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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

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The curve ${{C}_{1}}$ has equation ${{x}^{2}}-2x-4{{y}^{2}}=0$.</div><ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of ${{C}_{1}}$, stating clearly the coordinates of any points of intersection with the axes and the equations of any asymptotes.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By drawing a suitable graph ${{C}_{2}}$ on the same diagram, showing clearly the relative position with curve ${{C}_{1}}$, deduce the range of values of $k$ such that the equation ${{x}^{2}}-2x-4+\frac{4{{\left( x-2 \right)}^{2}}}{{{k}^{2}}}=0$ include non-positive root(s).</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Parameter Conditions for Graphs and Equations

Asymptotes and Discontinuities

Graphs of Conic Sections

2022 TJC J1 MYE Q11

Graphing Techniques

<p>A curve $C$ has parametric equations $x=\cos 2\theta $ and $y=\cos \mathrm{ec }\theta $, where $0 < \theta \le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{4}\cos \mathrm{e}{{\mathrm{c}}^{3}}\theta $. What can be said about $\frac{\mathrm{d}y}{\mathrm{d}x}$ as $\theta \to 0$?</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Sketch $C$, stating the equation of the asymptote and coordinates of the point of intersection with the axis.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the equation of the normal to $C$ at the point where $\theta =\frac{\pi }{6}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Show that the point(s) of intersection of the normal found in part <strong>(iii)</strong> and the curve $C$ satisfy the equation ${{\sin }^{3}}\theta +\frac{7}{4}\sin \theta -1=0$. Hence using a non-calculator method, show that this normal does not meet the curve again.</div><p style="text-align: right;">[4]</p></li></ol>

A curve $C$ has parametric equations $x=\cos 2\theta $ and $y=\cos \mathrm{ec }\theta $, where $0 < \theta \le \frac{\pi }{2}$.

Tangents and Normals to Parametric Curves

Derivatives of Reciprocal Trigonometric Functions

Differentiation of Parametric Equations

Sketching Parametric Curves

2020 TMJC Promo Q9

Techniques of Differentiation

Applications of Differentiation

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto 4+\frac{1}{{{\left( x-1 \right)}^{2}}+1}$, $x\ge 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760427682032-q1mhs21yc" data-list-item-path="item-1760427682032-q1mhs21yc"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, indicating clearly the coordinates of any stationary points, the value of the $y-$intercept, and the equation of any asymptotes.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \sin x$, $x\in \left[ 0,2\pi \right]$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760427706093-2z98vh9ys" data-list-item-path="item-1760427706093-2z98vh9ys"><div class="list-item-paragraph">State the exact coordinates of two points on the graph of $y=\mathrm{g}\left( x \right)$, which demonstrate that&nbsp;$\mathrm{g}$ does not have an inverse.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760427706699-4mo4xqh11" data-list-item-path="item-1760427706699-4mo4xqh11"><div class="list-item-paragraph">Explain clearly why the composite function $\mathrm{gf}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760427707741-u39pf5sa7" data-list-item-path="item-1760427707741-u39pf5sa7"><div class="list-item-paragraph">Using a suitably-labelled sketch of the graph of $y=\mathrm{g}\left( x \right)$, find the range of $\mathrm{gf}$.<span class="inline-marks">[3]</span></div></li></ol>

$\mathrm{f}:x\mapsto 4+\frac{1}{{{\left( x-1 \right)}^{2}}+1}$, $x\ge 0$.

$\mathrm{g}:x\mapsto \sin x$, $x\in \left[ 0,2\pi \right]$.

