required for one-quarter of the plants in the garden to become infected.</div></li></ol>

In a garden with $10\,000$ plants, a disease begins spreading after one plant is infected by virus. Assuming all plants are susceptible to the disease, the number of plants $N$ infected $t$ days after the disease begins spreading is modelled by the equation $N=\frac{10\,000}{1+9999{{\mathrm{e}}^{-0.2t}}}$.

<p>A hot liquid was left to cool according to the equation $T={{\mathrm{e}}^{4.5-0.1t}}$, where&nbsp;$T$ is the temperature of the liquid after $t$ minutes. After the liquid is cooled to $a{}^\circ $C for&nbsp;$b$ minutes, it is then immediately heated again according to the equation $T={{\mathrm{e}}^{0.5t-2.7}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the initial temperature of the liquid when it was left to cool.</div></li><li><div class="list-item-paragraph">Find the values of $a$ and $b$.</div></li></ol>

A hot liquid was left to cool according to the equation $T={{\mathrm{e}}^{4.5-0.1t}}$, where $T$ is the temperature of the liquid after $t$ minutes. After the liquid is cooled to $a{}^\circ $C for $b$ minutes, it is then immediately heated again according to the equation $T={{\mathrm{e}}^{0.5t-2.7}}$.

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