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PSLE

PSLE Foundation Mathematics

<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li data-list-item-id="ri-p2-q4-2-content-items-1" data-list-item-path="ri-p2-q4-2-content-items-1"><div class="list-item-paragraph">The curve $y=\mathrm{f}(x)$ passes through the origin and has gradient given by</div><div class="list-item-paragraph">\[\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x+y}{1+xy}.\]</div><ol class="i-list"><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-1" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-1"><div class="list-item-paragraph">Show that&nbsp;$\displaystyle(1+xy)\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\left(\dfrac{\mathrm{d}y}{\mathrm{d}x}\right)^2+(y-1)\dfrac{\mathrm{d}y}{\mathrm{d}x}=1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-2" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-2"><div class="list-item-paragraph">By further differentiation of the result in part <strong>(a)(i)</strong>, show that</div><div class="list-item-paragraph">\[(1+xy)\frac{\mathrm{d}^3y}{\mathrm{d}x^3}+\mathrm{p}(x)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)\left(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)+\mathrm{q}(y)\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+r\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2=0,\]</div><div class="list-item-paragraph">where the functions $\mathrm{p}(x)$, $\mathrm{q}(y)$ and the constant $r$ are to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-1-subItems-3" data-list-item-path="ri-p2-q4-2-content-items-1/ri-p2-q4-2-content-items-1-subItems-3"><div class="list-item-paragraph">Hence find the Maclaurin series for&nbsp;$y$ up to and including the term in $x^3$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="ri-p2-q4-2-content-items-2" data-list-item-path="ri-p2-q4-2-content-items-2"><div class="list-item-paragraph">The equation</div><div class="list-item-paragraph">\[2\sqrt{2}\cos\left(x-\frac\pi4\right)+3\sin2x-3=0\]</div><div class="list-item-paragraph">has a root $\theta$, which is close to zero.</div><ol class="i-list"><li data-list-item-id="ri-p2-q4-2-content-items-2-subItems-1" data-list-item-path="ri-p2-q4-2-content-items-2/ri-p2-q4-2-content-items-2-subItems-1"><div class="list-item-paragraph">Show that&nbsp;$\theta^2+a\theta+b\approx0$, where&nbsp;$a$ and $b$ are integers to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ri-p2-q4-2-content-items-2-subItems-2" data-list-item-path="ri-p2-q4-2-content-items-2/ri-p2-q4-2-content-items-2-subItems-2"><div class="list-item-paragraph">Hence find an approximation for&nbsp;$\theta$ in surd form.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

**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

<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

Substitution and Composition of Power Series

Derivatives of Reciprocal Trigonometric Functions

Accuracy of Power Series Approximations

2026 TJC P1 Q5

<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>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 $x^2$.

Products and Quotients of Power Series

Quotient of exponential and logarithmic 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 $x^3$.

Binomial Expansion as a Maclaurin Series

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

Derivatives of Inverse Trigonometric Functions

Maclaurin Series by Successive Differentiation

Problems involving Calculus

Deriving Series by Differentiation and Integration

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 $r = 1, 2, 3$.

Derivatives of Trigonometric Functions

Logarithmic Differentiation

Derivatives of Exponential Functions

<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

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

Implicit Differentiation

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

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.



General Terms and Coefficients in Power Series

<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 $x^3$.

Math Question Bank