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PSLE

PSLE Foundation Mathematics

<p>A curve $C$ has equation $y=x^{-3}\ln x$.</p>
<ol class="abc-list"><li data-list-item-id="p1q7-q-b2-i1" data-list-item-path="p1q7-q-b2-i1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="p1q7-q-b2-i2" data-list-item-path="p1q7-q-b2-i2"><div class="list-item-paragraph">Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.<span class="inline-marks">[5]</span></div></li></ol>

A curve $C$ has equation $y=x^{-3}\ln x$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1-3\ln x}{x^4}$ and hence find the coordinates of the turning point of $C$.

Find the exact area enclosed by $C$, the $x$-axis and the line $x=3$.

Derivatives of Natural Logarithmic Functions

Product Rule

Area Bounded by Curves

Stationary Points and Their Nature

N2022 P1 Q7

Techniques of Differentiation

Applications of Integration

Applications of Differentiation

Techniques of Integration

<p>It is given that&nbsp;$y=\sqrt{1+\ln(1+2x)}$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-0" data-list-item-path="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-0"><div class="list-item-paragraph"><p>Show that&nbsp;$y\left(\dfrac{ \mathrm{d}y}{ \mathrm{d}x}\right)=\dfrac{1}{1+2x}$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-1" data-list-item-path="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-1"><div class="list-item-paragraph">By further differentiation, obtain the Maclaurin series of $y$, up to and including the term in $x^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-2" data-list-item-path="h2-2025-rvhs-promo-q8-rvhs-q8-item-block-1-item-2"><div class="list-item-paragraph"><p>Hence, find the Maclaurin series of $\dfrac{1}{\left(1+\ln(1+2x)\right)^{\frac{3}{2}}},$ up to and including the term in $x^2$.</p><span class="inline-marks">[3]</span></div></li></ol>

It is given that $y=\sqrt{1+\ln(1+2x)}$.

By further differentiation, obtain the Maclaurin series of $y$, up to and including the term in $x^2$.

Maclaurin Series from Differential Equations

Substitution and Composition of Power Series

Higher Derivatives by Implicit Differentiation

Chain Rule

2025 RVHS Promo Q8

Maclaurin Series and Power Series

<p>Mathematical correction: the stationary point $P$ is a maximum. The earlier wording said minimum; the equation and assessed parts are unchanged.</p>
<p>A curve $y=\dfrac{1+3\ln x}{x^2}$ has a maximum point at $P$.</p>
<ol class="abc-list"><li data-list-item-id="jvss-p1-q9-q-1-content-items-0" data-list-item-path="jvss-p1-q9-q-1-content-items-0"><div class="list-item-paragraph"><p>Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="jvss-p1-q9-q-1-content-items-1" data-list-item-path="jvss-p1-q9-q-1-content-items-1"><div class="list-item-paragraph"><p>Show that the $x$-coordinate of $P$ is $\mathrm{e}^{\frac{1}{6}}$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="jvss-p1-q9-q-1-content-items-2" data-list-item-path="jvss-p1-q9-q-1-content-items-2"><div class="list-item-paragraph"><p>Find the equation of normal to the curve at the point $x=\mathrm{e}$. Leave your answer in terms of $\mathrm{e}$.</p><span class="inline-marks">[5]</span></div></li></ol>

Mathematical correction: the stationary point $P$ is a maximum. The earlier wording said minimum; the equation and assessed parts are unchanged.

A curve $y=\dfrac{1+3\ln x}{x^2}$ has a maximum point at $P$.

Tangents and Normals at a Point

Quotient Rule

2025 JVSS PRELIMS P1 Q9

Differentiation

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.<span class="inline-marks">[2]</span></div></li></ol>
<p>A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Using the result in part <strong>(a)</strong>, find the exact volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $x$-axis.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Find the volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.

A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.

Find the volume generated when $R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.

Volume of Revolution about the y-axis

Volume of Revolution about the x-axis

N2025 P1 Q7

<ol class="i-list"><li data-list-item-id="amint-q23-question-1" data-list-item-path="amint-q23-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\sin3x)=3\cot3x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q23-question-2" data-list-item-path="amint-q23-question-2"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\cos3x)=-3\tan3x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q23-question-3" data-list-item-path="amint-q23-question-3"><div class="list-item-paragraph">Show that&nbsp;$\cot6x=\dfrac12(\cot3x-\tan3x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q23-question-4" data-list-item-path="amint-q23-question-4"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_{\pi/18}^{\pi/12}\cot6x\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\sin3x)=3\cot3x$.

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\ln(\cos3x)=-3\tan3x$.

Show that $\cot6x=\dfrac12(\cot3x-\tan3x)$.

Hence evaluate the exact value of $\displaystyle\int_{\pi/18}^{\pi/12}\cot6x\,\mathrm{d}x$.

