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PSLE

PSLE Foundation Mathematics

<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

Maclaurin Series from Differential Equations

Substitution and Composition of Power Series

Derivatives of Natural Logarithmic Functions

Higher Derivatives by Implicit Differentiation

Chain Rule

2024 NYJC Promo Q8

Maclaurin Series and Power Series

Techniques of Differentiation

<p>Using standard series from the List of Formulae (MF 27), expand $\cos \left( {{e}^{x}} \right)$ up to and including the term in ${{x}^{2}}$.</p>

Using standard series from the List of Formulae (MF 27), expand $\cos \left( {{e}^{x}} \right)$ up to and including the term in ${{x}^{2}}$.

Maclaurin Series Standard Series Q6

<p>Using standard series from the List of Formulae (MF 27), expand ${{e}^{1-x}}\ln \left( 5+2x \right)$ up to and including the term in ${{x}^{2}}$.	</p>

Using standard series from the List of Formulae (MF 27), expand ${{e}^{1-x}}\ln \left( 5+2x \right)$ up to and including the term in ${{x}^{2}}$.	

Products and Quotients of Power Series

Maclaurin Series Standard Series Q4

<p>By using the standard series, derive the first four non-zero terms of the power series expansion of $\frac{\ln (1+2x)}{\sqrt{1+2x}}$. Write down the range of values of $x$ for the expansion to be valid</p>

By using the standard series, derive the first four non-zero terms of the power series expansion of $\frac{\ln (1+2x)}{\sqrt{1+2x}}$. Write down the range of values of $x$ for the expansion to be valid

Convergence and Validity of Power Series

Binomial Expansion as a Maclaurin Series

<p>It is given that&nbsp;$\mathrm{f}(x) = \ln(a + x)$, $x \in \mathbb{R}$, $x &gt; -a$, where&nbsp;$a$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-4-a" data-list-item-path="item-4-a"><div class="list-item-paragraph">Using the standard series from the List of Formulae (MF27), find the series expansion for&nbsp;$\mathrm{f}(x)$, up to and including the term in $x^3$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that&nbsp;$a=1$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-4-b" data-list-item-path="item-4-b"><div class="list-item-paragraph">Hence, or otherwise, show that the series expansion of $\sin[\mathrm{f}(x)]$, up to and including the term in $x^3$ is given by $x - \frac{1}{2}x^2 + \frac{1}{6}x^3$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4-c" data-list-item-path="item-4-c"><div class="list-item-paragraph">Deduce the Maclaurin series for&nbsp;$\cos[\mathrm{f}(x)]$ up to and including the term in $x^2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4-d" data-list-item-path="item-4-d"><div class="list-item-paragraph">Find $\displaystyle \displaystyle\int_{1}^{3} (x - \frac{1}{2}x^2 + \frac{1}{6}x^3) \, \mathrm{d}x$. Without the use of a calculator or any further calculation, explain, whether this value is a good approximation to the value of $\displaystyle \displaystyle\int_{1}^{3} \sin[\mathrm{f}(x)] \, \mathrm{d}x$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $\mathrm{f}(x) = \ln(a + x)$, $x \in \mathbb{R}$, $x > -a$, where $a$ is a constant.

Using the standard series from the List of Formulae (MF27), find the series expansion for $\mathrm{f}(x)$, up to and including the term in $x^3$.

Hence, or otherwise, show that the series expansion of $\sin[\mathrm{f}(x)]$, up to and including the term in $x^3$ is given by $x - \frac{1}{2}x^2 + \frac{1}{6}x^3$.

Deduce the Maclaurin series for $\cos[\mathrm{f}(x)]$ up to and including the term in $x^2$.

Find $\displaystyle \displaystyle\int_{1}^{3} (x - \frac{1}{2}x^2 + \frac{1}{6}x^3) \, \mathrm{d}x$. Without the use of a calculator or any further calculation, explain, whether this value is a good approximation to the value of $\displaystyle \displaystyle\int_{1}^{3} \sin[\mathrm{f}(x)] \, \mathrm{d}x$.

