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PSLE Foundation Mathematics

<p>It is given that&nbsp;$y=\mathrm{f}(x)$ is such that \[\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{\mathrm{d}y}{\mathrm{d}x}\left(2\tan x+p\sec^2x\right),\] where&nbsp;$p$ is a constant. The Maclaurin series for&nbsp;$\mathrm{f}(x)$ is given by \[y=1+x+\frac12x^2+qx^3+\cdots,\] where&nbsp;$q$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="tjc-q05-item01" data-list-item-path="tjc-q05-item01"><div class="list-item-paragraph">State the values of $\mathrm{f}(0)$ and $\mathrm{f}'(0)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="tjc-q05-item02" data-list-item-path="tjc-q05-item02"><div class="list-item-paragraph">Find the values of $p$ and $q$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="tjc-q05-item03" data-list-item-path="tjc-q05-item03"><div class="list-item-paragraph">Using the given series for&nbsp;$\mathrm{f}(x)$ and standard series from the List of Formulae (MF27), find the first three non-zero terms of the series expansion of $\sqrt{\mathrm{f}(x)}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="tjc-q05-item04" data-list-item-path="tjc-q05-item04"><div class="list-item-paragraph">The given Maclaurin series for&nbsp;$\mathrm{f}(x)$ is used to estimate $\displaystyle \int_0^{0.2}\mathrm{f}(x)\,\mathrm{d}x$ and $\displaystyle \int_{0.2}^{0.4}\mathrm{f}(x)\,\mathrm{d}x$. Without any further computations, determine which estimate has a smaller percentage error. Give an explanation for your choice.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $y=\mathrm{f}(x)$ is such that \[\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{\mathrm{d}y}{\mathrm{d}x}\left(2\tan x+p\sec^2x\right),\] where $p$ is a constant. The Maclaurin series for $\mathrm{f}(x)$ is given by \[y=1+x+\frac12x^2+qx^3+\cdots,\] where $q$ is a constant.

State the values of $\mathrm{f}(0)$ and $\mathrm{f}'(0)$.

Using the given series for $\mathrm{f}(x)$ and standard series from the List of Formulae (MF27), find the first three non-zero terms of the series expansion of $\sqrt{\mathrm{f}(x)}$.

The given Maclaurin series for $\mathrm{f}(x)$ is used to estimate $\displaystyle \int_0^{0.2}\mathrm{f}(x)\,\mathrm{d}x$ and $\displaystyle \int_{0.2}^{0.4}\mathrm{f}(x)\,\mathrm{d}x$. Without any further computations, determine which estimate has a smaller percentage error. Give an explanation for your choice.

Finding Parameters from Power Series Coefficients

Maclaurin Series from Differential Equations

Substitution and Composition of Power Series

Derivatives of Reciprocal Trigonometric Functions

Accuracy of Power Series Approximations

Higher Derivatives by Implicit Differentiation

2026 TJC P1 Q5

Maclaurin Series and Power Series

Techniques of Differentiation

<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 $x^2$.

Derivatives of Natural Logarithmic Functions

Chain Rule

2025 RVHS Promo Q8

<p>It is given that&nbsp;$\mathrm{e}^y = 1 + \sin 3x$.</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</div><div class="list-item-paragraph" style="text-align: center">$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + 9 = 9\mathrm{e}^{-y}.$ </div><div class="list-item-paragraph">Hence find the series expansion of $y$ in ascending powers of $x$, up to and including the term in $x^3$, simplifying your answer.<span class="inline-marks">[5]</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, verify that the series expansion obtained in <strong>part (a)</strong> is correct.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $\mathrm{e}^y = 1 + \sin 3x$.

Implicit Differentiation

Derivatives of Trigonometric Functions

Derivatives of Exponential Functions

Maclaurin Series by Implicit Differentiation

2024 RI Promo Q6

<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

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Using the standard series expansion, expand $\ln(1+\sin x)$ where&nbsp;$0 \le x &lt; \pi$, in ascending powers of $x$, up to and including the term in $x^3$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>The first two terms of the series in part <strong>(a)</strong> are equal to the first two terms in the series expansion of $\frac{ax}{(1+bx)^2}$, where&nbsp;$a$ and $b$ are non-zero constants. Find the values of $a$ and $b$. Hence write down the third term in the series expansion of $\frac{ax}{(1+bx)^2}$.</p><span class="inline-marks">[4]</span></div></li></ol>

Binomial Expansion as a Maclaurin Series

2024 MI PRU2 P1 Q4

<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>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 ${{\left( 1+x-{{x}^{2}} \right)}^{-5}}$ up to and including the term in ${{x}^{2}}$.</p>

