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>Given&nbsp;$h(x)=1-|x-2|$, where&nbsp;$0\le x\le4$, and $H(x)=\dfrac{1}{h(x)}$, the range of $H(x)$ is the set&nbsp;$R$ such that&nbsp;$R=\{y:y\ge a\text{ or}\hspace{0.5em}y\le b\}$ where&nbsp;$a,b\in\mathbb Z$.</p>
<p>Determine the values of $a$ and $b$.</p>

Given $h(x)=1-|x-2|$, where $0\le x\le4$, and $H(x)=\dfrac{1}{h(x)}$, the range of $H(x)$ is the set $R$ such that $R=\{y:y\ge a\text{ or }y\le b\}$ where $a,b\in\mathbb Z$.

Modulus Functions

Range of a Function

Reciprocal Graph, $y=\frac{1}{f(x)}$

Graphs of Modulus Function

Two branches in a reciprocal modulus range

Functions

Modulus Function

Transformations of Graphs

<p>For&nbsp;$-2\le x\le3$, define $p(x)=|2x-1|-3$ and $q(x)=\dfrac{1}{p(x)}$ wherever $p(x)\ne0$. The range of $q$ is $\{y:y\le a\text{ or}\hspace{0.5em}y\ge b\}$. Find the exact values of $a$ and $b$.<span class="inline-marks">[4]</span></p>

For $-2\le x\le3$, define $p(x)=|2x-1|-3$ and $q(x)=\dfrac{1}{p(x)}$ wherever $p(x)\ne0$. The range of $q$ is $\{y:y\le a\text{ or }y\ge b\}$. Find the exact values of $a$ and $b$.

Reciprocal range of a stretched modulus trough

<ol class="abc-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0"><div class="list-item-paragraph">The graphs of $y=2|5x-p|$ and $y=qx$, where&nbsp;$p,q&gt;0$, intersect at only one point.</div><ol class="i-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-0"><div class="list-item-paragraph"><p>On a single diagram, sketch the two graphs.</p><p>State the range of values of $q$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-1" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-1"><div class="list-item-paragraph"><p>Solve $2|5x-p|&lt;10x$. Deduce the solution set of the inequality $\left|\frac{10}{x}-2p\right|&lt;\frac{10}{x}$.</p><p>Leave both answers in terms of $p$.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0"><div class="list-item-paragraph"><p>The diagram shows a sketch of the curve $y=2\mathrm{f}(-x-1)$.</p><p>The curve has asymptotes with equation $x=-1$, $x=1$ and $y=0$.</p><p>The curve has a stationary point of inflexion at $(0,0)$, a maximum point at $(-2,3)$ and cuts the $x$-axis at $(-1.5,0)$.</p></div><div class="graph-block" data-graph-config="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" data-width="460" data-height="360" style="width:460px;height:360px;max-width:100%;display:block;margin:0 auto;"></div><div class="list-item-paragraph">On separate diagrams, sketch the graphs of</div><ol class="i-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-0"><div class="list-item-paragraph">$y=\frac{1}{2\mathrm{f}(-x-1)}$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-1" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-1"><div class="list-item-paragraph"><p>$y=\mathrm{f}(x)$,</p><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content">stating clearly the equation of any asymptotes, and the coordinates of any turning points and axial intercepts.</div></li></ol>

The graphs of $y=2|5x-p|$ and $y=qx$, where $p,q>0$, intersect at only one point.

Parameter Conditions for Graphs and Equations

Inequalities involving Substitution

Graphical Method

Recovering Original Graphs from Transformations

Inequalities involving Modulus Functions

2025 DHS Promo Q11

Graphing Techniques

Inequalities

<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch the curve with equation $y=\left|\frac{4}{p-x}\right|$, where&nbsp;$p$ is a positive constant. State, in terms of $p$, the equations of the asymptotes and the coordinates of any intersections with the coordinate axes. On the same diagram, sketch the line with equation $y=q(x-p)$, where&nbsp;$q$ is a positive constant.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find, in terms of $p$ and $q$, the root of the equation $\left|\frac{4}{p-x}\right|=q(x-p)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence solve the inequality $\left|\frac{4}{p-x}\right|&gt;q(x-p)$.<span class="inline-marks">[2]</span></div></li></ol>