Composite Functions

Sketching Rational Functions

One-to-One Functions and Inverse Existence

Existence of Composite Functions

Range of a Composite Function

Inverse Functions

2023 EJC P1 Q8

Functions

<p>It is given that&nbsp;$y=\frac{2{{x}^{2}}+3}{x-2},x\in \mathbb{R},x\ne 2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ and label clearly the equations of the asymptotes and intercepts if any.</div></li><li><div class="list-item-paragraph">The graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ cannot lie between values $p$ and $m$, where&nbsp;$p&gt;m$ . State the value of $p$ and $m$ .</div></li><li><div class="list-item-paragraph">By means of a graphical argument, state the maximum number of real roots the equation $kx\left( x-2 \right)-2{{x}^{2}}=3$ have.</div></li></ol>

It is given that $y=\frac{2{{x}^{2}}+3}{x-2},x\in \mathbb{R},x\ne 2$.

Graphical Root Counting

Finding Possible y-values of a Curve

ACJC Tutorial 2 Q8

<p>A curve $C$ has equation $y=\frac{2}{{{x}^{2}}-{{k}^{2}}}$ where&nbsp;$k\ge 2$. It is given that&nbsp;$C$ has only one stationary point.</p>
<ol class="i-list"><li data-list-item-id="item-1760439942706-1m9zyiuw3" data-list-item-path="item-1760439942706-1m9zyiuw3"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$. Hence find the coordinates of the stationary point in terms of $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760439944386-u9p7qij1o" data-list-item-path="item-1760439944386-u9p7qij1o"><div class="list-item-paragraph">Sketch the graph of $C$, stating the equations of any asymptotes. coordinates of any points of intersection with the axes and the coordinates of the stationary point.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760439945173-d4tyuqve2" data-list-item-path="item-1760439945173-d4tyuqve2"><div class="list-item-paragraph">By adding a suitable graph on the same diagram in part <strong>(ii)</strong>, find the range of values of $h$, in terms of $k$, where&nbsp;$h$ is a positive real constant and $h\ne k$ such that&nbsp;${{\left( hx \right)}^{2}}+{{\left( \frac{2k}{{{x}^{2}}-{{k}^{2}}} \right)}^{2}}={{\left( kh \right)}^{2}}$ has at least one real root.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $y=\frac{2}{{{x}^{2}}-{{k}^{2}}}$ where $k\ge 2$. It is given that $C$ has only one stationary point.

Chain Rule

Stationary Points and Points of Inflection

Stationary Points and Their Nature

2020 TMJC Promo Q4

<p>The curve $D$ has equation $y = \frac{2x - 5}{x^2 + 2x - 3}.$</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $D$, stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By considering a suitable curve, hence, find the exact range of values of $k$, where&nbsp;$k &gt; 0$, such that the equation $(x + 3)^2 + \left( \frac{2x - 5}{x^2 + 2x - 3} \right)^2 = k^2$ has at least one <strong>positive</strong> real root.<span class="inline-marks">[3]</span></div></li></ol>

The curve $D$ has equation $y = \frac{2x - 5}{x^2 + 2x - 3}.$

Sketch the graph of $D$, stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s).

By considering a suitable curve, hence, find the exact range of values of $k$, where $k > 0$, such that the equation \[(x + 3)^2 + \left( \frac{2x - 5}{x^2 + 2x - 3} \right)^2 = k^2\] has at least one **positive** real root.

Circle–Curve Intersection Conditions

2024 NYJC Promo Q3

<p>The curve $C$ has the equation $y=\frac{-{{x}^{2}}+ax}{x-2b}$, where $a$ and $b$ are non-zero constants. Given that $y=-x+1$ is an asymptote, find $a$ in terms of $b$.</p><p style="text-align: right;">[2]</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the range of values of $b$, such that the curve $C$ has $2$ stationary points.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $1 < b < 3$, sketch the curve $C$. Show clearly (if any), the equations of the asymptotes, and the coordinates of the axial intercepts and stationary points, in terms of $b$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Show that the intersection point of the asymptotes lies on the line $y=m(x-2b)+(1-2b)$, where $m$ is a constant. Hence, find the range of values of $m$ for which the line $y=m(x-2b)+(1-2b)$ cuts $C$ at $2$ distinct points.</div><p style="text-align: right;">[2]</p></li></ol>

The curve $C$ has the equation $y=\frac{-{{x}^{2}}+ax}{x-2b}$, where $a$ and $b$ are non-zero constants. Given that $y=-x+1$ is an asymptote, find $a$ in terms of $b$.