Derivatives of Trigonometric Functions

Definite Integration as the Reverse of Differentiation

Proving and Simplifying Trigonometric Identities

Cotangent integral from logarithmic derivatives

Integration

Trigonometry

<ol class="i-list"><li data-list-item-id="amint-q13-question-1" data-list-item-path="amint-q13-question-1"><div class="list-item-paragraph">Differentiate $x\tan x$ with respect to $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q13-question-2" data-list-item-path="amint-q13-question-2"><div class="list-item-paragraph">Differentiate $\ln(\cos x)$ with respect to $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q13-question-3" data-list-item-path="amint-q13-question-3"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_0^{\pi/4}x\sec^2x\,\mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Differentiate $x\tan x$ with respect to $x$.

Differentiate $\ln(\cos x)$ with respect to $x$.

Hence evaluate the exact value of $\displaystyle\int_0^{\pi/4}x\sec^2x\,\mathrm{d}x$.

Tangent product and logarithmic derivative

<ol class="i-list"><li data-list-item-id="amint-q21-question-1" data-list-item-path="amint-q21-question-1"><div class="list-item-paragraph">Show that&nbsp;$\dfrac{\mathrm{d}}{\mathrm{d}x}[(2x+1)\ln x]=2\ln x+2+\dfrac1x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q21-question-2" data-list-item-path="amint-q21-question-2"><div class="list-item-paragraph">Hence evaluate $\displaystyle\int_1^{\mathrm{e}}\ln x\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}[(2x+1)\ln x]=2\ln x+2+\dfrac1x$.

Hence evaluate $\displaystyle\int_1^{\mathrm{e}}\ln x\,\mathrm{d}x$.

Logarithmic integral between one and e

<p>Find the derivative of each of the following w.r.t $x$:</p>
<ol class="abc-list"><li data-list-item-id="item-1784097596054-iuq9f0oyv" data-list-item-path="item-1784097596054-iuq9f0oyv"><div class="list-item-paragraph"><p>$y = \ln{\left( (5x + 3)^2(3x^2 - 1) \right)} $</p></div></li><li data-list-item-id="item-1784097597557-iairj8pgi" data-list-item-path="item-1784097597557-iairj8pgi"><div class="list-item-paragraph"><p>$y = \ln \left( \frac{7x^2 - 8}{2 - x} \right) $</p></div></li><li data-list-item-id="item-1784097598086-2r5ar0fqh" data-list-item-path="item-1784097598086-2r5ar0fqh"><div class="list-item-paragraph"><p>$ y = \ln { \left( (x + 3)(x - 2)^2(5 - 3x^2) \right) }$</p></div></li></ol>

Find the derivative of each of the following w.r.t $x$:

<p>A curve $C$ is given by the equation</p>
<p style="text-align: center">\[e^{\frac{x^2}{2}}y=\ln x^2,\]</p>
<p>where&nbsp;$x\ne0$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that&nbsp;$x\ln x^2+e^{\frac{x^2}{2}}\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2}{x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point $P$ where&nbsp;$x=1$. Given that the tangent at $P$ meets $C$ again at the point $Q$, find the coordinates of $Q$, giving your answer correct to 5 significant figures.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Determine the acute angle between the tangents to $C$ at the points $P$ and $Q$.<span class="inline-marks">[3]</span></div></li></ol>

Implicit Differentiation

Derivatives of Exponential Functions

Tangents and Normals to Implicit Curves

Implicit Curve Tangents and Angle of Intersection

<p>Given that&nbsp;$y = \frac{3\ln 2x}{x^2}$ for&nbsp;$x &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1781506047585-uk90uikz0"><div class="list-item-paragraph"> Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{3}{x^3}(1 - 2\ln 2x)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1781506048125-wavvo5cvu"><div class="list-item-paragraph">Hence, find $\int \frac{\ln 2x}{x^3} \mathrm{d}x$.<span class="inline-marks">[3]</span></div></li></ol>

Given that $y = \frac{3\ln 2x}{x^2}$ for $x > 0$.

Integration as an Inverse Process of Differentiation

2024 FMSS PRELIMS P2 Q3

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{\ln 2x}{3x^2} \right) = \frac{1}{3x^3} - \frac{2\ln 2x}{3x^3}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Integrate $\frac{\ln 2x}{x^3}$ with respect to $x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Given that the curve $y = \mathrm{f}(x)$ passes through the point $ \left(\frac{1}{2}, 1 \right)$ and is such that&nbsp;$\mathrm{f}'(x) = \frac{\ln 2x}{x^3}$, find $\mathrm{f}(x)$.<span class="inline-marks">[3]</span></div></li></ol>

Finding the Equation of the Curve

2024 CGSS P1 Q12

<p>It is given that&nbsp;$y = \ln \left(\frac{2x}{5 - x} \right)$, where&nbsp;$0 &lt; x &lt; 5$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">If&nbsp;$y$ is decreasing at a rate of $10$ units/s, find the rate of change of $x$ when&nbsp;$y = \ln 8$.<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">The equation of the normal to the curve at point $A$ is $8y = 8\ln 2 + 25 - 10x$. Find the $x$-coordinate of point $A$.<span class="inline-marks">[4]</span></div></li></ol>

It is given that $y = \ln \left(\frac{2x}{5 - x} \right)$, where $0 < x < 5$.