Sketching Rational Functions

Inequalities involving Rational Functions

Asymptotes and Discontinuities

Modulus Rational Function and Linear Intersection

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\right|, \quad x \in \mathbb{R}, \quad x \neq 3.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \mathrm{f}(x)$, giving the equations of any asymptotes and the coordinates of the points where the curve meets the axes.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence state the range of $\mathrm{f}$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why the function $\mathrm{f}^2$ does not exist.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The domain of $\mathrm{f}$ is further restricted to $x \leqslant a$, where&nbsp;$a$ is a constant.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">State the greatest value of $a$ such that the function $\mathrm{f}^{-1}$ exists.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $\mathrm{f}^{-1}(x)$ and state its domain.</span><span class="inline-marks">[4]</span></div></li></ol>

The function $\mathrm{f}$ is defined by

\[\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\right|, \quad x \in \mathbb{R}, \quad x \neq 3.\]

Sketch the graph of $y = \mathrm{f}(x)$, giving the equations of any asymptotes and the coordinates of the points where the curve meets the axes.

Explain why the function $\mathrm{f}^2$ does not exist.

The domain of $\mathrm{f}$ is further restricted to $x \leqslant a$, where $a$ is a constant.

State the greatest value of $a$ such that the function $\mathrm{f}^{-1}$ exists.

Hence find $\mathrm{f}^{-1}(x)$ and state its domain.

Existence of Composite Functions

Restricting a Domain to Obtain an Inverse

Inverse Functions

N2023 P1 Q7

<p>Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$y = 2x^3 - 5x^2 + x - 2$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = 4x^2 e^x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{\ln x}{2x}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = |2x - x^2|$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$-1 &lt; a &lt; 0$ and $0 &lt; b &lt; 1$;</span></div></li><li><div class="list-item-content"><span class="list-item-text">$a &lt; -1$ and $b &gt; 1$.</span></div></li></ol></li></ol>
<p>What could be said about the graph if&nbsp;$ab + 1 = 0$?</p>

Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.

$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$

$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by

What could be said about the graph if $ab + 1 = 0$?

Sketching Exponential Functions and Solving Equations Graphically

Sketching Logarithmic Functions and Solving Equations Graphically

Polynomial Functions and Key Features

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">On the same diagram, sketch the graphs of $y = |x - a|$ and $y = \frac{1}{|x - a|}$, where&nbsp;$a$ is a real constant such that&nbsp;$a &gt; 1$. You should show clearly the equations of any asymptotes and axial intercepts of both graphs.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, solve the inequality $|x - a| &gt; \frac{1}{|x - a|}$.</span><span class="inline-marks">[3]</span></div></li></ol>

On the same diagram, sketch the graphs of $y = |x - a|$ and $y = \frac{1}{|x - a|}$, where $a$ is a real constant such that $a > 1$. You should show clearly the equations of any asymptotes and axial intercepts of both graphs.

Hence, or otherwise, solve the inequality $|x - a| > \frac{1}{|x - a|}$.

2025 TMJC P1 Q1

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The graph of $y=\mathrm{f}(x)$ has turning points at $A(-a, 0)$, $B(a, b)$ and $C(c, -b)$ and intersects the $x$-axis at the points $D(d, 0)$ and $E(e, 0)$. The graph of $y=\mathrm{f}(x)$ is as shown in the diagram below.<span class="inline-marks">[3]</span></div></li></ol>
<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/1769223141925-ir1uwhrzoth.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkyMjMxNDE5MjUtaXIxdXdocnpvdGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTIxMCwiZXhwIjoxNzkyODM5NTEwfQ.zWXvr_j1aEfo3c-nTeCUIScNAixSMlV7uc_Jp_lDM_c" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Sketch the graph of $y=\mathrm{f}(a-x)$, labelling the coordinates of the corresponding points of $A$, $B$, $C$, $D$, and $E$ clearly.<span class="inline-marks">[3]</span></p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The graph of $y=\mathrm{g}(x)$ has asymptotes $y=0$, $y=-m$ and $x=-n$, and intersects the $y$-axis at $P(0, p)$, where&nbsp;$p &lt; m$. The graph of $y=\mathrm{g}(x)$ is as shown in the diagram below.</div></li></ol>
<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/1769223158996-vh24fdfkh2.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkyMjMxNTg5OTYtdmgyNGZkZmtoMi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MjEwLCJleHAiOjE3OTI4Mzk1MTB9.W7lh7BjDz81ECQX179NaSW2ff3xze77ga0keaekC5fY" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Sketch on separate diagrams the graphs of</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y=|\mathrm{g}(x)|$ and<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">$y=\mathrm{g}(|x|)$,<span class="inline-marks">[2]</span></div></li></ol>
<p>labelling clearly the asymptote(s) and coordinates of $P$ on each graph.</p>