Accuracy of Power Series Approximations

Deriving Series by Differentiation and Integration

2025 VJC P2 Q1

<p>It is given that $y={{e}^{{{\tan }^{-1}}4x}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $\left( 1+16{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+k\frac{\mathrm{d}y}{\mathrm{d}x}\left( 8x-1 \right)=0$, where $k$ is a constant to be found.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)</strong>, find the Maclaurin series for $y$ up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The first three non-zero terms of the Maclaurin expansion of ${{e}^{{{\tan }^{-1}}4x}}$ are equal to the first three non-zero terms of the series expansion of $\ln \left( \frac{a}{1-bx} \right)$, where $a$ and $b$ are constants. Using standard series from the List of Formulae (MF27), find the exact values of $a$ and $b$.</div><p style="text-align: right;">[2]</p></li></ol>

Show that $(1+16x^2)\frac{d^2y}{dx^2}+k\frac{dy}{dx}(8x-1)=0$, where $k$ is a constant to be found.

By further differentiation of the result in part (a), find the Maclaurin series for $y$ up to and including the term in $x^3$.

The first three non-zero terms of the Maclaurin expansion of $e^{\tan^{-1}4x}$ are equal to the first three non-zero terms of the series expansion of $\ln\left(\frac{a}{1-bx}\right)$, where $a$ and $b$ are constants. Using standard series from the List of Formulae (MF27), find the exact values of $a$ and $b$.

Finding Parameters from Power Series Coefficients

Derivatives of Inverse Trigonometric Functions

Maclaurin Series by Successive Differentiation

Second and Higher Derivatives

Derivatives of Exponential Functions

2025 DHS Promo Q6

<p>It is given that&nbsp;$y=\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760870612493-0lhphq5um" data-list-item-path="item-1760870612493-0lhphq5um"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=4{{\mathrm{e}}^{-y}}-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760870613475-t4r53w0xb" data-list-item-path="item-1760870613475-t4r53w0xb"><div class="list-item-paragraph">Hence find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760870614360-lu2pah4wx" data-list-item-path="item-1760870614360-lu2pah4wx"><div class="list-item-paragraph"><p>Using standard series from the List of Formulae (MF26), expand $\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$ as far as the term in ${{x}^{2}}$, and use this expansion as a check on the correctness of the series found in part <strong>(b)</strong>.</p><span class="inline-marks">[4]</span></div></li></ol>

It is given that $y=\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$.

2023 EJC Promo Q9

<p>Using standard series from the List of Formulae (MF 27), expand ${\sin\left(2x - \frac{\pi}{4}\right)}$ up to and including the term in ${x^{3}}$.</p>

Using standard series from the List of Formulae (MF 27), expand ${\sin\left(2x - \frac{\pi}{4}\right)}$ up to and including the term in ${x^{3}}$.

Maclaurin Series Standard Series Q5

<p>Using standard series from the List of Formulae (MF 27), expand ${e^{3x}\ln(2+x)}$ up to and including the term in ${x^{2}}$.</p>

Using standard series from the List of Formulae (MF 27), expand ${e^{3x}\ln(2+x)}$ up to and including the term in ${x^{2}}$.

Maclaurin Series Standard Series Q3

<p>By considering the series expansions of ${\ln \left( 1-{{x}^{2}} \right)}$ and ${\ln \left( \frac{1+x}{1-x} \right)}$, or otherwise, find the exact value of the following series.</p>
<p style="text-align: center">${\sum\limits_{r=1}^{\infty }{\left[ \left( \frac{1}{2r}+\frac{1}{2r+1} \right){{\left( \frac{1}{4} \right)}^{r}} \right]}}$.</p>

By considering the series expansions of ${\ln \left( 1-{{x}^{2}} \right)}$ and ${\ln \left( \frac{1+x}{1-x} \right)}$, or otherwise, find the exact value of the following series.

${\sum\limits_{r=1}^{\infty }{\left[ \left( \frac{1}{2r}+\frac{1}{2r+1} \right){{\left( \frac{1}{4} \right)}^{r}} \right]}}$.

