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H2 Math

IB AA HL Math

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IB AI SL Math

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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The curve $C$ has equation $y = \frac{1}{4x - x^2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$. Hence, find the $x$-coordinate, $x = x_1$, of the turning point on $C$ and determine its nature.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Using calculus, find the exact area of the region between $C$, the $x$-axis and the lines with equations $x = 1$ and $x = x_1$.</span><span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation $y = \frac{1}{4x - x^2}$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$. Hence, find the $x$-coordinate, $x = x_1$, of the turning point on $C$ and determine its nature.

Using calculus, find the exact area of the region between $C$, the $x$-axis and the lines with equations $x = 1$ and $x = x_1$.

Area Bounded by Curves

Chain Rule

Stationary Points and Points of Inflection

Stationary Points and Their Nature

2025 JPJC P1 Q1

Graphing Techniques

Applications of Integration

Applications of Differentiation

Techniques of Differentiation

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MDIzNzc3MDYtMGZuNmZnenYxMWo3LnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTM1NTYzOTMsImV4cCI6MTc5MzU1NjY5M30.c-AfSLFe68XdXbDkqqNtY5QFJ4EViU8XUzsF9V9_oso" 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

Parameter Conditions for Graphs and Equations

Increasing and Decreasing Functions

Graphs of Modulus Function

2025 DHS P1 Q5 

Transformations of Graphs

<p>A curve $C$ has equation $y=2ax+\frac{b}{x}$ where&nbsp;$a$ and $b$ are non-zero real constants and $x\ne 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760765552949-17xocwmpx" data-list-item-path="item-1760765552949-17xocwmpx"><div class="list-item-paragraph">Using differentiation, determine whether $C$ has any stationary points if&nbsp;$ab &lt; 0$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$b=\frac{1}{2}a$ where&nbsp;$a&gt;0$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760765589976-7l9zd6yvg" data-list-item-path="item-1760765589976-7l9zd6yvg"><div class="list-item-paragraph">Sketch $C$, stating the equations of any asymptotes, and the coordinates of any stationary points and points where&nbsp;$C$ crosses the axes (if any).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760765593674-qsqu3k010" data-list-item-path="item-1760765593674-qsqu3k010"><div class="list-item-paragraph">State the range of values of $k$, in terms of $a$, for which the equation $2ax+\frac{b}{x}=kx$ has no real roots.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760765594477-wx20pfe6v" data-list-item-path="item-1760765594477-wx20pfe6v"><div class="list-item-paragraph">The region bounded by $C$, the axes, the lines $x=\frac{1}{2}$ and $y=4a$ is rotated about the $y$-axis through $2\pi $ radians. Show that the volume generated is given by $\frac{1}{2}a\pi +\frac{\pi }{8{{a}^{2}}}\displaystyle\displaystyle \displaystyle\int_{2a}^{4a}{{{y}^{2}}}-2{{a}^{2}}-y\sqrt{{{y}^{2}}-4{{a}^{2}}}\ \mathrm{d}y$.</div><div class="list-item-paragraph">Hence, find, in terms of $a$ and $\pi $, the exact volume generated.<span class="inline-marks">[6]</span></div></li></ol>

A curve $C$ has equation $y=2ax+\frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x\ne 0$.

It is now given that $b=\frac{1}{2}a$ where $a>0$.