The graph of $y=\mathrm{f}(x)$ has turning points at $A(-a, 0)$, $B(a, b)$ and $C(c, -b)$ and intersects the $x$-axis at the points $D(d, 0)$ and $E(e, 0)$. The graph of $y=\mathrm{f}(x)$ is as shown in the diagram below.

Sketch the graph of $y=\mathrm{f}(a-x)$, labelling the coordinates of the corresponding points of $A$, $B$, $C$, $D$, and $E$ clearly.

The graph of $y=\mathrm{g}(x)$ has asymptotes $y=0$, $y=-m$ and $x=-n$, and intersects the $y$-axis at $P(0, p)$, where $p < m$. The graph of $y=\mathrm{g}(x)$ is as shown in the diagram below.

Sketch on separate diagrams the graphs of

labelling clearly the asymptote(s) and coordinates of $P$ on each graph.

Graph Transformations of $f(ax+b)$

Sketching Graphs of $f(\lvert x\rvert)$

2025 NYJC P1 Q4

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find, in terms of $a$, the roots of the equation $\frac{1}{(x - a)^2} = |x - a|$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">On the same axes, sketch the curves with equations $y = \frac{1}{(x - a)^2}$ and $y = |x - a|$, where&nbsp;$a &gt; 1$.</span></div></li></ol>
<p>Hence solve the inequality $\frac{1}{(x - a)^2} &gt; |x - a|$.<span class="inline-marks">[3]</span></p>

Find, in terms of $a$, the roots of the equation $\frac{1}{(x - a)^2} = |x - a|$.

On the same axes, sketch the curves with equations $y = \frac{1}{(x - a)^2}$ and $y = |x - a|$, where $a > 1$.

Hence solve the inequality $\frac{1}{(x - a)^2} > |x - a|$.

2025 JPJC P1 Q2

<p>The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below.</p>
<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/1768702377706-0fn6fgzv11j7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MDIzNzc3MDYtMGZuNmZnenYxMWo3LnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI4MzkyMTAsImV4cCI6MTc5MjgzOTUxMH0.RM9sWS2ZxWWR7GM5nX5eljz5Pz3WmNTlJRhceWrgZpY" style="max-width: min(800px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">State the nature of all turning point(s) of the graph of $y=\mathrm{f}(x)$.<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-paragraph">State the range of values of $x$ where&nbsp;$\mathrm{f}$ is decreasing.<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">Sketch the graph of $y=\mathrm{f}(x)$ indicating clearly the equations of the asymptotes, coordinates of the turning point(s) and the intersections with the axes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">On the copy of the graph of $y=|\mathrm{f}(x)|$ in the Printed Answer Booklet, sketch and label a line $y=kx+3k$, where&nbsp;$k$ is a constant. Hence state the range of values of $k$ for which there is no real solution to the equation $|\mathrm{f}(x)|=kx+3k$.<span class="inline-marks">[2]</span></div></li></ol>

The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below.

State the nature of all turning point(s) of the graph of $y=\mathrm{f}(x)$.

State the range of values of $x$ where $\mathrm{f}$ is decreasing.

Sketch the graph of $y=\mathrm{f}(x)$ indicating clearly the equations of the asymptotes, coordinates of the turning point(s) and the intersections with the axes.

On the copy of the graph of $y=|\mathrm{f}(x)|$ in the Printed Answer Booklet, sketch and label a line $y=kx+3k$, where $k$ is a constant. Hence state the range of values of $k$ for which there is no real solution to the equation $|\mathrm{f}(x)|=kx+3k$.

