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PSLE

PSLE Foundation Mathematics

<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

Substitution and Composition of Power Series

Higher Derivatives by Implicit Differentiation

Derivatives of Exponential Functions

Maclaurin Series by Implicit Differentiation

2024 RI Promo Q6

Maclaurin Series and Power Series

Techniques of Differentiation

<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

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">In a triangle $ABC$, $AB = 2$, angle $CAB = x$ radians and angle $CBA = \frac{\pi}{6}$ radians.</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">Show that&nbsp;$AC = \frac{2}{\cos x + \sqrt{3} \sin x}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><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">\[AC \approx a + bx + cx^2,\]</div><div class="list-item-paragraph">where&nbsp;$a$, $b$ and $c$ are constants to be determined.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">It is given that&nbsp;$2xy + \ln y = \ln 3$. Show that</div><div class="list-item-paragraph" style="text-align: center">\[(2xy^2 + y)\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 4y^2\frac{\mathrm{d}y}{\mathrm{d}x} - \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 = 0.\]</div><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">[5]</span></div></li></ol>

Small Angle Approximations

Derivatives of Natural Logarithmic Functions

2025 RI P2 Q2

<p>It is given that&nbsp;$y=\mathrm{f}\left( x \right)$, where&nbsp;${{y}^{3}}+8=3xy$.</p>
<p>Find the Maclaurin series for&nbsp;$\mathrm{f}\left( x \right)$ up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[6]</span></p>
<p>Hence write down the equation of the normal at the point on the curve of $y=\mathrm{f}\left( x \right)$ at $x=0$.<span class="inline-marks">[1]</span></p>

It is given that $y=\mathrm{f}\left( x \right)$, where ${{y}^{3}}+8=3xy$.

Find the Maclaurin series for $\mathrm{f}\left( x \right)$ up to and including the term in ${{x}^{3}}$.

Hence write down the equation of the normal at the point on the curve of $y=\mathrm{f}\left( x \right)$ at $x=0$.

Maclaurin Series by Successive Differentiation

Tangents and Normals to Implicit Curves

2024 DHS P2 Q2

Applications of Differentiation

<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 $\ln \left( 1+y \right)=\tan x$.

Approximations Using Power Series

2024 CJC P1 Q6

<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>

Derivatives of Inverse Trigonometric Functions

2013 HCI P2 Q2

<ol class="i-list"><li><div class="list-item-paragraph">It is given that $\ln y=2\sin x$. Show that $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=-y\ln y+\frac{1}{y}{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the first four terms of the Maclaurin series for $y$ in ascending powers of $x$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Using appropriate expansions from the List of Formulae (MF26), verify the expansion found in part <strong>(ii)</strong>.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $x$ is sufficiently small for ${{x}^{4}}$ and higher powers of $x$ to be neglected, deduce an approximation for ${{\mathrm{e}}^{\left( 2\sin x \right)-\ln \left( \sec x \right)}}$ in ascending powers of $x$.</div><p style="text-align: right;">[2]</p></li></ol>

Maclaurin Series Using Standard Series

Products and Quotients of Power Series

2017 HCI P1 Q7

<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 $c$.

Finding Parameters from Power Series Coefficients

2022 NJC J2 CT P1 Q3

<p>The equation of a curve $C$ is given by</p>
<p style="text-align: center">${{y}^{3}}+xy-{{x}^{2}}=1$.</p>
<ol class="i-list"><li data-list-item-id="item-1758949145700-obd88ifwu" data-list-item-path="item-1758949145700-obd88ifwu"><div class="list-item-paragraph">Find the $x$- coordinate of the point at which the tangent is parallel to the $x$-axis.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1758949147590-geuz5gs2j" data-list-item-path="item-1758949147590-geuz5gs2j"><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$\left( 3{{y}^{2}}+x \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+6y{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}+2\frac{\mathrm{d}y}{\mathrm{d}x}-2=0$.</span></div><div class="list-item-paragraph">Given that the value of $\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}$ when&nbsp;$x=0$ is $\frac{20}{27}$, find the Maclaurin series for&nbsp;$y$, up to and including the term in ${{x}^{3}}$.<span class="inline-marks">[3]</span></div></li></ol>

Find the $x$- coordinate of the point at which the tangent is parallel to the $x$-axis.

Show that $\left( 3{{y}^{2}}+x \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+6y{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}+2\frac{\mathrm{d}y}{\mathrm{d}x}-2=0$.

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