Sketching Rational Functions

Parameter Conditions for Stationary Points

Integration by the Reverse Chain Rule

Volume of Revolution about the y-axis

Asymptotes and Discontinuities

2024 RI P1 Q9

Techniques of Integration

<p>The function $\mathrm{f}$ is given by</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto a+\frac{3a}{\left( x+3 \right)\left( x-1 \right)}$, where&nbsp;$a &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760756659892-xuinyrf9p" data-list-item-path="item-1760756659892-xuinyrf9p"><div class="list-item-paragraph">Using calculus, find the $x$-coordinate of the stationary point of $y=\mathrm{f}\left( x \right)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760756660972-8b0t4whz8" data-list-item-path="item-1760756660972-8b0t4whz8"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$, clearly stating the coordinates of the stationary point and equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760756661573-8nfcta32z" data-list-item-path="item-1760756661573-8nfcta32z"><div class="list-item-paragraph">State the range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is increasing and concave downwards.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760756663289-qtm77oaqp" data-list-item-path="item-1760756663289-qtm77oaqp"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$, clearly stating the coordinates of the points where the graph crosses the axes and equations of any asymptotes. Leave your answers in terms of $a$ where appropriate.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is given that&nbsp;$\mathrm{g}\left( x \right)=\mathrm{f}\left( x \right)$ for&nbsp;$x\ge 2$. Suppose the composite function $\mathrm{gg}$ exists.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-1760756702537-yzpn5lfnw" data-list-item-path="item-1760756702537-yzpn5lfnw"><div class="list-item-paragraph">Deduce, with a reason, the condition on $a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760756703234-rdqxsegk9" data-list-item-path="item-1760756703234-rdqxsegk9"><div class="list-item-paragraph">Find the exact range of $\mathrm{gg}$ for&nbsp;$a=5$.<span class="inline-marks">[2]</span></div></li></ol>

$\mathrm{f}:x\mapsto a+\frac{3a}{\left( x+3 \right)\left( x-1 \right)}$, where $a > 0$.

It is given that $\mathrm{g}\left( x \right)=\mathrm{f}\left( x \right)$ for $x\ge 2$. Suppose the composite function $\mathrm{gg}$ exists.

Composite Functions

Existence of Composite Functions

Range of a Composite Function

Sketching Derivative Graphs

2023 DHS P1 Q11

Functions

<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.

2020 TMJC Promo Q4

<p>The curve $C$ has equation $y={{x}^{3}}{{\mathrm{e}}^{-x}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Using differentiation, find the coordinates of the stationary points on $C$ and determine their nature.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">The region bounded by the curve $C$, the line $y=-x+2$ and the $x$-axis is rotated through $2\pi $ radians about the $x$-axis. Find the volume of the solid obtained.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation $y={{x}^{3}}{{\mathrm{e}}^{-x}}$.

Stationary Points of Inflection

Volume of Revolution about the x-axis

2023 MI PRU2 P1 Q4

<p>A curve $D$ has parametric equations</p>
<p style="text-align: center">$x=k\left( 2\cos \theta -\cos 2\theta \right)$</p>
<p style="text-align: center">$y=-k\sin 2\theta $</p>
<p>for $0<\theta \le \pi $, where $k$ is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find, in terms of $k$, the exact coordinates of the stationary points of $D$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Sketch the graph of $D$, labelling, in terms of $k$, the points where $D$ meets the $x$-axis.</div><p style="text-align: right;">[2]</p></li></ol>

$x=k\left( 2\cos \theta -\cos 2\theta \right)$

for $0<\theta \le \pi $, where $k$ is a positive constant.

Differentiation of Parametric Equations

Sketching Parametric Curves

2022 NJC Promo Q9

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x={{\sin }^{3}}\theta $, $y=3{{\sin }^{2}}\theta \cos \theta $, $0\le \theta \le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=2\cot \theta -\tan \theta $.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Show that $C$ has a turning point when $\tan \theta =\sqrt{k}$, where $k$ is an integer to be determined.</div><div class="list-item-paragraph">Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Show that the area of the region bounded by $C$ and the $x$-axis is given by</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{0}^{\frac{1}{2}\pi }{9{{\sin }^{4}}\theta {{\cos }^{2}}\theta \mathrm{ d}\theta }$.</div><div class="list-item-paragraph">Use your calculator to find the area, giving your answer correct to $3$ decimal places.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The line with equation $y=ax$, where $a$ is a positive constant, meets $C$ at the origin and at the point $P$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Show that $\tan \theta =\frac{3}{a}$ at $P$. Find the exact value of $a$ such that the line passes through the maximum point of $C$.</div><p style="text-align: right;">[3]</p></li></ol>

$x={{\sin }^{3}}\theta $, $y=3{{\sin }^{2}}\theta \cos \theta $, $0\le \theta \le \frac{\pi }{2}$.

The line with equation $y=ax$, where $a$ is a positive constant, meets $C$ at the origin and at the point $P$.

