A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

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

<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

<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 $y={{e}^{{{\tan }^{-1}}4x}}$.</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 $b$.

Maclaurin Series Using Standard Series

Derivatives of Inverse Trigonometric Functions

Maclaurin Series by Successive Differentiation

Second and Higher Derivatives

Derivatives of Exponential Functions

Chain Rule

2025 DHS Promo Q6

<p>In this question you may use expansions from the List of Formulae (MF26).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the first four non-zero terms of the Maclaurin series of ${{\mathrm{e}}^{-x}}\left( 1+\cos 3x \right)$, in ascending powers of $x$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">It is given that the first three terms of this series are equal to the first three terms in the series expansion, in ascending powers of $x$, of $\frac{1}{a+bx}+c{{x}^{2}}$. Find the values of $a$, $b$ and $c$.</div><p style="text-align: right;">[4]</p></li></ol>

In this question you may use expansions from the List of Formulae (MF26).

2023 RI Promo Q5

<ol class="i-list"><li><div class="list-item-paragraph">Given that $y=\tan \left( {{\mathrm{e}}^{2x}}-1 \right)$, show that $\frac{\mathrm{d}y}{\mathrm{d}x}=k{{\mathrm{e}}^{2x}}\left( 1+{{y}^{2}} \right)$, where $k$ is to be found. Hence find the values of $\frac{\mathrm{d}y}{\mathrm{d}x}$, $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}$ and $\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}$ when $x=0$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Write down the first three non-zero terms in the Maclaurin series for $\tan \left( {{\mathrm{e}}^{2x}}-1 \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">The first two non-zero terms in the Maclaurin series for $\tan \left( {{\mathrm{e}}^{2x}}-1 \right)$ are equal to the first two non-zero terms in the series expansion of ${{\mathrm{e}}^{ax}}\ln \left( 1+nx \right)$. By using appropriate expansions from the List of Formulae (MF26), find the constants $a$ and $n$. Hence find the third non-zero term of the series expansion of ${{\mathrm{e}}^{ax}}\ln \left( 1+nx \right)$ for these values of $a$ and $n$.</div><p style="text-align: right;">[4]</p></li></ol>

Derivatives of Trigonometric Functions

RI Maclaurin Series Tutorial Q4

<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathrm{f}\left( x \right)={{\sec }^{2}}x$, find $\mathrm{f}'\mathrm{ }\left( x \right)$ and $\mathrm{f}''\mathrm{ }\left( x \right)$. Hence, find the Maclaurin series for&nbsp;$\mathrm{f}\left( x \right)$, up to and including the term in ${x}^{2}$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Using standard series from the List of Formulae (MF27), expand ${{\left( 1+ax \right)}^{-\,4}}$ as far as the term in ${{x}^{2}}$, where&nbsp;$a$ is a non- zero constant. Hence find the value of $a$ for which the coefficients of the $x$ and ${{x}^{2}}$ terms in the expansion are equal.<span class="inline-marks">[3]</span></div></li></ol>

2023 MI PU3 P2 Q2

<div class="graph-block" data-graph-config="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" data-width="420" data-height="200" style="width:420px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<p>In the triangle $ABC$, $AC=1$, angle $BAC=x$ radians and angle $ACB=\frac{1}{6}\pi $ radians (see diagram).</p>
<ol class="abc-list"><li data-list-item-id="item-1760349711336-rlv7rkmgn" data-list-item-path="item-1760349711336-rlv7rkmgn"><div class="list-item-paragraph">Show that&nbsp;$AB=\frac{1}{\cos x+\sqrt{3}\sin x}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760349724409-q7gyr6zn1" data-list-item-path="item-1760349724409-q7gyr6zn1"><div class="list-item-paragraph">Given that&nbsp;$x$ is a sufficiently small angle, show that</div><div class="list-item-paragraph" style="text-align: center">$AB\approx 1+ax+b{{x}^{2}}$,</div><div class="list-item-paragraph">for constants $a$ and $b$ to be determined exactly.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760349743720-yntyhxtg9" data-list-item-path="item-1760349743720-yntyhxtg9"><div class="list-item-paragraph"><p>The first two terms in the Maclaurin series for&nbsp;$\ln \left( px+q \right)$ are equal to the first two terms in the series expansion in part <strong>(b)</strong>. Using standard series from the List of Formulae <strong>(MF26)</strong>, find the exact values of $p$ and $q$.</p><span class="inline-marks">[4]</span></div></li></ol>

In the triangle $ABC$, $AC=1$, angle $BAC=x$ radians and angle $ACB=\frac{1}{6}\pi $ radians (see diagram).

