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N(A) E Math

PSLE

PSLE Foundation Mathematics

**Do not use a calculator in answering this question.**

Small Angle Approximations

Approximations Using Power Series

Maclaurin Series from Differential Equations

Higher Derivatives by Implicit Differentiation

2026 RI P2 Q4

Techniques of Differentiation

Maclaurin Series and Power Series

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Using standard series from the List of Formula (MF27), expand $\frac{\cos ax}{x^2 + a^2}$ as far as the term in $x^4$, where&nbsp;$a$ is a positive constant. Give your answer in the form $c_1 + c_2 x^2 + c_3 x^4$ where&nbsp;$c_1$, $c_2$ and $c_3$ are in terms of $a$.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the range of values of $x$, in terms of $a$, for which the expansion in part <strong>(a)</strong> is valid.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Using your answer in part <strong>(a)</strong> and a suitable value of $a$, show that&nbsp;$\int_{0}^{1} \frac{\cos 2x}{x^2 + 4} \,\mathrm{d}x \approx \frac{119}{960}$.<span class="inline-marks">[3]</span></div></li></ol>

Convergence and Validity of Power Series

Binomial Expansion as a Maclaurin Series

Deriving Series by Differentiation and Integration

Products and Quotients of Power Series

2024 VJC Promo Q9

<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

Substitution and Composition of Power Series

Derivatives of Natural Logarithmic Functions

Chain Rule

2024 NYJC Promo Q8

<p>Find the expansion of ${{\left( \frac{1+2{{x}^{2}}}{4-x} \right)}^{\frac{1}{2}}}$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[4]</span></p>
<ol class="i-list"><li data-list-item-id="item-1770181391847-uq8gnjufc" data-list-item-path="item-1770181391847-uq8gnjufc"><div class="list-item-paragraph">Find the set of values of $x$ for which the expansion is valid.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1770181403164-gdk9rpc3l" data-list-item-path="item-1770181403164-gdk9rpc3l"><div class="list-item-paragraph">By putting $x=\frac{1}{4}$, show that&nbsp;$\sqrt{30}\approx \frac{a}{b}$, where&nbsp;$a$ and $b$ are integers to be determined.<span class="inline-marks">[2]</span></div></li></ol>

Find the expansion of ${{\left( \frac{1+2{{x}^{2}}}{4-x} \right)}^{\frac{1}{2}}}$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$.

2013 IJC P1 Q8

<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

2025 VJC P2 Q1

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the expansion of $(8 - x)^{\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^2$. State the set of values of $x$ for which this expansion is valid.</span></div></li><li><div class="list-item-content"><span class="list-item-text">By substituting $x=-1$, find an approximation of $\sqrt[3]{3}$, leaving your answer in the form $\frac{m}{n}$, where $m$ and $n$ are integers to be determined.</span></div></li><li><div class="list-item-content"><span class="list-item-text">The substitution $x=5$ can also be used to find an approximation for $\sqrt[3]{3}$. Explain whether this will give a better approximation than using $x=-1$.</span></div></li></ol>

Find the expansion of $(8 - x)^{\frac{1}{3}}$ in ascending powers of $x$ up to and including the term in $x^2$. State the set of values of $x$ for which this expansion is valid.

By substituting $x=-1$, find an approximation of $\sqrt[3]{3}$, leaving your answer in the form $\frac{m}{n}$, where $m$ and $n$ are integers to be determined.

The substitution $x=5$ can also be used to find an approximation for $\sqrt[3]{3}$. Explain whether this will give a better approximation than using $x=-1$.

<p>It is given that&nbsp;$y = \mathrm{e}^{2\sin^{-1} x}$.</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;$\sqrt{1 - x^2} \frac{\mathrm{d}y}{\mathrm{d}x} = 2y$.<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"><p>By further differentiation of the result in part <strong>(i)</strong>, find the Maclaurin series for&nbsp;$y$ up to and including the term in $x^2$.</p><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">Using your result from part <strong>(ii)</strong>, find an approximate value for&nbsp;$\displaystyle \displaystyle\int_{0}^{0.1} x\mathrm{e}^{2\sin^{-1} x} \, \mathrm{d}x$, giving your answer to $4$ significant figures.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \mathrm{e}^{2\sin^{-1} x}$.