Interpreting Gradients and Concavity

Increasing and Decreasing Functions

Stationary Points and Points of Inflection

2025 DHS P1 Q5 

Applications of Differentiation

<ol class="abc-list"><li><div class="list-item-paragraph">Solve the following integral.</div><ol class="i-list"><li><div class="list-item-paragraph">$\displaystyle \displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}} \, \mathrm{d}x$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$\displaystyle \displaystyle\int x \ln(2-x^2) \, \mathrm{d}x$, where&nbsp;$-\sqrt{2} &lt; x &lt; \ \sqrt{2}$.<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">A function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \mathrm{e}^x - 2$.</div><ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y = |\mathrm{f}(x)|$, indicating clearly the equation(s) of asymptote(s), if any.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find $\displaystyle \displaystyle\int_0^{\ln 3} |\mathrm{f}(x)| \, \mathrm{d}x$, giving your answer in exact form.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Integration by Splitting the Numerator

Integration by Parts

Area Bounded by Curves

Integration by Completing the Square

Standard Integrals with Quadratic Denominators

2025 ASRJC P2 Q5

Applications of Integration

Techniques of Integration

<div class="graph-block" data-graph-config="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" data-width="401" data-height="290" style="width:401px;height:290px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Sketch $y = -f\left( \frac{1}{2}x \right)$</p>
<p>Sketch $y = \left| f(3 - x) \right|$</p>
<p>Sketch $y = f\left( 1 - |x| \right)$</p>

Sketch $y = -f\left( \frac{1}{2}x \right)$

Composite Transformations

<ol class="abc-list"><li data-list-item-id="item-1761383465097-lhs8n3c5h" data-list-item-path="item-1761383465097-lhs8n3c5h"><div class="list-item-paragraph">On the same diagram, sketch the graphs $y=\left| 2x-1 \right|$ and $y={{x}^{2}}-4x-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383465907-xaflj7mr2" data-list-item-path="item-1761383465907-xaflj7mr2"><div class="list-item-paragraph">Hence find the set of values of $x$ for which $\left| 2x-1 \right|&lt;{{x}^{2}}-4x-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383466491-iqg4ujtg8" data-list-item-path="item-1761383466491-iqg4ujtg8"><div class="list-item-paragraph">Without using a calculator, find the range of values of $k$ for which $\left| 2x-1 \right|={{x}^{2}}-4x-2+k$ has only positive real roots.<span class="inline-marks">[4]</span></div></li></ol>

2025 YIJC P1 Q7

<p>Functions $\mathrm{f}$ and $\mathrm{g}$ are defined as:</p>
<p style="text-align: center">${\mathrm{f}(x) = 4a^2 - x^2,\quad x\in\mathbb{R},}$<br>${\mathrm{g}(x) = \left\vert{}4x - a\right\vert{},\quad x\in\mathbb{R},}$</p>
<p>where&nbsp;$a$ is a constant such that&nbsp;$a\ge 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-1761381825391-laso2kvsk" data-list-item-path="item-1761381825391-laso2kvsk"><div class="list-item-paragraph">Sketch on the same diagram the graph of $\mathrm{f}\left( x \right)$ and the graph of $\mathrm{g}\left( x \right)$. The sketch must include the coordinates of any points where each graph meets the coordinate axes and the exact $x$ coordinates of intersections in terms of $a$.</div></li><li data-list-item-id="item-1761381826051-7bvnu9kpd" data-list-item-path="item-1761381826051-7bvnu9kpd"><div class="list-item-paragraph">Find, in exact form where appropriate, the solutions of the equation:</div><div class="list-item-paragraph" style="text-align: center">$4-{{x}^{2}}=\left| 4x-1 \right|$.</div></li></ol>

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined as:

${\mathrm{f}(x) = 4a^2 - x^2,\quad x\in\mathbb{R},}$
${\mathrm{g}(x) = \left\vert{}4x - a\right\vert{},\quad x\in\mathbb{R},}$

where $a$ is a constant such that $a\ge 1$.