N2015 P1 Q11

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x={{\sin }^{2}}\theta $, $y=2\sin \theta -{{\sin }^{3}}\theta $,</p>
<p>where $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value(s) of the parameter $\theta$ at the point(s) where the curve $C$ is stationary.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Show that the equation of the tangent to the curve at the point where the gradient is $\frac{5}{4}$ is given by $y=\frac{5}{4}x+\frac{9}{16}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Determine whether the tangent $y=\frac{5}{4}x+\frac{9}{16}$ meets the curve $C$ again.</div><p style="text-align: right;">[3]</p></li></ol>

$x={{\sin }^{2}}\theta $, $y=2\sin \theta -{{\sin }^{3}}\theta $,

where $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$.

Tangents and Normals to Parametric Curves

Derivatives of Trigonometric Functions

2016 MI P1 Q11

<p>A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\cos x$, for&nbsp;$0\le x\le \frac{1}{2}\pi $.</p>
<ol class="abc-list"><li data-list-item-id="item-1760091646738-5ojj0bz2n" data-list-item-path="item-1760091646738-5ojj0bz2n"><div class="list-item-paragraph">Using calculus, find the stationary point of $\mathrm{f}\left( x \right)$ and determine its nature.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1760091647683-j170tny5e" data-list-item-path="item-1760091647683-j170tny5e"><div class="list-item-paragraph">Integrate by parts twice to show that</div><div class="list-item-paragraph" style="text-align: center"><p>$\int \mathrm{e}^{2x} \cos 2x \mathrm{d}x = \frac{1}{4} \mathrm{e}^{2x} (\sin 2x + \cos 2x) + c.$</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760091648223-4stjq6k1n" data-list-item-path="item-1760091648223-4stjq6k1n"><div class="list-item-paragraph">The graph of $y=\mathrm{f}\left( x \right)$ is rotated completely about the $x$-axis. Find the exact volume generated.<span class="inline-marks">[4]</span></div></li></ol>

A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)={{\mathrm{e}}^{x}}\cos x$, for $0\le x\le \frac{1}{2}\pi $.

Integrating Powers of Sine and Cosine

Integration by Parts

Integrable Forms of Trigonometric Functions

Cyclic Integration by Parts

N2021 P1 Q9

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+3x+b}{x+a}$ $(x\ne -a)$, where $a$ and $b$ are constants. It is given that the line $y=x+4$is an asymptote of $C$, and that $C$ has a stationary point at $x=2$. Show that $a=-1$, and find the value of $b$.</p><p style="text-align: right;">[6]</p>
<p>Using the same values of $a$ and $b$,</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $C$ has exactly two stationary points, and find the coordinates of each stationary point,</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Draw a sketch of $C$,</div><p style="text-align: right;">[2]</p></li></ol>
<p>Deduce that, for all positive values of $k$, the equation ${{x}^{3}}+3{{x}^{2}}+(b-k)x-ka=0$ has exactly one negative root.</p>

The curve $C$ has equation $y=\frac{{{x}^{2}}+3x+b}{x+a}$ $(x\ne -a)$, where $a$ and $b$ are constants. It is given that the line $y=x+4$is an asymptote of $C$, and that $C$ has a stationary point at $x=2$. Show that $a=-1$, and find the value of $b$.

Deduce that, for all positive values of $k$, the equation ${{x}^{3}}+3{{x}^{2}}+(b-k)x-ka=0$ has exactly one negative root.

Graphical Root Counting

FM 395 P1 Q1

<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/converted/1759839630427-l8bh9gkaeb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk4Mzk2MzA0MjctbDhiaDlna2FlYi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTU2MzkzLCJleHAiOjE3OTM1NTY2OTN9.WzzLkvHXna4WV7QPSBvhHdDgLMjNhiN6UgoClBoB4Sk" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>A sketch of the curve $\mathrm{f}(x)=\frac{a{{x}^{2}}+bx}{2{{x}^{2}}-k}$, where&nbsp;$a$, $b$ and $k$ are non-zero constants, is shown above. </p>
<p>Explain why $a=4$ and $k&lt;0$. Using the given graph,</p>
<ol class="i-list"><li data-list-item-id="item-1759839644799-3qirbx8et" data-list-item-path="item-1759839644799-3qirbx8et"><div class="list-item-paragraph">Sketch the graph of $\mathrm{f}(x)=\frac{a{{x}^{2}}-bx}{2{{x}^{2}}-k}$for all values of $x$.</div></li><li data-list-item-id="item-1759839645491-3l09h0a1v" data-list-item-path="item-1759839645491-3l09h0a1v"><div class="list-item-paragraph">Determine the number of stationary points of the curve whose equation is $y={{[\mathrm{f}(x)]}^{2}}$.</div></li></ol>