Small Angle Approximations

2023 YIJC P2 Q5

<p>A curve $C$ has equation $y=\mathrm{f}\left( x \right)$. The equation of the tangent to the curve $C$ at the point where $x=0$ is given by $2x-ay=3$ where $a$ is a positive constant.</p>
<p>It is also given that $y=\mathrm{f}\left( x \right)$satisfies the equation $\left( 1+2x \right)\frac{{{\mathrm{d}}^{2}}y}{{{\mathrm{d}}^{2}}x}+y\frac{\mathrm{d}y}{\mathrm{d}x}=0$ and that the third term in the Maclaurin’s expansion of $\mathrm{f}\left( x \right)$is $\frac{1}{3}{{x}^{2}}$.</p>
<p>Find the value of $a$. Hence, find the Maclaurin’s series for $\mathrm{f}\left( x \right)$in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.</p><p style="text-align: right;">[7]</p>

A curve $C$ has equation $y=\mathrm{f}\left( x \right)$. The equation of the tangent to the curve $C$ at the point where $x=0$ is given by $2x-ay=3$ where $a$ is a positive constant.

It is also given that $y=\mathrm{f}\left( x \right)$satisfies the equation $\left( 1+2x \right)\frac{{{\mathrm{d}}^{2}}y}{{{\mathrm{d}}^{2}}x}+y\frac{\mathrm{d}y}{\mathrm{d}x}=0$ and that the third term in the Maclaurin’s expansion of $\mathrm{f}\left( x \right)$is $\frac{1}{3}{{x}^{2}}$.

Find the value of $a$. Hence, find the Maclaurin’s series for $\mathrm{f}\left( x \right)$in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.

2017 AJC P1 Q5

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Using standard series from the List of Formulae (MF26), expand $\frac{\cos 3x}{4-x}$ as far as the term in ${{x}^{3}}$. Give the coefficients as exact fractions in their simplest form.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">It is given that the third and fourth terms found in part (i) are equal to the third and fourth terms in the series expansion of ${{\left( a+bx \right)}^{5}}$, in ascending powers of $x$ respectively. Find the values of the constants $a$ and $b$.</span></div><p style="text-align: right;">[4]</p></li></ol>

Using standard series from the List of Formulae (MF26), expand $\frac{\cos 3x}{4-x}$ as far as the term in ${{x}^{3}}$. Give the coefficients as exact fractions in their simplest form.

It is given that the third and fourth terms found in part (i) are equal to the third and fourth terms in the series expansion of ${{\left( a+bx \right)}^{5}}$, in ascending powers of $x$ respectively. Find the values of the constants $a$ and $b$.

2021 RI Promo Q5

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\frac{p{{x}^{2}}+\left( 4p-q \right)x+\left( 4p+q \right)}{\left( 1-x \right){{\left( 2+x \right)}^{2}}}=\frac{p}{1-x}+\frac{q}{{{\left( 2+x \right)}^{2}}}$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the values of $p$, $q$ and $r$ such that the series expansion of</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{p{{x}^{2}}+\left( 4p-q \right)x+\left( 4p+q \right)}{\left( 1-x \right){{\left( 2+x \right)}^{2}}}+r\ln \left( 1-3x \right)$</div><div class="list-item-paragraph">up to and including the term in ${{x}^{2}}$ is $11-3x+{{x}^{2}}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the range of the values of $x$ for which the expansion is valid.</span></div><p style="text-align: right;">[2]</p></li></ol>

Show that $\frac{p{{x}^{2}}+\left( 4p-q \right)x+\left( 4p+q \right)}{\left( 1-x \right){{\left( 2+x \right)}^{2}}}=\frac{p}{1-x}+\frac{q}{{{\left( 2+x \right)}^{2}}}$.

