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PSLE Foundation Mathematics

<p>Rowan is investigating the spread of an infection through a population of rabbits.</p>
<p>He models $N$, the number (in thousands) of infected rabbits in the population, at time $t$ days.</p>
<p>Rowan models the spread of the infection by the differential equation $\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)$, where&nbsp;$t\geq0$ and $k\in\mathbb{R}^+$.</p>
<p>Initially, $1500$ rabbits are infected.</p>
<ol class="abc-list"><li data-list-item-id="p1-tza-q12-item01" data-list-item-path="p1-tza-q12-item01"><div class="list-item-paragraph">Express $\frac1{-3+4x-x^2}$ in the form $\frac{A}{x-1}+\frac{B}{3-x}$, where&nbsp;$A,B\in\mathbb{R}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="p1-tza-q12-item02" data-list-item-path="p1-tza-q12-item02"><div class="list-item-paragraph">Hence, show that&nbsp;$\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p$, where&nbsp;$p\in\mathbb{R}^+$ is a constant to be determined.<span class="inline-marks">[8]</span></div></li><li data-list-item-id="p1-tza-q12-item03" data-list-item-path="p1-tza-q12-item03"><div class="list-item-paragraph">Find an expression for&nbsp;$N$ in the form $N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}$, where&nbsp;$a,b,c\in\mathbb{Z}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="p1-tza-q12-item04" data-list-item-path="p1-tza-q12-item04"><div class="list-item-paragraph">According to Rowan’s model, the number of infected rabbits approaches a limit, $L$, over the long term.</div><div class="list-item-paragraph">Find the value of $L$.<span class="inline-marks">[2]</span></div></li></ol>

Rowan is investigating the spread of an infection through a population of rabbits.

He models $N$, the number (in thousands) of infected rabbits in the population, at time $t$ days.

Rowan models the spread of the infection by the differential equation $\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)$, where $t\geq0$ and $k\in\mathbb{R}^+$.

Initially, $1500$ rabbits are infected.

Express $\frac1{-3+4x-x^2}$ in the form $\frac{A}{x-1}+\frac{B}{3-x}$, where $A,B\in\mathbb{R}$.

Hence, show that $\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p$, where $p\in\mathbb{R}^+$ is a constant to be determined.

Find an expression for $N$ in the form $N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}$, where $a,b,c\in\mathbb{Z}$.

Solving Differential Equations by Separation of Variables

Integration Using Partial Fractions

Partial Fractions

IB AA HL May 2026 P1 TZA Q12

Differential Equations

Techniques of Integration

Polynomials, Cubic Equations and Partial Fractions

<ol class="abc-list"><li data-list-item-id="jvss-p1-q7-q-0-content-items-0" data-list-item-path="jvss-p1-q7-q-0-content-items-0"><div class="list-item-paragraph">Express $\frac{5x^2+3x+7}{(x+1)(x-2)^2}$ in partial fractions.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="jvss-p1-q7-q-0-content-items-1" data-list-item-path="jvss-p1-q7-q-0-content-items-1"><div class="list-item-paragraph">Hence find $\int \frac{5x^2+3x+7}{(x+1)(x-2)^2}\, dx$.<span class="inline-marks">[3]</span></div></li></ol>

Express $\frac{5x^2+3x+7}{(x+1)(x-2)^2}$ in partial fractions.

Hence find $\int \frac{5x^2+3x+7}{(x+1)(x-2)^2}\, dx$.

2025 JVSS PRELIMS P1 Q7

Integration

<p>The spread of an infection through a population of rabbits is modelled by a differential equation.</p>
<p>At time $t$ days, $N$ thousand rabbits are infected.</p>
<p>The model is $\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)$, where&nbsp;$t\geq0$ and $k$ is a positive constant.</p>
<p>Initially, $1500$ rabbits are infected.</p>
<ol class="abc-list"><li data-list-item-id="h2-rabbits-6" data-list-item-path="h2-rabbits-6"><div class="list-item-paragraph">Express $\frac1{-3+4x-x^2}$ in the form $\frac{A}{x-1}+\frac{B}{3-x}$, where&nbsp;$A,B\in\mathbb{R}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-rabbits-8" data-list-item-path="h2-rabbits-8"><div class="list-item-paragraph">By separating the variables and using part (a), show that&nbsp;$\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p$, where&nbsp;$p$ is a positive constant to be determined. You may assume that&nbsp;$1&lt;N&lt;3$.<span class="inline-marks">[8]</span></div></li><li data-list-item-id="h2-rabbits-10" data-list-item-path="h2-rabbits-10"><div class="list-item-paragraph">Hence express $N$ explicitly in terms of $t$ and $k$, in the form $N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}$, where&nbsp;$a$, $b$ and $c$ are integers to be determined.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="h2-rabbits-12" data-list-item-path="h2-rabbits-12"><div class="list-item-paragraph">According to the model, the number of infected rabbits approaches a limiting value, $L$, as&nbsp;$t$ becomes large.</div><div class="list-item-paragraph">Find the value of $L$.<span class="inline-marks">[2]</span></div></li></ol>

The spread of an infection through a population of rabbits is modelled by a differential equation.