Show that $\sqrt{1 - x^2} \frac{\mathrm{d}y}{\mathrm{d}x} = 2y$.

Derivatives of Inverse Trigonometric Functions

Derivatives of Exponential Functions

2025 MI P1 Q8

<p>Find the first three terms in the binomial expansion of ${{\left( 1-8x \right)}^{12}}$.  Hence, use the expansion obtained to evaluate ${{\left( 0.992 \right)}^{12}}$, correct to three decimal places.</p>

Find the first three terms in the binomial expansion of ${{\left( 1-8x \right)}^{12}}$.  Hence, use the expansion obtained to evaluate ${{\left( 0.992 \right)}^{12}}$, correct to three decimal places.

Binomial Expansion for Positive Integer Powers

Binomial Theorem

<p>Expand ${{\left( 1-2x \right)}^{8}}$ in ascending powers of $x$, up to and including the term ${{x}^{3}}$. Hence, evaluate ${{\left( 0.96 \right)}^{8}}$, correct to $3$ decimal places.</p>

Expand ${{\left( 1-2x \right)}^{8}}$ in ascending powers of $x$, up to and including the term ${{x}^{3}}$. Hence, evaluate ${{\left( 0.96 \right)}^{8}}$, correct to $3$ decimal places.

<p>Obtain the first $4$ terms in the expansion, in ascending powers of $x$, of ${{\left( 1-4x \right)}^{5}}$. Hence, find the value of ${{\left( 0.96 \right)}^{5}}$ correct to $3$ decimal places.</p>

Obtain the first $4$ terms in the expansion, in ascending powers of $x$, of ${{\left( 1-4x \right)}^{5}}$. Hence, find the value of ${{\left( 0.96 \right)}^{5}}$ correct to $3$ decimal places.

<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>

Series from Products, Substitution, Integration and Differentiation

2024 MI P2 Q5

<p>It is given that&nbsp;$\mathrm{f}\left( x \right)={{\mathrm{e}}^{{{\tan }^{-1}}x}}$, where&nbsp;${{\tan }^{-1}}x$ denotes the principal values.</p>
<ol class="i-list"><li data-list-item-id="item-1760747941731-9vyv1phos" data-list-item-path="item-1760747941731-9vyv1phos"><div class="list-item-paragraph">Show that&nbsp;$\left( 1+{{x}^{2}} \right)\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)=\mathrm{f}\left( x \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760747942428-8ls2y4o9j" data-list-item-path="item-1760747942428-8ls2y4o9j"><div class="list-item-paragraph">By further differentiation of the above result, find the Maclaurin series for&nbsp;$\mathrm{f}(x)$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760747943185-0r00ysebb" data-list-item-path="item-1760747943185-0r00ysebb"><div class="list-item-paragraph">Using the series in part <strong>(ii)</strong> find an approximate value of $\displaystyle\displaystyle \displaystyle\int_{0}^{0.5}{\mathrm{f}\left( x \right)\,}\mathrm{d}x$, giving your answer to $4$ significant figures.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760747944566-nz3f4hhzo" data-list-item-path="item-1760747944566-nz3f4hhzo"><div class="list-item-paragraph">Comment on the suitability of substituting $x=1$ into the series in part <strong>(ii)</strong> to estimate the value of ${{\mathrm{e}}^{\frac{\pi }{4}}}$.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $\mathrm{f}\left( x \right)={{\mathrm{e}}^{{{\tan }^{-1}}x}}$, where ${{\tan }^{-1}}x$ denotes the principal values.

Maclaurin Series by Successive Differentiation

2024 JPJC P2 Q3

<p>It is given that&nbsp;$\ln \left( 1+y \right)=\tan x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760706591850-i302r80vw" data-list-item-path="item-1760706591850-i302r80vw"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{\mathrm{d}y}{\mathrm{d}x}\left( 2\tan x+{{\sec }^{2}}x \right)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760706592383-z6af7y2id" data-list-item-path="item-1760706592383-z6af7y2id"><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)</strong>, find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760706593619-3s8feuomy" data-list-item-path="item-1760706593619-3s8feuomy"><div class="list-item-paragraph">Hence find an estimate for the value of $\mathrm{e}$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $\ln \left( 1+y \right)=\tan x$.

