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PSLE

PSLE Foundation Mathematics

<p>It is given that</p>
<p style="text-align: center">\[\mathrm{f}(x) = \begin{cases} \tan \left( \frac{\pi}{4a} x \right) & \text{for}\hspace{0.5em} -a &lt; x \le a, \\ \sin \left( \frac{\pi}{2a} x \right) & \text{for}\hspace{0.5em} a &lt; x \le 3a, \end{cases}\]</p>
<p>and that&nbsp;$\mathrm{f}(x) = \mathrm{f}(x - 4a)$ for all real values of $x$, where&nbsp;$a$ is a positive real constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}(x)$ for&nbsp;$-4a \le x \le 7a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph"><p>By using the sketch in part <strong>(a)</strong>, write down an equation relating $\mathrm{f}(x)$ and $\mathrm{f}(-x)$.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">Show that&nbsp;$\int_{-2a}^{4a} \mathrm{f}(x) \,\mathrm{d}x = \frac{ka}{\pi} (1 + \ln m)$, where&nbsp;$k$ and $m$ are constants to be determined.<span class="inline-marks">[5]</span></div></li></ol>

\[\mathrm{f}(x) = \begin{cases} \tan \left( \frac{\pi}{4a} x \right) & \text{for } -a < x \le a, \\ \sin \left( \frac{\pi}{2a} x \right) & \text{for } a < x \le 3a, \end{cases}\]

and that $\mathrm{f}(x) = \mathrm{f}(x - 4a)$ for all real values of $x$, where $a$ is a positive real constant.

Sketch the graph of $y = \mathrm{f}(x)$ for $-4a \le x \le 7a$.

Show that $\int_{-2a}^{4a} \mathrm{f}(x) \,\mathrm{d}x = \frac{ka}{\pi} (1 + \ln m)$, where $k$ and $m$ are constants to be determined.

Piecewise-Defined Functions

Integrable Forms of Trigonometric Functions

Periodic Functions

Odd and Even Functions

2026 HCI P1 Q7 [Modified]

Functions

Techniques of Integration

<p>Consider $h(x)=(ax)^2\cos x,\ x\in\mathbb{R}$. It is given that&nbsp;$a$ is a real constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1783668286410-82w6cr0ks" data-list-item-path="item-1783668286410-82w6cr0ks"><div class="list-item-paragraph">Show that&nbsp;$h(x)$ is an even function.</div></li><li data-list-item-id="item-1783668286410-8qbyb8vq5" data-list-item-path="item-1783668286410-8qbyb8vq5"><div class="list-item-paragraph">It is given that&nbsp;$h(\pi)=-4$. Find the possible value(s) of $a$.</div></li></ol>

Consider $h(x)=(ax)^2\cos x,\ x\in\mathbb{R}$. It is given that $a$ is a real constant.

<ol class="abc-list"><li data-list-item-id="even-odd-q2a" data-list-item-path="even-odd-q2a"><div class="list-item-paragraph">Determine if&nbsp;$f(x)=x^4\cos x$ is an even function.</div></li><li data-list-item-id="even-odd-q2b" data-list-item-path="even-odd-q2b"><div class="list-item-paragraph">Determine if&nbsp;$g(x)=x^3\cos x$ is an odd function.</div></li></ol>

<p>A curve $C$ is defined parametrically by</p>
<p style="text-align: center">\[x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} &lt; t &lt; \frac{\pi}{4},\]</p>
<p>where&nbsp;$a$ is a positive constant.</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;$\frac{\tan t}{\sqrt{1 + \tan^2 t}} = \sin t$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">By using part <strong>(a)</strong> or otherwise, find the cartesian equation of $C$ in the form $y = \mathrm{f}(x)$, simplifying your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}(-x) = -\mathrm{f}(x)$. Hence sketch $C$.<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">Find the exact area of the region bounded by $C$, the $x$-axis and the lines $x = -A$ and $x = A$, where&nbsp;$0 &lt; A &lt; a$, leaving your answer in terms of $A$ and $a$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ is defined parametrically by

\[x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},\]

Show that $\frac{\tan t}{\sqrt{1 + \tan^2 t}} = \sin t$.

Show that $\mathrm{f}(-x) = -\mathrm{f}(x)$. Hence sketch $C$.

Find the exact area of the region bounded by $C$, the $x$-axis and the lines $x = -A$ and $x = A$, where $0 < A < a$, leaving your answer in terms of $A$ and $a$.

Area Bounded by Curves

Sketching Parametric Curves

Converting Parametric Equations to Cartesian Form

2025 DHS P1 Q6

Graphing Techniques

Applications of Integration

<p>It is given that $\mathrm{f}\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx$, where $a$, $b$ and $c$ are non-zero real constants.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $\mathrm{f}\left( -x \right)=-\mathrm{f}\left( x \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">It is given that $\mathrm{f}\left( x \right)=0$ has only one real root and one of the non-real roots is $p+q\mathbf{i}$, where $p$ and $q$ are non-zero real constants. Find, in terms of $p$ and $q$, all the other non-real roots of $\mathrm{f}\left( x \right)=0$, justifying your answers.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $\mathop{\displaystyle\int }_{0}^{3}\mathrm{f}\left( x \right)\mathrm{d}x=-5$, state the values of $\displaystyle\int_{-3}^{3}{\mathrm{f}\left( x \right)\mathrm{d}x}$ and $\displaystyle\int_{-3}^{3}{\mathrm{f}\left( \left| x \right| \right)\mathrm{d}x}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Let $a=1$ and $b=3$ .</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">By considering $f\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$, find the range of values of $c$ such that the curve with equation $y=\mathrm{f}\left( x \right)$ has 2 stationary points, showing your working clearly.</div><p style="text-align: right;">[3]</p></li></ol>

It is given that $\mathrm{f}\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx$, where $a$, $b$ and $c$ are non-zero real constants.