At time $t$ days, $N$ thousand rabbits are infected.

The model is $\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)$, where $t\geq0$ and $k$ is a positive constant.

Equilibrium Solutions and Long-Term Behaviour

IB AA HL May 2026 P1 TZA Q12 (H2 adaptation)

<ol class="i-list"><li data-list-item-id="amint-q02-question-1" data-list-item-path="amint-q02-question-1"><div class="list-item-paragraph">Express $\dfrac{3x^2+3x+3}{(x-1)(x+2)^2}$ in partial fractions.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q02-question-2" data-list-item-path="amint-q02-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_2^4 \dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Express $\dfrac{3x^2+3x+3}{(x-1)(x+2)^2}$ in partial fractions.

Hence evaluate the exact value of $\displaystyle\int_2^4 \dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\,\mathrm{d}x$.

Definite Integral

Repeated-factor logarithmic integral

<ol class="i-list"><li data-list-item-id="amint-q01-question-1" data-list-item-path="amint-q01-question-1"><div class="list-item-paragraph">Express $\dfrac{3x^2+3x+6}{x(x+1)(x+3)}$ in partial fractions.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q01-question-2" data-list-item-path="amint-q01-question-2"><div class="list-item-paragraph">Hence evaluate $\displaystyle\int_1^2 \dfrac{3x^2+3x+6}{x(x+1)(x+3)}\,\mathrm{d}x$, giving your answer as a single logarithm.<span class="inline-marks">[4]</span></div></li></ol>

Express $\dfrac{3x^2+3x+6}{x(x+1)(x+3)}$ in partial fractions.

Hence evaluate $\displaystyle\int_1^2 \dfrac{3x^2+3x+6}{x(x+1)(x+3)}\,\mathrm{d}x$, giving your answer as a single logarithm.

Three-factor logarithmic integral

<ol class="i-list"><li data-list-item-id="amint-q04-question-1" data-list-item-path="amint-q04-question-1"><div class="list-item-paragraph">Express $\dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}$ in partial fractions.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q04-question-2" data-list-item-path="amint-q04-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_1^2 \dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Express $\dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}$ in partial fractions.

Hence evaluate the exact value of $\displaystyle\int_1^2 \dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}\,\mathrm{d}x$.

Repeated linear factors and an exact integral

<ol class="i-list"><li data-list-item-id="amint-q03-question-1" data-list-item-path="amint-q03-question-1"><div class="list-item-paragraph">Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where&nbsp;$a$, $b$, $c$ and $d$ are integers.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q03-question-2" data-list-item-path="amint-q03-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where $a$, $b$, $c$ and $d$ are integers.

Hence evaluate the exact value of $\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x$.

Integrals Involving Long Division

Polynomial Division and Remainders

Improper rational definite integral

<p>Express $\dfrac{4x^2-11x+12}{(x+1)(x-2)^2}$ in partial fractions.</p>

Express $\dfrac{4x^2-11x+12}{(x+1)(x-2)^2}$ in partial fractions.

Partial fractions with a repeated linear factor

<p>Express $\frac{x + 7}{(x^2 - 9)(x - 3)}$ as the sum of $3$ partial fractions.<span class="inline-marks">[6]</span></p>

Express $\frac{x + 7}{(x^2 - 9)(x - 3)}$ as the sum of $3$ partial fractions.

2024 GESS PRELIMS P1 Q3

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Express $\frac{8x + 13}{(1 + 2x)(2 + x)^2}$ in partial fractions.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Hence, evaluate $\int_{1}^{2} \frac{8x + 13}{(1 + 2x)(2 + x)^2} \,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

2024 FMSS PRELIMS P1 Q6

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Explain why $\frac{2x^3 - 4x^2 + 13x - 10}{(x^2 + 4)(x - 2)}$ is considered an improper fraction.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Express $\frac{2x^3 - 4x^2 + 13x - 10}{(x^2 + 4)(x - 2)}$ in the form of $A + \frac{Bx + C}{(x^2 + 4)(x - 2)}$ and hence, express $\frac{2x^3 - 4x^2 + 13x - 10}{(x^2 + 4)(x - 2)}$ in partial fractions.<span class="inline-marks">[6]</span></div></li></ol>

2024 CCHY P1 Q5

<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Express $\frac{1}{x^3 + x}$ as the sum of its partial fractions.</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">Hence find $\int \frac{1}{x^3 + x} \mathrm{d}x$</span></div></li></ol>

Express $\frac{1}{x^3 + x}$ as the sum of its partial fractions.