Maclaurin and Power Series Q1

<ol class="i-list"><li><div class="list-item-paragraph">Using standard result given in the List of Formulae (MF26), find the Maclaurin’s series for&nbsp;$\ln (1+x+{{x}^{2}})$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine the set of values of $x$ for which the expansion in part <strong>(i)</strong> is valid.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, or otherwise, obtain the series expansion of $\frac{1+2x}{1+x+{{x}^{2}}}$ up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

Series from Products, Substitution, Integration and Differentiation

<ol class="i-list"><li><div class="list-item-paragraph">Using standard result given in the List of Formulae (MF26), find the Maclaurin’s series for&nbsp;$\ln (1-2x+{{x}^{2}})$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine the set of values of $x$ for which the expansion in part <strong>(i)</strong> is valid.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, or otherwise, obtain the series expansion of $\frac{x-1}{1-2x+{{x}^{2}}}$ up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

<p>By using the standard series, find the following expansion up to and including the term ${{x}^{4}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760835305042-5tlobfpvy" data-list-item-path="item-1760835305042-5tlobfpvy"><div class="list-item-paragraph">$\ln \left( 2+x \right)$</div></li><li data-list-item-id="item-1760835305582-tei19v71k" data-list-item-path="item-1760835305582-tei19v71k"><div class="list-item-paragraph">$\ln \left( 5-x \right)$</div></li><li data-list-item-id="item-1760835306324-rd1gb2h8b" data-list-item-path="item-1760835306324-rd1gb2h8b"><div class="list-item-paragraph">${{e}^{1-x}}$</div></li><li data-list-item-id="item-1760835308313-liqz6j2zm" data-list-item-path="item-1760835308313-liqz6j2zm"><div class="list-item-paragraph">${{e}^{2+x}}$</div></li><li data-list-item-id="item-1760835308867-in7v42vc0" data-list-item-path="item-1760835308867-in7v42vc0"><div class="list-item-paragraph">$\sin \left( 2x+1 \right)$</div></li><li data-list-item-id="item-1760835310089-t762vyezz" data-list-item-path="item-1760835310089-t762vyezz"><div class="list-item-paragraph">$\cos \left( \frac{\pi }{3}-x \right)$</div></li></ol>

By using the standard series, find the following expansion up to and including the term ${{x}^{4}}$.

<p>By using the standard series, derive the first four non-zero terms of the power series expansion of ${{e}^{2x+1}}\cos 2x$. Write down the range of values of $x$ for the expansion to be valid.</p>

By using the standard series, derive the first four non-zero terms of the power series expansion of ${{e}^{2x+1}}\cos 2x$. Write down the range of values of $x$ for the expansion to be valid.

<p>By using the standard series, derive the first four non-zero terms of the power series expansion of ${{e}^{3{{x}^{2}}}}\ln (1-3x)$. Write down the range of values of $x$ for the expansion to be valid.</p>

By using the standard series, derive the first four non-zero terms of the power series expansion of ${{e}^{3{{x}^{2}}}}\ln (1-3x)$. Write down the range of values of $x$ for the expansion to be valid.

<p>Find the Maclaurin’s series for&nbsp;$y={{e}^{3x}}\ln \left( 1+\frac{x}{2} \right)$ up to the term ${{x}^{3}}$.</p>

Find the Maclaurin’s series for $y={{e}^{3x}}\ln \left( 1+\frac{x}{2} \right)$ up to the term ${{x}^{3}}$.

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Maclaurin Expansion </p>
<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/1770177155889-wk9lxx8lv3m.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzAxNzcxNTU4ODktd2s5bHh4OGx2M20ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MjYxNywiZXhwIjoxNzkzNTQyOTE3fQ.RxUXcMzkiF7CmTOYO2S1IUBD0P5zlioSPQzwMOhFBsg" style="max-width: min(600px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The following standard series are stated in MF27. Their validity conditions are part of the result and must be carried into any substituted or combined expansion.

When a standard series is used with an inner expression, apply its condition to that whole expression. For example, $\ln(1-3x)$ requires $-1<-3x\leq1$, so $-\dfrac13\leq x<\dfrac13$. For a product or combination of series, use the intersection of all relevant validity intervals.