Modulus Graphs and Inequalities Q1

<ol class="i-list"><li><div class="list-item-paragraph">State a sequence of three transformations which transform the graph with equation ${{x}^{2}}+{{y}^{2}}=1$ to the graph with equation $\frac{{{(x+1)}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch, on the same diagram, the graphs of $y=\left| 2x-1 \right|$ and $9{{\left( x+1 \right)}^{2}}+4{{y}^{2}}=36$ for&nbsp;$y\ge 0$. State clearly the axial intercepts of both the graphs and the coordinates of the intersections between the two graphs.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Hence solve the inequality $\sqrt{36-9{{\left( x+1 \right)}^{2}}}\le 2\left| 2x-1 \right|$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Using your answer in <strong>(iii)</strong>, find the exact range of values of $x$ for which $\sqrt{36-9{{\left( \frac{1}{x}+1 \right)}^{2}}}\le 2\left| \frac{2}{x}-1 \right|$.<span class="inline-marks">[2]</span></div></li></ol>

Graphs of Conic Sections

2023 ASRJC Promo Q9

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\left| b+\frac{a}{x+b} \right|$, where&nbsp;$0 &lt; a &lt; b$, stating the equations of any asymptotes and the coordinates of the points that cross the axes.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence solve the inequality $-x\le \left| b+\frac{a}{x+b} \right|$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Using part <strong>(ii)</strong>, given that&nbsp;$a=1$, $b=2$, solve the inequality ${{\mathrm{e}}^{x}}\le \left| b+\frac{a}{b-{{\mathrm{e}}^{x}}} \right|$.<span class="inline-marks">[2]</span></div></li></ol>

2023 ACJC Promo Q2

<ol class="i-list"><li><div class="list-item-paragraph">On the same axes, sketch the graphs of $y=x+2+\frac{1}{x-1}$ and $y=\left| 2x+2 \right|$, stating the coordinates of any points of intersection with the axes, turning points and the equations of any asymptotes.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Hence solve the inequality</div><div class="list-item-paragraph" style="text-align: center">$x+2+\frac{1}{x-1}&gt;\left| 2x+2 \right|$,</div><div class="list-item-paragraph">giving your answers in exact form.<span class="inline-marks">[5]</span></div></li></ol>

2021 RI Promo Q6

<ol class="abc-list"><li><div class="list-item-paragraph">Without using a calculator, solve the inequality $\frac{4x}{{{x}^{2}}-3x-4}\ge -x$. Hence solve the inequality $\frac{4\left| x \right|}{{{x}^{2}}-3\left| x \right|-4}&lt;-\left| x \right|$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">The diagrams below show the graphs of $y=\left| \mathrm{f}(x) \right|$ and $y=\mathrm{f}'(x)$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1772447541701-4on7e987hzh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzI0NDc1NDE3MDEtNG9uN2U5ODdoemgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTIxMCwiZXhwIjoxNzkyODM5NTEwfQ.D89FTi00QQ_mrVBMr-mBXXGay1OdHzTutnaz-8KS6go" alt="List item image" class="aligncenter custom-width" style="max-width: min(500px, 100%); height: auto;" /></div><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}(x)$, labelling the equations of any asymptotes and the coordinates of the turning point and the points where the curve crosses the axes.<span class="inline-marks">[3]</span></div></li></ol>

2020 NJC Promo Q6

<ol class="i-list"><li data-list-item-id="item-1761051218078-vv88pbkhs" data-list-item-path="item-1761051218078-vv88pbkhs"><div class="list-item-paragraph">Sketch the graph of $y=\left| {{2}^{x}}-10 \right|$, giving the exact values of any points where the curve meets the axes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761051219083-aa3zpf31j" data-list-item-path="item-1761051219083-aa3zpf31j"><div class="list-item-paragraph">Without using a calculator, and showing all your working, find the exact interval, or intervals, for which $\left| {{2}^{x}}-10 \right|\le 6$. Give your answer in its simplest form.<span class="inline-marks">[3]</span></div></li></ol>

N2019 P1 Q4

<ol class="i-list"><li><div class="list-item-paragraph">On the same axes, sketch the graphs of $y=\frac{x-b}{x}$ and $y=\frac{1}{a}\left| x-b \right|$, where $0&lt;a&lt;1$ and $b&gt;1$. State the equations of any asymptotes and the coordinates of the points where the curves cross the axes.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence solve the inequality $\frac{x-b}{x}\le \frac{1}{a}\left| x-b \right|$.<span class="inline-marks">[3]</span></div></li></ol>

Math Question Bank