Implicit Differentiation

Maclaurin Series by Implicit Differentiation

2024 CJC P1 Q6

<p>It is given that&nbsp;$y={{3}^{\left( {{x}^{2}}+2x \right)}}$</p>
<ol class="i-list"><li data-list-item-id="item-1760700301254-i57tkd6d0" data-list-item-path="item-1760700301254-i57tkd6d0"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{1}{y}{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}+2y\ln 3$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760700301801-0spy3h3cg" data-list-item-path="item-1760700301801-0spy3h3cg"><div class="list-item-paragraph">Find Maclaurin’s series for&nbsp;$y$, up to and including the term in ${{x}^{2}}$, leaving the coefficients in exact form.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760700303129-d83sph6t1" data-list-item-path="item-1760700303129-d83sph6t1"><div class="list-item-paragraph">Hence estimate the value of $\displaystyle\displaystyle \displaystyle\int_{0}^{1}{{{3}^{\left( {{x}^{2}}+2x \right)}}\,}\mathrm{d}x$, giving your answer in exact form. Explain how the accuracy of the estimation can be improved.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $y={{3}^{\left( {{x}^{2}}+2x \right)}}$

Second and Higher Derivatives

2013 MI P1 Q2

<p>It is given that&nbsp;$y=\frac{1}{3+\sin 2x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760690577236-g5ojqpjt3" data-list-item-path="item-1760690577236-g5ojqpjt3"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-4{{y}^{2}}\sin 2x+4y\frac{\mathrm{d}y}{\mathrm{d}x}\cos 2x=0$. Hence find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760690578376-mdfu6pvwh" data-list-item-path="item-1760690578376-mdfu6pvwh"><div class="list-item-paragraph"><p>Given further that&nbsp;$x$ is small so that&nbsp;${{x}^{3}}$ and higher powers of $x$ can be neglected, use appropriate expansions from the List of Formulae <strong>(MF26)</strong> to verify the correctness of the series of $y$ in part <strong>(a)</strong>.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760690578909-9polo9nl0" data-list-item-path="item-1760690578909-9polo9nl0"><div class="list-item-paragraph"><p>Use the series expansion found in part <strong>(a)</strong> and integration to find an approximate value for&nbsp;$\int_0^{1.5} \frac{1}{3+\sin 2x}\mathrm{d}x.$. </p><span class="inline-marks">[2]</span></div></li></ol>

It is given that $y=\frac{1}{3+\sin 2x}$.

Derivatives of Trigonometric Functions

2023 ASRJC P1 Q8

<p>It is given that&nbsp;$y={{\sin }^{-1}}x$, $-1\le x\le 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760430006158-aony6z2na" data-list-item-path="item-1760430006158-aony6z2na"><div class="list-item-paragraph">Show that&nbsp;$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=x{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430006939-x6lgv1h1z" data-list-item-path="item-1760430006939-x6lgv1h1z"><div class="list-item-paragraph">Using the result in part <strong>(a)</strong>, find the Maclaurin expansion for&nbsp;$y$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430007540-7hyj4gagf" data-list-item-path="item-1760430007540-7hyj4gagf"><div class="list-item-paragraph">By substituting $x=0.5$ into the expansion found in part <strong>(b)</strong>, find an approximate value of $\pi $.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y={{\sin }^{-1}}x$, $-1\le x\le 1$.

2023 CJC Promo Q5

<ol class="abc-list"><li><div class="list-item-paragraph">Using standard series, find the Maclaurin series expansion of $\frac{1}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}$ up to and including the term in ${{x}^{4}}$, where $a$ is a positive constant. Find also the range of values of $x$, in terms of $a$, for which the expansion is valid.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Hence, by using a suitable value for $a$, show that ${{\cos }^{-1}}x\approx \frac{\pi }{2}-x-\frac{1}{6}{{x}^{3}}-\frac{3}{40}{{x}^{5}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Using the result in part <strong>(b)</strong>, estimate $\displaystyle\int_{0}^{\frac{1}{2}}{{{\cos }^{-1}}x\,\,\mathrm{d}x}$ to $~5$ significant figures.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Comparing your estimate in <strong>(c)</strong> with the actual value of $\displaystyle\int_{0}^{\frac{1}{2}}{{{\cos }^{-1}}x\,\,\mathrm{d}x}$, comment on the accuracy of your estimate and suggest how it can be further improved.</div><p style="text-align: right;">[2]</p></li></ol>