Parameter Conditions for Stationary Points

2009 AJC Promo Q13

<p>The curve $C$ has equation $y=2+\frac{3a}{x-a}$, where $a$ is a positive constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Describe fully a sequence of transformations which would transform the curve $y=\frac{1}{x}$ onto the curve $C$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\left| 2+\frac{3a}{x-a} \right|$, labelling the equations of the asymptotes and coordinates of the axial intercepts.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Use an algebraic method to solve $\left| 2+\frac{3}{x-1} \right|=3$ and state the solution for $\left| 2+\frac{3}{x-1} \right|<3$.</div><p style="text-align: right;">[4]</p></li></ol>

The curve $C$ has equation $y=2+\frac{3a}{x-a}$, where $a$ is a positive constant.

Composite Transformations

Graphs of Modulus Function

Inequalities involving Modulus Functions

2023 TMJC P1 Q4

Inequalities

Transformations of Graphs

<p>A curve $C$ has equation $y=\frac{{{x}^{2}}-2x+7}{x-3}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Using an algebraic method, find the range of values that&nbsp;$y$ cannot take. Leave your answer in exact form.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Sketch the curve $C$, stating the equations of any asymptotes and the exact coordinates of any points where&nbsp;$C$ crosses the axes and of any turning points.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">By considering a suitable graph, find the range of values of $k$ such that the equation ${{x}^{4}}-2{{x}^{3}}+7{{x}^{2}}=kx-3k$ has exactly one positive root and one negative root.<span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has equation $y=\frac{{{x}^{2}}-2x+7}{x-3}$.

2022 VJC Promo Q8

<p>The curve ${{C}_{1}}$ has equation $y=\frac{{{x}^{2}}-5}{x+3}$.</p>
<p>The curve ${{C}_{2}}$ has equation $\frac{{{\left( x+5 \right)}^{2}}}{4}+\frac{{{\left( y+6 \right)}^{2}}}{{{b}^{2}}}=1$, where&nbsp;$b&gt;0$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">For&nbsp;${{C}_{1}}$, find algebraically the range of values that&nbsp;$y$ can take.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$b=2$, sketch ${{C}_{1}}$ and ${{C}_{2}}$ on the same graph, giving the exact coordinates of any points where&nbsp;${{C}_{1}}$ or ${{C}_{2}}$ crosses the $x$-axis and the equations of any asymptotes.<span class="inline-marks">[5]</span></div></li><li><ol class="i-list"><li><div class="list-item-paragraph">State the range of values of $m$ such that&nbsp;$y+6=m\left( x+3 \right)$ does not intersect ${{C}_{1}}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">State the range of values of $b$ such that there are two distinct points of intersection between ${{C}_{1}}$ and ${{C}_{2}}$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

The curve ${{C}_{1}}$ has equation $y=\frac{{{x}^{2}}-5}{x+3}$.

The curve ${{C}_{2}}$ has equation $\frac{{{\left( x+5 \right)}^{2}}}{4}+\frac{{{\left( y+6 \right)}^{2}}}{{{b}^{2}}}=1$, where $b>0$.