A sketch of the curve $\mathrm{f}(x)=\frac{a{{x}^{2}}+bx}{2{{x}^{2}}-k}$, where $a$, $b$ and $k$ are non-zero constants, is shown above. 

Explain why $a=4$ and $k<0$. Using the given graph,

Translations, Reflections and Scalings of Graphs

2008 AJC P1 Q9

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Given $x$ and $y$ are related by the differential equation ${{x}^{2}}\frac{\mathrm{d}y}{\mathrm{d}x}+xy=k$ for $k\in \mathbb{R}$, show that $y=\frac{k\left( \ln x+\alpha \right)}{x}$ is a solution of the differential equation where $\alpha $ is an arbitrary constant.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence, show that $\left( {{\mathrm{e}}^{1-\alpha }},k{{\mathrm{e}}^{\alpha -1}} \right)$ is a stationary point of the curve $y=\frac{k\left( \ln x+\alpha \right)}{x}$.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">It is given that $x$ and $y$ are related by the differential equation $y\frac{\mathrm{d}y}{\mathrm{d}x}+x=\sqrt{{{x}^{2}}+{{y}^{2}}}$ and that $y=0$ when $x=-2$.</div><ol class="i-list"><li><div class="list-item-paragraph">By substituting $v={{x}^{2}}+{{y}^{2}}$, show that the differential equation can be written as $\frac{\mathrm{d}v}{\mathrm{d}x}=2\sqrt{v}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $v$ in terms of $x$ and hence show that ${{y}^{2}}=\mathrm{f}\left( x \right)$ where $\mathrm{f}\left( x \right)$ is to be determined.</div><p style="text-align: right;">[4]</p></li></ol></li></ol>

Solving Differential Equations by Separation of Variables

Solving Differential Equations by Substitution

Verifying Solutions of Differential Equations

2020 EJC P1 Q9

Differential Equations

<p>Consider the curve $y=\frac{{{x}^{2}}+a}{4-x}$, where a is a constant and $a\ne 16$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the equations of the asymptotes.</div></li><li><div class="list-item-paragraph">Find, in terms of $a$, the coordinates where the curve intersects the axes.</div></li><li><div class="list-item-paragraph">Given the curve has one of its turning points at $x=-1$, show that the value of $a=9$. State also the coordinates of the other turning point.</div></li></ol>
<p>Hence or otherwise, state the range of values of $k$ for which the equation $\frac{{{x}^{2}}+9}{4-x}-k=0$ has no real roots.</p>

Consider the curve $y=\frac{{{x}^{2}}+a}{4-x}$, where a is a constant and $a\ne 16$.

Hence or otherwise, state the range of values of $k$ for which the equation $\frac{{{x}^{2}}+9}{4-x}-k=0$ has no real roots.

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+5}{\lambda x+1}$, where $\lambda $ is a non-zero constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Obtain the equations of the asymptotes of $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that $C$ has exactly two points at which $\frac{\mathrm{d}y}{\mathrm{d}x}=0$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Draw a sketch of $C$ to illustrate the case $\lambda >0$, and a separate of $C$ to illustrate the case $\lambda <0$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the set of values of $\lambda $ for which the line $y=4x$ and $C$ has at least one point in common.</span></div></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+5}{\lambda x+1}$, where $\lambda $ is a non-zero constant.

Obtain the equations of the asymptotes of $C$.

Show that $C$ has exactly two points at which $\frac{\mathrm{d}y}{\mathrm{d}x}=0$.

Draw a sketch of $C$ to illustrate the case $\lambda >0$, and a separate of $C$ to illustrate the case $\lambda <0$.

Find the set of values of $\lambda $ for which the line $y=4x$ and $C$ has at least one point in common.