2023 EJC P1 Q9

<p>It is given that&nbsp;$\mathrm{f}(x)=\sin \left[ \ln \left( 1+x \right) \right]$, $x\in \mathbb{R}$, $x&gt;-1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;${{(1+x)}^{2}}\mathrm{f}''(x)+(1+x)\,\mathrm{f}'(x)+\mathrm{f}(x)=0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By further differentiation of the result in <strong>(i)</strong>, find the first three non-zero terms of the Maclaurin series for&nbsp;$\mathrm{f}(x)$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Verify the correctness of the series found in part <strong>(ii)</strong> using the standard series from the List of Formulae (MF26).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Use the series in part <strong>(ii)</strong> to approximate the value of  $\displaystyle\int_{0}^{2}{\mathrm{f}(x)}\,\mathrm{d}x$.</div><div class="list-item-paragraph">Use your calculator to find an accurate value of $\displaystyle\int_{0}^{2}{\mathrm{f}(x)}\,\mathrm{d}x$. Why is the approximated value not very good?<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using the series obtained in part <strong>(ii)</strong>, deduce the Maclaurin series for&nbsp;$\cos \left[ \ln (1+x) \right]$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $\mathrm{f}(x)=\sin \left[ \ln \left( 1+x \right) \right]$, $x\in \mathbb{R}$, $x>-1$.

2020 TJC P2 Q3

<ol class="i-list"><li><div class="list-item-paragraph">Given that $\ln \left( ky \right)={{\tan }^{-1}}\left( kx \right)$, where $k$ is a non-zero constant, show that $\left( 1+{{k}^{2}}{{x}^{2}} \right)\frac{\mathrm{d}y}{\mathrm{d}x}=ky$. By further differentiation of this result, find the Maclaurin’s series for $y$, in terms of $k$, up to and including the term in ${{x}^{3}}$.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Let $k=1$. By using a suitable substitution for $~x$, find, in terms of $\sqrt{3}$, an approximate value of ${{\mathrm{e}}^{\frac{\pi }{6}}}$.</div><p style="text-align: right;">[4]</p></li></ol>

2013 HCI P2 Q2

<p>Given that&nbsp;$y={{({{\sin }^{-1}}x)}^{2}}$, show that</p>
<p style="text-align: center">\[\left( 1-{{x}^{2}} \right){{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=4y\]</p>
<p style="text-align: center">and $\left( 1-{{x}^{2}} \right)\frac{{{\mathrm{d}}^{\text{2}}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2$.<span class="inline-marks">[3]</span></p>
<p>By further differentiation of these results, find the Maclaurin series of $y$ up to including the term in ${{x}^{4}}$.<span class="inline-marks">[3]</span></p>
<p>The series found above is to be used to estimate a value for&nbsp;${{\pi }^{2}}$. Explain with the aid of a sketch why using $x=\frac{1}{2}$ in the series gives a better approximation than $x=1$, and hence find an approximation to ${{\pi }^{2}}$, giving your answer in fraction form. <span class="inline-marks">[4]</span></p>

Given that $y={{({{\sin }^{-1}}x)}^{2}}$, show that

\[\left( 1-{{x}^{2}} \right){{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=4y\]

and $\left( 1-{{x}^{2}} \right)\frac{{{\mathrm{d}}^{\text{2}}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2$.

By further differentiation of these results, find the Maclaurin series of $y$ up to including the term in ${{x}^{4}}$.

The series found above is to be used to estimate a value for ${{\pi }^{2}}$. Explain with the aid of a sketch why using $x=\frac{1}{2}$ in the series gives a better approximation than $x=1$, and hence find an approximation to ${{\pi }^{2}}$, giving your answer in fraction form. 

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