Connected Rates of Change

Equation of a Straight Line

2024 CGSS P1 Q9

Coordinate Geometry

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that&nbsp;$ \mathrm{f}(x) = \frac{2x + 3}{\sqrt{4x - 3}}$, find $ \mathrm{f}'(x)$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the set of values of $x$ for&nbsp;$y = \ln\left(\frac{3 + 8x}{8x - 5}\right)$ to be a decreasing function.</div></li></ol>

Increasing and Decreasing Functions

2024 CCHY P1 Q4

<p>Evaluate the following: </p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(4^{2x}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\ln(2x)}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)$</div></li></ol>

<p>Given that</p>
<p style="text-align: center">\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]</p>
<p>where&nbsp;$A$ and $B$ are constants with $x &gt; -1$.</p>
<p>Show that&nbsp;$(x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0$.<span class="inline-marks">[5]</span></p>

\[y = A \cos [\ln(x+1)] + B \sin [\ln(x+1)],\]

where $A$ and $B$ are constants with $x > -1$.

Show that $(x+1)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (x+1) \frac{\mathrm{d}y}{\mathrm{d}x} + y = 0$.

Second and Higher Derivatives

2018 AJC C1BT1 Q3

<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(5^{\tan x}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(2^{x^2}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(3^{\ln x}\right)$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}}{\mathrm{d}x}\left(\pi^{\sin 2x^3}\right)$</div></li></ol>

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that&nbsp;$y = \sqrt{\ln(x + \mathrm{e})}$ for&nbsp;$x &gt; 1 - \mathrm{e}$, show that</div><div class="list-item-paragraph" style="text-align: center">\[2y\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{e}^{-y^2}.\]</div><div class="list-item-paragraph">Hence find the first three terms of the Maclaurin expansion of $y$. Give the coefficients as exact fractions in their simplest form.<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-paragraph">Using standard series from the List of Formulae (MF27), expand $\ln(x + \mathrm{e})$ as far as the term in $x^2$.</div><div class="list-item-paragraph">Hence, verify the correctness of your expansion in <strong>(a)</strong>, showing your working clearly.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use your expansion in part <strong>(a)</strong> to find an approximation to $\sqrt{\ln\left(\frac{1 + 10\mathrm{e}}{10}\right)}$, leaving your answer in terms of $\mathrm{e}$.<span class="inline-marks">[2]</span></div></li></ol>

Maclaurin Series Using Standard Series

Approximations Using Power Series

2024 NYJC Promo Q8

<p>Differentiate each of the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\cos^{-1}\left(3x^2\right)$,<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-paragraph">$\ln\left(\frac{\sqrt{2x+1}}{x^3}\right)$.<span class="inline-marks">[3]</span></div></li></ol>

Differentiate each of the following expressions with respect to $x$.

Derivatives of Inverse Trigonometric Functions

2024 JPJC Promo Q1

<p>It is given that&nbsp;$y = \tan[\ln(1+2x)]$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$(1+2x)\frac{\mathrm{d}y}{\mathrm{d}x} = 2(1+y^2)$.<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-paragraph">By differentiating the result in part <strong>(a)</strong>, find the Maclaurin expansion of $\tan[\ln(1+2x)]$ up to and including the term in $x^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Hence, find the Maclaurin expansion of $\mathrm{e}^{\frac{1}{2}y}$, up to and including the term in $x^2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Using your answer in part <strong>(c)</strong>, evaluate $\lim \limits_{x \to 0} \frac{\mathrm{e}^{\frac{1}{2}y}-1}{3x}$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \tan[\ln(1+2x)]$.

2024 EJC Promo Q8

<p>The equation of a curve is $y=\frac{ \ln x}{x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1773466434665-oon3h3sb1"><div class="list-item-paragraph">Find the coordinates of the stationary point of this curve. <br>Leave your answer in terms of e.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773466436825-f0ta69f8n"><div class="list-item-paragraph">Determine the nature of the stationary point.<span class="inline-marks">[3]</span></div></li></ol>

The equation of a curve is $y=\frac{\ln x}{x}$.

Find the coordinates of the stationary point of this curve. Leave your answer in terms of $e$.

Determine the nature of the stationary point.

Math Question Bank