2022 YIJC Promo Q7

<p>Sigmoid functions are used to model many natural processes such as population growth of virus. One example of a Sigmoid function $\mathrm{f}$ is given by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \frac{1}{1+{{\mathrm{e}}^{-x}}}$, $x\in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760244734514-9vpp0u6r3" data-list-item-path="item-1760244734514-9vpp0u6r3"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, indicating clearly the equation(s) of any asymptote(s) and the coordinates of any points where the curve crosses the axes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760244735053-5fgp6y18q" data-list-item-path="item-1760244735053-5fgp6y18q"><div class="list-item-paragraph">Find ${{\mathrm{f}}^{-1}}\left( x \right)$ in similar form.<span class="inline-marks">[3]</span></div></li></ol>
<p>Another function $\mathrm{g}$ is given by $g:x\mapsto 3x-1$, $x\in \mathbb{R}$, $0\le x\le 2$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760244762000-66i8i3qnm" data-list-item-path="item-1760244762000-66i8i3qnm"><div class="list-item-paragraph">Show that&nbsp;$\mathrm{fg}$ exists and find the range of $\mathrm{fg}$, expressing your answer in terms of $\mathrm{e}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760244762668-aa5zc39ca" data-list-item-path="item-1760244762668-aa5zc39ca"><div class="list-item-paragraph">Describe a sequence of transformations which transform the graph of $y=\mathrm{f}\left( x \right)$ onto the graph of $y=\mathrm{fg}\left( x \right)$.<span class="inline-marks">[2]</span></div></li></ol>

Sigmoid functions are used to model many natural processes such as population growth of virus. One example of a Sigmoid function $\mathrm{f}$ is given by

$\mathrm{f}:x\mapsto \frac{1}{1+{{\mathrm{e}}^{-x}}}$, $x\in \mathbb{R}$.

Another function $\mathrm{g}$ is given by $g:x\mapsto 3x-1$, $x\in \mathbb{R}$, $0\le x\le 2$.

Sketching Exponential Functions and Solving Equations Graphically

Graph Transformations of $f(ax+b)$

2021 CJC Promo Q11

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=\ln (2t)$, $y={{\tan }^{-1}}(2t)$, where $t>0$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Describe the behaviour of the curve $C$ as $x\to \infty $ and $x\to -\infty $. Sketch $C$, showing clearly the asymptotes and the axial intercepts.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Show that the gradient of $C$ at the point where $y=p$ is $\frac{1}{2}\sin 2p$.</div><p style="text-align: right;">[3]</p><div class="list-item-paragraph">Hence find the area of triangle bounded by the $x$-axis, the tangent and normal to $C$ when $p=\frac{\pi }{4}$.</div><p style="text-align: right;">[5]</p></li></ol>

$x=\ln (2t)$, $y={{\tan }^{-1}}(2t)$, where $t>0$.

Derivatives of Inverse Trigonometric Functions

2008 AJC P1 Q12 [Modified]

<ol class="abc-list" start="2"><li data-list-item-id="item-1760089055321-9u6qo54ss" data-list-item-path="item-1760089055321-9u6qo54ss"><div class="list-item-paragraph">The curve $C$ has equation $y=\frac{3{{x}^{2}}+9x+7}{x+2}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760089064231-okkk8zez5" data-list-item-path="item-1760089055321-9u6qo54ss/subitem-1760089064231-okkk8zez5"><div class="list-item-paragraph">By expressing the equation of $C$ in the form $y=3x+a+\frac{b}{x+2}$, where&nbsp;$a$ and $b$ are non-zero constants, write down the equations of the asymptotes of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760089077251-rosdo5nv8" data-list-item-path="item-1760089055321-9u6qo54ss/subitem-1760089077251-rosdo5nv8"><div class="list-item-paragraph">Describe a pair of transformations that will transform $C$ to the graph of $y=3x+\frac{1}{x}$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2018 ACJC P1 Q9

<p>The curve $C$ has equation $y=\frac{a{{x}^{2}}-2ax+a-2}{x-2}$, $a\in \mathbb{R}$, $a\ne 0$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the range of values of $a$ for which $C$ has no stationary points.<span class="inline-marks">[3]</span></div></li></ol>
<p>For&nbsp;$a=1$,</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Sketch the curve $C$, indicating the equations of asymptotes, coordinates of intersections with the axes and turning points, if any.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By sketching a suitable curve on the diagram in <strong>(iii)</strong>, state the number of positive and negative roots of the equation ${{x}^{3}}-2{{x}^{2}}-x=k\left( x-2 \right)$ where&nbsp;$k&gt;0$.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $y=\frac{a{{x}^{2}}-2ax+a-2}{x-2}$, $a\in \mathbb{R}$, $a\ne 0$.