Standard Series and Validity

<ol class="abc-list"><li data-list-item-id="item-1760748073985-6zkh4y147" data-list-item-path="item-1760748073985-6zkh4y147"><div class="list-item-paragraph">Find the series expansion for&nbsp;$\frac{\cos 2x}{{{\left( 1+\tan x \right)}^{2}}}$ up to and including the term in ${{x}^{2}}$, given that&nbsp;$x$ is a sufficiently small angle.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760748074735-fzd6yxb7p" data-list-item-path="item-1760748074735-fzd6yxb7p"><div class="list-item-paragraph">It is given that&nbsp;$\mathrm{f}(x)={{\left( 1-x \right)}^{\frac{1}{2}}}\ln \sqrt{1+x}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760748088122-r4auxec5s" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748088122-r4auxec5s"><div class="list-item-paragraph">Find the numerical value of $\mathrm{f}'(x)$ when&nbsp;$x=0.01$, leaving the answer in $5$ decimal places.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760748095885-ni5ilr8hq" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748095885-ni5ilr8hq"><div class="list-item-paragraph">Using the standard series from the List of Formulae (MF27), expand $\mathrm{f}\left( x \right)$ as far as the term in ${{x}^{3}}$. State the range of values of $x$ for which the expansion is valid.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1760748103077-oufpzpbxf" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748103077-oufpzpbxf"><div class="list-item-paragraph">Denote the answer in part <strong>(b)(ii)</strong> as&nbsp;$\mathrm{g}\left( x \right)$. Hence find $\mathrm{g }'\mathrm{ }\left( x \right)$ when&nbsp;$x=0.01$, leaving the answer in $5$ decimal places.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760748118528-hw1tpiuno" data-list-item-path="item-1760748074735-fzd6yxb7p/subitem-1760748118528-hw1tpiuno"><div class="list-item-paragraph">Comparing your answer in part <strong>(b)(i)</strong> against <strong>(b)(iii)</strong>, comment on the validity of this answer.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Small Angle Approximations

2024 MI P2 Q5

<p>For a right-angled triangle $ABC$ with $BC=a$ and $AC=b$, $\measuredangle CAB=x$ and $\measuredangle ABC=\frac{\pi }{2}$ as shown below, a sector with radius $a$ is extended from the vertex $C$. This sector meets the side $AC$ at the point $Y$.</p>
<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/converted/1760707177047-d4so0iwxo4a.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3MDcxNzcwNDctZDRzbzBpd3hvNGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MjYxNywiZXhwIjoxNzkzNTQyOTE3fQ.waf8Wnn7pltOqcDhNxO8Okksa1ypWJfnruIMj0MIzEc" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1760707176669-hpgk7x371" data-list-item-path="item-1760707176669-hpgk7x371"><div class="list-item-paragraph">Show that&nbsp;${{\left( \frac{AC}{AY} \right)}^{2}}=\frac{1}{{{\left( 1-\sin x \right)}^{2}}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760707177209-c5ixvvjxc" data-list-item-path="item-1760707177209-c5ixvvjxc"><div class="list-item-paragraph">Using appropriate expansions from the List of Formulae (MF26), find the Maclaurin series for&nbsp;$\mathrm{f}(x)={{\left( 1-\sin x \right)}^{-2}}$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760707178147-p1fikf4k6" data-list-item-path="item-1760707178147-p1fikf4k6"><div class="list-item-paragraph">Deduce the series expansion for&nbsp;$\frac{2\cos x}{{{\left( 1-\sin x \right)}^{3}}}$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760707178831-sau53jtmy" data-list-item-path="item-1760707178831-sau53jtmy"><div class="list-item-paragraph">Using your answer in part <strong>(b)</strong> or otherwise, show that&nbsp;$\frac{AC}{AY}\approx 1+x+{{x}^{2}}$, when&nbsp;$x$ is sufficiently small.<span class="inline-marks">[3]</span></div></li></ol>

For a right-angled triangle $ABC$ with $BC=a$ and $AC=b$, $\measuredangle CAB=x$ and $\measuredangle ABC=\frac{\pi }{2}$ as shown below, a sector with radius $a$ is extended from the vertex $C$. This sector meets the side $AC$ at the point $Y$.

2024 EJC P1 Q7

<p>It is given that $y=\sin \left[ \ln \left( 1+2x \right) \right]$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that ${{\left( 1+2x \right)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2\left( 1+2x \right)\frac{\mathrm{d}y}{\mathrm{d}x}+ky=0$, where $k$ is a constant to be found.</div><div class="list-item-paragraph">Hence, find the first three non-zero terms of the Maclaurin expansion for $y$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Using standard series from the List of Formulae, verify the correctness of your result from part <strong>(i)</strong> up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">Explain why the expansion is not valid when $x=-\frac{1}{2}$.</div><p style="text-align: right;">[1]</p></li></ol>

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

Math Question Bank