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

Maclaurin Series Standard Series Q2

<ol class="abc-list"><li data-list-item-id="item-1770181546354-i4tlclt5t" data-list-item-path="item-1770181546354-i4tlclt5t"><div class="list-item-paragraph">In the expansion of ${{(1+2x+3{{x}^{2}})}^{n}}$ in ascending powers of $x$, the coefficient of ${{x}^{2}}$ is $105$. Given that&nbsp;$n$ is a 	positive integer, find the value of $n$.</div></li><li data-list-item-id="item-1770181553179-qunwza7h2" data-list-item-path="item-1770181553179-qunwza7h2"><div class="list-item-paragraph">Obtain the expansion of $\frac{1-4x}{3+{{x}^{2}}}$ up to and including the term in ${{x}^{4}}$. Find the set of values of $x$ for which the 	expansion is valid.</div></li></ol>

Convergence and Validity of Power Series

Products and Quotients of Power Series

<p>It is given that&nbsp;$e^y = 1 + e^x$.</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;$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = \frac{\mathrm{d}y}{\mathrm{d}x}.$<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-content"><span class="list-item-text">Hence find the first four non-zero terms of the Maclaurin series for&nbsp;$y$.</span><span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Hence find the Maclaurin series for&nbsp;$\frac{e^{2x}}{1 + e^{2x}}$, up to and including the term in $x^3$.</span><span class="inline-marks">[3]</span></div></li></ol>

Show that \[\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = \frac{\mathrm{d}y}{\mathrm{d}x}.\]

Hence find the first four non-zero terms of the Maclaurin series for $y$.

Hence find the Maclaurin series for $\frac{e^{2x}}{1 + e^{2x}}$, up to and including the term in $x^3$.

Problems involving Calculus

Deriving Series by Differentiation and Integration

2025 YIJC P2 Q5

<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 $\displaystyle \displaystyle\int_{1}^{3} \sin[\mathrm{f}(x)] \, \mathrm{d}x$.

2025 VJC P2 Q1

<p>It is given that&nbsp;$y = \ln \sqrt{1 + \mathrm{e}^{3x}}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$2\mathrm{e}^{2y} \frac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x}$.</span><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 further differentiation of the result in part <strong>(a)</strong>, find the Maclaurin series for&nbsp;$y$, up to and including the term in $x^2$, giving the coefficients in exact form.<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-content"><span class="list-item-text">Deduce the Maclaurin series for&nbsp;$\ln \left( \frac{\sqrt{1 + \mathrm{e}^{-3x}}}{2} \right)$, up to and including the term in $x^2$.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \ln \sqrt{1 + \mathrm{e}^{3x}}$.

Show that $2\mathrm{e}^{2y} \frac{\mathrm{d}y}{\mathrm{d}x} = 3\mathrm{e}^{3x}$.

Deduce the Maclaurin series for $\ln \left( \frac{\sqrt{1 + \mathrm{e}^{-3x}}}{2} \right)$, up to and including the term in $x^2$.

2025 TJC P2 Q1

<p>It is given that&nbsp;$y = \ln \left[ \sin \left( x + \frac{\pi}{4} \right) \right]$, where&nbsp;$-\frac{\pi}{4} &lt; x &lt; \frac{3\pi}{4}$.</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;$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2 + 1 = 0$. Hence find the first four non-zero terms of the Maclaurin expansion of $y$, leaving your answer in exact form.<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"><p>Verify the result obtained in part <strong>(a)</strong> using standard series from the List of Formulae (MF27).</p><span class="inline-marks">[5]</span></div></li></ol>

It is given that $y = \ln \left[ \sin \left( x + \frac{\pi}{4} \right) \right]$, where $-\frac{\pi}{4} < x < \frac{3\pi}{4}$.

Show that $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)^2 + 1 = 0$. Hence find the first four non-zero terms of the Maclaurin expansion of $y$, leaving your answer in exact form.

Second and Higher Derivatives

2025 HCI P2 Q2

<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 $y=\ln \left( 2-{{\mathrm{e}}^{-2x}} \right)$.

Maclaurin Series by Successive Differentiation

2023 EJC Promo Q9

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

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

Maclaurin Series Standard 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

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3MDcxNzcwNDctZDRzbzBpd3hvNGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU2NDgwNiwiZXhwIjoxNzkzNTY1MTA2fQ.M1vlmSrGa3o8hLtEjHThWglkfHpQGcvgRXLm2TK2Gf4" 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 $Y$.

Small Angle Approximations

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

RI Maclaurin Series Tutorial Q3

<ol class="abc-list"><li data-list-item-id="item-1760706771310-0lbd2knc3" data-list-item-path="item-1760706771310-0lbd2knc3"><div class="list-item-paragraph">Using standard series from the List of Formulae (MF26), find the Maclaurin expansion of $\frac{1}{{{\left( 1+\cos x \right)}^{2}}}$ in ascending powers of $x$ up to and including the term in ${{x}^{4}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760706771895-x94ud5bna" data-list-item-path="item-1760706771895-x94ud5bna"><div class="list-item-paragraph">Find the set of values of $x$ for which $\frac{1}{{{\left( 1+\cos x \right)}^{2}}}$ is within $\pm \,0.5$ of the polynomial found in part <strong>(a)</strong>, where&nbsp;$0\le x &lt; \pi $.<span class="inline-marks">[3]</span></div></li></ol>

2024 HCI P2 Q5

<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 $y=\frac{1}{3+\sin 2x}$.

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