Quotient Rule

2024 ACJC Promo Q10

<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 Logarithmic Functions and Solving Equations Graphically

Polynomial Functions and Key Features

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+b}{c-x}$. The vertical asymptote of $C$ is $x=-2$, and the coordinates of the turning points are $\left( -4,2 \right)$ and $\left( 0,-6 \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $a$, $b$ and $c$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Sketch $C$, stating the equations of the asymptotes.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">By drawing an appropriate graph on the sketch of $C$, find the range of values of $k$$\left( k>0 \right)$ such that the equation ${{\left( x+4 \right)}^{2}}+{{\left( \frac{{{x}^{2}}+ax+b}{c-x} \right)}^{2}}={{k}^{2}}$ has no real roots.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">When $C$ undergoes a translation in the direction of $y$-axis by $p$ units, $C$ intersects the line $y=-1$ at one negative value of $x$. State the set of values for $p$.</span></div><p style="text-align: right;">[2]</p></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+b}{c-x}$. The vertical asymptote of $C$ is $x=-2$, and the coordinates of the turning points are $\left( -4,2 \right)$ and $\left( 0,-6 \right)$.

Sketch $C$, stating the equations of the asymptotes.

By drawing an appropriate graph on the sketch of $C$, find the range of values of $k$$\left( k>0 \right)$ such that the equation ${{\left( x+4 \right)}^{2}}+{{\left( \frac{{{x}^{2}}+ax+b}{c-x} \right)}^{2}}={{k}^{2}}$ has no real roots.

When $C$ undergoes a translation in the direction of $y$-axis by $p$ units, $C$ intersects the line $y=-1$ at one negative value of $x$. State the set of values for $p$.

Determining Model Coefficients Using Linear Systems

2013 PJC P2 Q5

<p>Sketch the graph of $y=\frac{2}{x\left( x-\mu \right)}$, where $\mu $ is a positive constant, indicating clearly any axial intercept(s), asymptote(s) and coordinates of turning point(s).</p>
<p>Given $a=\frac{\mu }{2}$, describe the range of values of $b$ in terms of $\mu $, such that the graphs of $b{{\left( x-a \right)}^{2}}+{{a}^{2}}{{y}^{2}}={{a}^{2}}b$ and $y=\frac{2}{x\left( x-\mu \right)}$ intersect twice, where $a$ and $b$ are positive real numbers.</p>

Sketch the graph of $y=\frac{2}{x\left( x-\mu \right)}$, where $\mu $ is a positive constant, indicating clearly any axial intercept(s), asymptote(s) and coordinates of turning point(s).

Given $a=\frac{\mu }{2}$, describe the range of values of $b$ in terms of $\mu $, such that the graphs of $b{{\left( x-a \right)}^{2}}+{{a}^{2}}{{y}^{2}}={{a}^{2}}b$ and $y=\frac{2}{x\left( x-\mu \right)}$ intersect twice, where $a$ and $b$ are positive real numbers.

2010 NJC Promo Q6

<p>Given that&nbsp;$b$ is a real constant such that&nbsp;$0 &lt; b &lt; 4$, describe fully a sequence of transformations that transforms the curve $y=x^2$ to the curve $y=4x^2+bx+1$.<span class="inline-marks">[4]</span></p>
<p>Sketch the curve $y=\frac{1}{4x^2+bx+1}$. Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of $b$ where appropriate.<span class="inline-marks">[3]</span></p>

Given that $b$ is a real constant such that $0 < b < 4$, describe fully a sequence of transformations that transforms the curve $y=x^2$ to the curve $y=4x^2+bx+1$.