Hence find $\int \frac{1}{x^3 + x} \mathrm{d}x$

Logarithmic Integrals of $\frac{f'(x)}{f(x)}$

<p>Express $\frac{15x^2-13x+18}{(x-2)(3x^2+1)}$ in partial fractions.<span class="inline-marks">[6]</span></p>

Express $\frac{15x^2-13x+18}{(x-2)(3x^2+1)}$ in partial fractions.

2025 MSHS PRELIMS P1 Q4

<ol class="i-list"><li data-list-item-id="cchy-p1-q11-q-0-content-items-0" data-list-item-path="cchy-p1-q11-q-0-content-items-0"><div class="list-item-paragraph">Express $\frac{11x+12}{x^2(x+4)}$ in partial fractions.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="cchy-p1-q11-q-0-content-items-1" data-list-item-path="cchy-p1-q11-q-0-content-items-1"><div class="list-item-paragraph">Hence find $\int \frac{11x+12}{x^2(x+4)}\, dx$.<span class="inline-marks">[2]</span></div></li></ol>

Express $\frac{11x+12}{x^2(x+4)}$ in partial fractions.

Hence find $\int \frac{11x+12}{x^2(x+4)}\, dx$.

(A Math) 2025 CCHY PRELIMS P1 Q11

<p>Express $\frac{2x^3 - 2x^2 + 7x - 12}{x^3 + 4x}$ in partial fractions.<span class="inline-marks">[6]</span></p>

Express $\dfrac{2x^3-2x^2+7x-12}{x^3+4x}$ in partial fractions.

2025 AHS PRELIMS P1 Q4

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $\frac{x^2 + 3x + 2}{x^2(2x + 1)}$ in partial fractions.</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence integrate $\frac{x^2 + 3x + 2}{x^2(2x + 1)}$ with respect to $x$.</span><span class="inline-marks">[3]</span></div></li></ol>

Express $\frac{x^2+3x+2}{x^2(2x+1)}$ in partial fractions.

Hence integrate $\frac{x^2+3x+2}{x^2(2x+1)}$ with respect to $x$.

2025 DSS PRELIMS P1 Q4

<p>Express $\frac{2x^3 - 5}{x^2(x - 1)}$ in partial fractions.<span class="inline-marks">[6]</span></p>

Express $\dfrac{2x^3-5}{x^2(x-1)}$ in partial fractions.

2025 CGSS PRELIMS P1 Q3

<p>Express $\frac{3{{x}^{3}}+10{{x}^{2}}+x+1}{{{x}^{3}}+3{{x}^{2}}}$ in partial fractions.<span class="inline-marks">[6]</span></p>

Express $\frac{3{{x}^{3}}+10{{x}^{2}}+x+1}{{{x}^{3}}+3{{x}^{2}}}$ in partial fractions.

IP3 Class Test (May 2025) Q7 (2023 Jurong Sec AM Prelim P1 Q2)

<p>Express $\frac{3x^{2} + 6x + 1}{(x + 1)^{2}(x^{2} + 1)}$ in partial fractions.</p>
<p>Hence find the derivative of $\frac{3x^{2} + 6x + 1}{(x + 1)^{2}(x^{2} + 1)}$.</p>

Express $\frac{3x^{2} + 6x + 1}{(x + 1)^{2}(x^{2} + 1)}$ in partial fractions.

Hence find the derivative of $\frac{3x^{2} + 6x + 1}{(x + 1)^{2}(x^{2} + 1)}$.

Quotient Rule

Chain Rule

Differentiation

<p>Express $\frac{1 - 2x + x^2}{(x^2 - 4)(2x + 3)}$ in partial fractions.</p>
<p>Hence find the derivative of $\frac{1 - 2x + x^2}{(x^2 - 4)(2x + 3)}$.</p>

Express $\frac{1 - 2x + x^2}{(x^2 - 4)(2x + 3)}$ in partial fractions.

Hence find the derivative of $\frac{1 - 2x + x^2}{(x^2 - 4)(2x + 3)}$.

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