Find the values of $p$, $q$ and $r$ such that the series expansion of

Find the range of the values of $x$ for which the expansion is valid.

Partial Fractions

2019 NJC P1 Q4

Techniques of Integration

<ol class="abc-list"><li data-list-item-id="item-1759215319532-l5rfxu6ty" data-list-item-path="item-1759215319532-l5rfxu6ty"><div class="list-item-paragraph">Expand ${{\left( b-\frac{x}{2} \right)}^{n}}$ in ascending powers of $x$, where&nbsp;$b$ is a positive constant and $n$ is a negative constant, up to and including the term in ${{x}^{2}}$.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">It is given that the coefficient of $x$ is four times the coefficient of ${{x}^{2}}$ and the constant term in the expansion is $\frac{1}{2}$. Find the integer values of $b$ and $n$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759215320139-999u6fsuj" data-list-item-path="item-1759215320139-999u6fsuj"><ol class="i-list"><li data-list-item-id="subitem-1759215344342-kg0sbhw0v" data-list-item-path="item-1759215320139-999u6fsuj/subitem-1759215344342-kg0sbhw0v"><div class="list-item-paragraph">Explain why it is not possible to obtain a Maclaurin series for&nbsp;$\ln \left( 2{{x}^{2}} \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1759215362042-cxfoul2s3" data-list-item-path="item-1759215320139-999u6fsuj/subitem-1759215362042-cxfoul2s3"><div class="list-item-content"><span class="list-item-text">A Taylor series is an expansion of a real function $\mathrm{f}\left( x \right)$ about a point $x=a$ and it is defined by</span></div><div class="list-item-paragraph" style="text-align: center">$\mathrm{f}\left( x \right)=\mathrm{f}\left( a \right)+\mathrm{f}'\left( a \right)\left( x-a \right)+\frac{\mathrm{f}''\left( a \right)}{2!}{{\left( x-a \right)}^{2}}+...+\frac{{{\mathrm{f}}^{\left( n \right)}}\left( a \right)}{n!}{{\left( x-a \right)}^{n}}+...$</div><div class="list-item-paragraph">where&nbsp;${{\mathrm{f}}^{\left( n \right)}}\left( a \right)$ is the value of the $n$th derivative of $\mathrm{f}\left( x \right)$ when&nbsp;$x=a$.</div><div class="list-item-paragraph">Find the first three exact non-zero terms of the Taylor series for&nbsp;$\ln \left( 2{{x}^{2}} \right)$ about the point $x=2$.</div><div class="list-item-paragraph">[You need not simplify your answer.]<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Expand ${{\left( b-\frac{x}{2} \right)}^{n}}$ in ascending powers of $x$, where $b$ is a positive constant and $n$ is a negative constant, up to and including the term in ${{x}^{2}}$.

Derivatives of Natural Logarithmic Functions

Existence of a Maclaurin Expansion

Taylor Expansions about a Nonzero Point

2021 TMJC P2 Q4

<p>It is given that ${{\tan }^{-1}}y=a{{\tan }^{-1}}x+b$, for constants $a$ and $b$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\left( 1+{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=2\left( ay-x \right)\frac{\mathrm{d}y}{\mathrm{d}x}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Suppose that the Maclaurin series for $y$ is $1+cx+8{{x}^{2}}+20{{x}^{3}}+...$. Find the exact values of $a$,$b$ and $c$.</span></div><p style="text-align: right;">[5]</p></li></ol>

It is given that ${{\tan }^{-1}}y=a{{\tan }^{-1}}x+b$, for constants $a$ and $b$.

Show that $\left( 1+{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=2\left( ay-x \right)\frac{\mathrm{d}y}{\mathrm{d}x}$.

Suppose that the Maclaurin series for $y$ is $1+cx+8{{x}^{2}}+20{{x}^{3}}+...$. Find the exact values of $a$,$b$ and $c$.

Math Question Bank