Sketch the curve $y=\frac{1}{4x^2+bx+1}$. Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of $b$ where appropriate.

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

2025 HCI P1 Q6

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+kx+1}{x-2}$, where $k$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that $y=x$ is an asymptote of $C$, find the value of $k$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Let $k=-2$. Using an algebraic method, find the set of values of $y$ for which there are no points on $C$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y=\frac{{{x}^{2}}+kx+1}{x-2}$ for $k=-2$, stating clearly the equations of any asymptotes, coordinates of any turning points and axial intercepts.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">On the same diagram, sketch the graph of $9{{y}^{2}}=9{{a}^{2}}-{{a}^{2}}{{\left( x-3 \right)}^{2}}$, where $a$ is a positive constant, showing clearly the coordinates of the axial intercepts.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Deduce the value of $a$ if the equation $9{{\left( x-1 \right)}^{4}}={{a}^{2}}{{\left( x-2 \right)}^{2}}\left( 9-{{\left( x-3 \right)}^{2}} \right)$ has exactly $3$ real roots.</span></div><p style="text-align: right;">[2]</p></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+kx+1}{x-2}$, where $k$ is a constant.

Given that $y=x$ is an asymptote of $C$, find the value of $k$.

Let $k=-2$. Using an algebraic method, find the set of values of $y$ for which there are no points on $C$.

Sketch the graph of $y=\frac{{{x}^{2}}+kx+1}{x-2}$ for $k=-2$, stating clearly the equations of any asymptotes, coordinates of any turning points and axial intercepts.

On the same diagram, sketch the graph of $9{{y}^{2}}=9{{a}^{2}}-{{a}^{2}}{{\left( x-3 \right)}^{2}}$, where $a$ is a positive constant, showing clearly the coordinates of the axial intercepts.

Deduce the value of $a$ if the equation $9{{\left( x-1 \right)}^{4}}={{a}^{2}}{{\left( x-2 \right)}^{2}}\left( 9-{{\left( x-3 \right)}^{2}} \right)$ has exactly $3$ real roots.

2012 NYJC Promo Q11

<p>A curve $C$ has equation $y=\frac{a{{x}^{2}}+bx+c}{x-d}$, where&nbsp;$a$, $b$, $c$ and $d$ are constants. It is given that two of its asymptotes are $y=x+2$ and $x=1$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760426275355-bukz3khu0" data-list-item-path="item-1760426275355-bukz3khu0"><div class="list-item-paragraph">State the value of $d$, and show that&nbsp;$a=b=1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760426276240-kdqji10ob" data-list-item-path="item-1760426276240-kdqji10ob"><div class="list-item-paragraph">Using differentiation, find the range of values of $c$ such that the graph of $C$ contains two stationary points.<span class="inline-marks">[4]</span></div></li></ol>
<p>Use $c=14$ for the rest of the question.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760426294465-nw8b3xfmx" data-list-item-path="item-1760426294465-nw8b3xfmx"><div class="list-item-paragraph">Sketch $C$, showing clearly the equations of asymptotes and the coordinates of the turning points.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760426295223-p01rj862t" data-list-item-path="item-1760426295223-p01rj862t"><div class="list-item-paragraph">State the maximum number of roots to the equation</div><div class="list-item-paragraph" style="text-align: center">${{k}^{2}}{{\left( x-5 \right)}^{2}}+{{\left( \frac{a{{x}^{2}}+bx+c}{x-d}-3 \right)}^{2}}={{k}^{2}}$, where&nbsp;$k&gt;0$.</div><div class="list-item-paragraph">Deduce the range of values of $k$ for the maximum number of roots to occur.<span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has equation $y=\frac{a{{x}^{2}}+bx+c}{x-d}$, where $a$, $b$, $c$ and $d$ are constants. It is given that two of its asymptotes are $y=x+2$ and $x=1$.

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