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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

Polynomials, Cubic Equations and Partial Fractions

Techniques of Integration

<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-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.

Definite Integral

Three-factor logarithmic 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

<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-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 \(\displaystyle\int_2^4 \dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\,\mathrm{d}x\).

Repeated-factor logarithmic integral

<p>\(\displaystyle \displaystyle\int{\frac{4{{x}^{2}}+2}{2{{x}^{2}}-x-1}}\,\mathrm{d}x\)</p>

\(\displaystyle \displaystyle\int{\frac{4{{x}^{2}}+2}{2{{x}^{2}}-x-1}}\,\mathrm{d}x\)

<p>\(\displaystyle \displaystyle\int{\frac{4x-5}{{{x}^{2}}+3x+2}}\,\mathrm{d}x\)</p>

\(\displaystyle \displaystyle\int{\frac{4x-5}{{{x}^{2}}+3x+2}}\,\mathrm{d}x\)

<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="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 \(\frac{f'(x)}{f(x)}\)

<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>\[\int \frac{1}{\sin^3x+\cos^3x} \, \mathrm{d}x\]</p>

\[\int \frac{1}{\sin^3x+\cos^3x} \, \mathrm{d}x\]

Integrable Forms of Trigonometric Functions

Integration by Substitution

Standard Integrals with Quadratic Denominators

<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

<ol class="abc-list"><li data-list-item-id="item-a-3" data-list-item-path="item-a-3"><ol class="i-list"><li data-list-item-id="item-a-3-i" data-list-item-path="item-a-3/item-a-3-i"><div class="list-item-paragraph"><p>Show that&nbsp;\(\frac{3x^2}{(x+1)(3x^2+x+1)}\) can be expressed as<br></p></div><div class="list-item-paragraph"><p style="text-align: center;">\(\frac{A}{x+1} + \frac{Bx+C}{3x^2+x+1}\),<br></p></div><div class="list-item-paragraph"><p>where&nbsp;\(A\), \(B\) and \(C\) are constants to be determined.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-3-ii" data-list-item-path="item-a-3/item-a-3-ii"><div class="list-item-paragraph">Hence, find \(\displaystyle \displaystyle\int \frac{3x^2}{(x+1)(3x^2+x+1)} \, \mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-3" data-list-item-path="item-b-3"><div class="list-item-paragraph">The region \(R\) is bounded by the curve \(y = \frac{3x^2}{(x+1)(3x^2+x+1)}\), the line \(x=4\) and the \(x\)-axis.</div><ol class="i-list"><li data-list-item-id="item-b-3-i" data-list-item-path="item-b-3/item-b-3-i"><div class="list-item-paragraph">Sketch the graph of \(y = \frac{3x^2}{(x+1)(3x^2+x+1)}\) for&nbsp;\(x \ge 0\), giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region \(R\) on your sketch.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3-ii" data-list-item-path="item-b-3/item-b-3-ii"><div class="list-item-paragraph">Find the exact area of \(R\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3-iii" data-list-item-path="item-b-3/item-b-3-iii"><div class="list-item-paragraph">Find the volume of the solid generated when&nbsp;\(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Sketching Rational Functions

Integration by Splitting the Numerator

Volume of Revolution about the x-axis

Area Bounded by Curves

Asymptotes and Discontinuities

Integration by Completing the Square

2025 CJC P1 Q11

Graphing Techniques

Applications of Integration

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express \(\frac{27x^2-6x+9}{(4x-1)(x^2+3)}\) in partial fractions.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Differentiate \(\ln(x^2+3)\) with respect of \(x\).</span></div></li><li><div class="list-item-paragraph">Using the results of <strong>(a)</strong> and <strong>(b)</strong>, find \(\displaystyle\int \frac{27x^2-6x+9}{(4x-1)(x^2+3)} \, \mathrm{d}x\).</div></li></ol>

Express \(\frac{27x^2-6x+9}{(4x-1)(x^2+3)}\) in partial fractions.

Differentiate \(\ln(x^2+3)\) with respect of \(x\).

Using the results of **(a)** and **(b)**, find \(\displaystyle\displaystyle \displaystyle\int \frac{27x^2-6x+9}{(4x-1)(x^2+3)} \, \mathrm{d}x\).

Derivatives of Natural Logarithmic Functions

Integration as an Inverse Process of Differentiation

Chain Rule

Differentiation

<p>Given that \(\frac{6+5x-8x^2}{(2x-1)(x^2+3)} = \frac{A}{2x-1} + \frac{Bx+C}{x^2+3}\), where \(A\), \(B\) and \(C\) are constants</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find the values of \(A\), \(B\) and \(C\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">differentiate \(\ln(x^2 + 3)\) with respect to \(x\).</span></div></li><li><div class="list-item-paragraph">Using the results from parts <strong>(a)</strong> and <strong>(b)</strong>, find \(\displaystyle\int \frac{6+5x-8x^2}{(2x-1)(x^2+3)} \, \mathrm{d}x\).</div></li></ol>

Given that \(\frac{6+5x-8x^2}{(2x-1)(x^2+3)} = \frac{A}{2x-1} + \frac{Bx+C}{x^2+3}\), where \(A\), \(B\) and \(C\) are constants

find the values of \(A\), \(B\) and \(C\),

differentiate \(\ln(x^2 + 3)\) with respect to \(x\).

Using the results from parts **(a)** and **(b)**, find \(\displaystyle\displaystyle \displaystyle\int \frac{6+5x-8x^2}{(2x-1)(x^2+3)} \, \mathrm{d}x\).

<p>Express \(\frac{3x^2-5x-3}{x^3-3x^2}\) in partial fractions. Hence, evaluate \(\displaystyle\int_{4}^{6} \frac{3x^2-5x-3}{x^3-3x^2} \, \mathrm{d}x\).</p>

Express \(\frac{3x^2-5x-3}{x^3-3x^2}\) in partial fractions. Hence, evaluate \(\displaystyle\displaystyle \displaystyle\int_{4}^{6} \frac{3x^2-5x-3}{x^3-3x^2} \, \mathrm{d}x\).

<p>Express \(\frac{8x^2+2x+1}{x(2x+1)^2}\) in partial fractions. Hence, find \(\displaystyle\int_1^2 \frac{8x^2+2x+1}{x(2x+1)^2} \, \mathrm{d}x\).</p>

Express \(\frac{8x^2+2x+1}{x(2x+1)^2}\) in partial fractions. Hence, find \(\displaystyle\displaystyle \displaystyle\int_1^2 \frac{8x^2+2x+1}{x(2x+1)^2} \, \mathrm{d}x\).

<p>Express \(\frac{5x-1}{x^2-1}\) in partial fractions. Hence, evaluate \(\displaystyle\int_{2}^{3} \frac{5x-1}{x^2-1} \, \mathrm{d}x\).</p>

Express \(\frac{5x-1}{x^2-1}\) in partial fractions. Hence, evaluate \(\displaystyle\displaystyle \displaystyle\int_{2}^{3} \frac{5x-1}{x^2-1} \, \mathrm{d}x\).

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