A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

<p>Using standard series from the List of Formulae (MF27), expand $\frac{e^{2x}-1}{\ln(1+2x)}$, for&nbsp;$x\ne0$, up to and including the term in $x^2$.<span class="inline-marks">[5]</span></p>

Using standard series from the List of Formulae (MF27), expand $\frac{e^{2x}-1}{\ln(1+2x)}$, for $x\ne0$, up to and including the term in $x^2$.

Maclaurin Series and Power Series - General

Quotient of exponential and logarithmic series

Maclaurin Series and Power Series

<p>Using standard series from the List of Formulae (MF27), expand $e^{-2x}\sqrt{1+3x}$ up to and including the term in $x^3$.<span class="inline-marks">[3]</span></p>

Using standard series from the List of Formulae (MF27), expand $e^{-2x}\sqrt{1+3x}$ up to and including the term in $x^3$.

Product of exponential and binomial series

<p>It is given that&nbsp;$y = \sin^{-1} px$, where&nbsp;$p$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph"><p>By further differentiation of the result in part <strong>(a)</strong> find, in terms of $p$, the Maclaurin series for&nbsp;$y$ up to and including the term in $x^3$.</p><span class="inline-marks">[5]</span></div></li><li data-list-item-id="imported-item-block-1-item-2" data-list-item-path="imported-item-block-1-item-2"><div class="list-item-paragraph"><p>Use your answer to part <strong>(b)</strong> to find the series expansion of $\frac{1}{\sqrt{1 - 9x^2}}$, up to and including the term in $x^2$.</p><span class="inline-marks">[3]</span></div></li></ol>

It is given that $y = \sin^{-1} px$, where $p$ is a constant.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = p \sec y$.

Maclaurin Series by Successive Differentiation

Problems involving Calculus

N2025 P1 Q9

<p>It is given that&nbsp;$y = \mathrm{e}^{\tan x}$, for&nbsp;$-\frac{\pi}{2} &lt; x &lt; \frac{\pi}{2}$, and $y_r = \frac{\mathrm{d}^r y}{\mathrm{d}x^r}$, for&nbsp;$r = 1, 2, 3$.</p>
<ol class="abc-list"><li data-list-item-id="part-a" data-list-item-path="part-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{y_1}{y} = 1 + \tan^2 x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="part-b" data-list-item-path="part-b"><div class="list-item-paragraph">By expressing $\ln y_1$ in terms of $\tan x$, show that&nbsp;$\frac{y_2}{y_1} = (1 + \tan x)^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="part-c" data-list-item-path="part-c"><div class="list-item-paragraph">Find $\frac{y_3}{y_2}$ in terms of $\tan x$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="part-d" data-list-item-path="part-d"><div class="list-item-paragraph">Hence obtain the Maclaurin series for&nbsp;$\mathrm{e}^{\tan x}$ in ascending powers of $x$ up to and including the term in $x^3$.<span class="inline-marks">[2]</span></div></li></ol>

It is given that $y = \mathrm{e}^{\tan x}$, for $-\frac{\pi}{2} < x < \frac{\pi}{2}$, and $y_r = \frac{\mathrm{d}^r y}{\mathrm{d}x^r}$, for $r = 1, 2, 3$.

<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>

2024 VJC Promo Q9

<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 $\mathrm{e}^y = 1 + \sin 3x$.

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>

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>

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 ${{e}^{1-x}}\ln \left( 5+2x \right)$ 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 ${{x}^{2}}$.	

Maclaurin Series Standard Series Q4

<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>

<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>Express $\mathrm{f}(x)=\frac{16-9x}{(1-3x)(4+x)}$ in partial fractions and hence obtain $\mathrm{f}(x)$ as a series in ascending powers of $x$, giving the first $3$ non-zero terms of this expansion. Find the coefficient of${{x}^{r}}$ in the expansion.<br><br></p>

Express $\mathrm{f}(x)=\frac{16-9x}{(1-3x)(4+x)}$ in partial fractions and hence obtain $\mathrm{f}(x)$ as a series in ascending powers of $x$, giving the first $3$ non-zero terms of this expansion. Find the coefficient of${{x}^{r}}$ in the expansion.



<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

<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 $x^3$.

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;$\mathrm{f}(\theta) = a\theta^2 + b\theta + c$, where&nbsp;$a$, $b$ and $c$ are real constants. The curve with equation $y = \mathrm{f}(\theta)$ passes through the point with coordinates $(0, 1)$ and has a turning point at $\left( -\frac{\sqrt{3}}{3}, \frac{1}{2} \right)$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the exact values of $a$, $b$ and $c$.<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">In a triangle $ABC$, $AB = 1$, $AC = \sqrt{3}$ and angle $BAC = \theta + \frac{\pi}{6}$ radians. Given that&nbsp;$\theta$ is sufficiently small for&nbsp;$\theta^3$ and higher powers of $\theta$ to be neglected, show that&nbsp;$BC^2 \approx \mathrm{f}(\theta)$.<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, show that&nbsp;$BC \approx 1 + \frac{\sqrt{3}}{2}\theta + \frac{3}{8}\theta^2$.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $\mathrm{f}(\theta) = a\theta^2 + b\theta + c$, where $a$, $b$ and $c$ are real constants. The curve with equation $y = \mathrm{f}(\theta)$ passes through the point with coordinates $(0, 1)$ and has a turning point at $\left( -\frac{\sqrt{3}}{3}, \frac{1}{2} \right)$.

Find the exact values of $a$, $b$ and $c$.

In a triangle $ABC$, $AB = 1$, $AC = \sqrt{3}$ and angle $BAC = \theta + \frac{\pi}{6}$ radians. Given that $\theta$ is sufficiently small for $\theta^3$ and higher powers of $\theta$ to be neglected, show that $BC^2 \approx \mathrm{f}(\theta)$.

Hence, show that $BC \approx 1 + \frac{\sqrt{3}}{2}\theta + \frac{3}{8}\theta^2$.

2025 TMJC P2 Q2

<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 $x^2$.

Math Question Bank