Modulus Functions

Parameter Conditions for Stationary Points

2024 HCI P1 Q7

Applications of Differentiation

<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/pasted/1782726703319-nxij2mb895.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI3MjY3MDMzMTktbnhpajJtYjg5NS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODQyLCJleHAiOjE3OTM1MDUxNDJ9.tiSEK4GGudprxDd_ArkAzn2ZEZA47wFARtXMU0YzjYo" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>It is given that&nbsp;$\mathrm{f}\left( x \right)={{x}^{6}}-3{{x}^{4}}-7.$ The diagram shows the curve with equation $y=\mathrm{f}\left( x \right)$and the line with equation$y=-7,$for&nbsp;$x\ge 0.$ The curve crosses the positive $x$-axis at $x=\alpha ,$and the curve and the line meet where&nbsp;$x=0$ and $x=\beta .$</p>
<ol class="i-list"><li data-list-item-id="item-1760361993163-4jowgrq6y" data-list-item-path="item-1760361993163-4jowgrq6y"><div class="list-item-paragraph">Find the value of $\alpha ,$giving your answer correct to $3$ decimal places, and find the exact value of $\beta .$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760361993881-5ooy9ogs6" data-list-item-path="item-1760361993881-5ooy9ogs6"><div class="list-item-paragraph">Evaluate $\displaystyle\displaystyle \displaystyle\int_{\beta }^{\alpha }{\mathrm{f}\left( x \right)\mathrm{d}x},$giving your answer correct to $3$ decimal places.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760361996206-upwdhlc7q" data-list-item-path="item-1760361996206-upwdhlc7q"><div class="list-item-paragraph">Find, in terms of $\sqrt{3},$the area of the finite region bounded by the curve and the line, for&nbsp;$x\ge 0.$<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760361996919-u75ajitdv" data-list-item-path="item-1760361996919-u75ajitdv"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{f}\left( x \right)=\mathrm{f}\left( -x \right).$What can be said about the six roots of the equation $\mathrm{f}\left( x \right)=0?$<span class="inline-marks">[4]</span></div></li></ol>

It is given that $\mathrm{f}\left( x \right)={{x}^{6}}-3{{x}^{4}}-7.$ The diagram shows the curve with equation $y=\mathrm{f}\left( x \right)$and the line with equation$y=-7,$for $x\ge 0.$ The curve crosses the positive $x$-axis at $x=\alpha ,$and the curve and the line meet where $x=0$ and $x=\beta .$

N2014 P1 Q7

<p>A function $\mathrm{f}\left( x \right)$ is said to be even if $\mathrm{f}\left( -x \right)=\mathrm{f}\left( x \right)$ for all $x\in {{\mathrm{D}}_{\mathrm{f}}}$.</p>
<p>Let $\mathrm{f}\left( x \right)=\frac{px+q}{x+r}$ where $p$, $q$ and $r$ are real constants, $r\ne 0$ and $x\in \mathbb{R}\mathrm{ }\!\!\backslash\!\!\mathrm{ }\left\{ \pm r \right\}$.</p>
<p>Prove that if $\mathrm{f}\left( x \right)$ is an even function on the domain $\mathbb{R}\mathrm{ }\!\!\backslash\!\!\mathrm{ }\left\{ \pm r \right\}$, then $\mathrm{f}\left( x \right)=k$ for some real constant $k$ which is to be determined in terms of $p$, $q$ or $r$.</p>

A function $\mathrm{f}\left( x \right)$ is said to be even if $\mathrm{f}\left( -x \right)=\mathrm{f}\left( x \right)$ for all $x\in {{\mathrm{D}}_{\mathrm{f}}}$.

Let $\mathrm{f}\left( x \right)=\frac{px+q}{x+r}$ where $p$, $q$ and $r$ are real constants, $r\ne 0$ and $x\in \mathbb{R}\mathrm{ }\!\!\backslash\!\!\mathrm{ }\left\{ \pm r \right\}$.

Prove that if $\mathrm{f}\left( x \right)$ is an even function on the domain $\mathbb{R}\mathrm{ }\!\!\backslash\!\!\mathrm{ }\left\{ \pm r \right\}$, then $\mathrm{f}\left( x \right)=k$ for some real constant $k$ which is to be determined in terms of $p$, $q$ or $r$.

Proving Property Of An Even Function

<p>The curve $y=\mathrm{f}\left( x \right)$ for $0\le x<3$ is shown below. It is given that $\mathrm{f}$ is an odd function, $\mathrm{f}\left( -x \right)=-\mathrm{f}\left( x \right)$, and $\mathrm{f}\left( x+6 \right)=\mathrm{f}\left( x \right)$ for all real values of $x$. Sketch the curve of $y=\mathrm{f}\left( x \right)$ for $-3<x\le 6$.</p><p style="text-align: right;">[4]</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759052558917-69mdilb66ea.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkwNTI1NTg5MTctNjltZGlsYjY2ZWEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg0MiwiZXhwIjoxNzkzNTA1MTQyfQ.ds1zqRzjXP2LEBwJ6KtXXTcP8mH3-eTFhm5tsI-5g5M" width="400" class="aligncenter" /></figure>

The curve $y=\mathrm{f}\left( x \right)$ for $0\le x<3$ is shown below. It is given that $\mathrm{f}$ is an odd function, $\mathrm{f}\left( -x \right)=-\mathrm{f}\left( x \right)$, and $\mathrm{f}\left( x+6 \right)=\mathrm{f}\left( x \right)$ for all real values of $x$. Sketch the curve of $y=\mathrm{f}\left( x \right)$ for $-3<x\le 6$.

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