N1999 P1 Q1 (FM)

<p>The curve $C$ has parametric equation $x=3-2\cos \,t$, $y=-\tan \,t$ for $0 < t < \frac{\pi }{2}$. Using differentiation, show that the curve $C$ does not have any stationary point.</p><p style="text-align: right;">[2]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">A point $Q$ on $C$ has parameter $t=\frac{\pi }{6}$. Find the equation of the tangent at $Q$, leaving your answer in exact form.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the locus of the mid-point of $(3-2\cos \,t,\,-\tan \,t)$ and $(-3,0)$ as $t$ varies.</span></div><p style="text-align: right;">[3]</p></li></ol>

The curve $C$ has parametric equation $x=3-2\cos \,t$, $y=-\tan \,t$ for $0 < t < \frac{\pi }{2}$. Using differentiation, show that the curve $C$ does not have any stationary point.

A point $Q$ on $C$ has parameter $t=\frac{\pi }{6}$. Find the equation of the tangent at $Q$, leaving your answer in exact form.

Find the cartesian equation of the locus of the mid-point of $(3-2\cos \,t,\,-\tan \,t)$ and $(-3,0)$ as $t$ varies.

Locus of a Midpoint

2016 ACJC MYE Q4

<p>The curve $C$ has the equation of $y=\frac{{{x}^{2}}-8}{x-2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$C$ has no stationary points.</div></li><li><div class="list-item-paragraph">Sketch the curve $C$, showing clearly the asymptotes and coordinates of any intersections with the coordinate axes.</div></li><li><div class="list-item-paragraph">By considering the equation of an appropriate line drawn on the same diagram with the curve $C$, find the range of values of $k$ for the equation ${{x}^{2}}-8=k{{(x-2)}^{2}}+4(x-2)$ to have no real roots.</div></li></ol>

The curve $C$ has the equation of $y=\frac{{{x}^{2}}-8}{x-2}$.

Sketch the curve $C$, showing clearly the asymptotes and coordinates of any intersections with the coordinate axes.

By considering the equation of an appropriate line drawn on the same diagram with the curve $C$, find the range of values of $k$ for the equation ${{x}^{2}}-8=k{{(x-2)}^{2}}+4(x-2)$ to have no real roots.

2013 ACJC J1 MYE Q4

<p>A sketch of the curve $y=\frac{a{{x}^{2}}+bx+c}{x+d}$, where&nbsp;$a$, $b$, $c$ and $d$ are constants, is shown, not to scale, in the diagram. The equations of the asymptotes, also shown in the diagram, are $x=2$ and $y=3-2x$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1783131710522-rjfb6yzii" data-list-item-path="item-1783131710522-rjfb6yzii"><div class="list-item-paragraph">Write the value of $d$.</div></li><li data-list-item-id="item-1783131711410-k1q9f1uih" data-list-item-path="item-1783131711410-k1q9f1uih"><div class="list-item-paragraph">State the value of $a$ and show that&nbsp;$b=7$.</div></li><li data-list-item-id="item-1783131712034-qykspf6b4" data-list-item-path="item-1783131712034-qykspf6b4"><div class="list-item-paragraph">Given that the curve has a stationary point where&nbsp;$x=1$, find the value of $c$ and the $x$-coordinate of the other stationary point.</div></li><li data-list-item-id="item-1783131712650-r3fu0jzop" data-list-item-path="item-1783131712650-r3fu0jzop"><div class="list-item-paragraph">Copy the above sketch and, by drawing a sketch of another suitable curve on the same diagram, show that the equation $2{{x}^{4}}-7{{x}^{3}}+8{{x}^{2}}+x-2=0$ has exactly two real roots.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="375" data-height="237" style="width:375px;height:237px;max-width:100%;display:block;margin:0 auto;"></div></div></div></div>

A sketch of the curve $y=\frac{a{{x}^{2}}+bx+c}{x+d}$, where $a$, $b$, $c$ and $d$ are constants, is shown, not to scale, in the diagram. The equations of the asymptotes, also shown in the diagram, are $x=2$ and $y=3-2x$.

N1998 P1 Q1

<p>A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d$. The graph of $y=\mathrm{f}\left( x \right)$ passes through the points $\left( 1,5 \right)$ and $\left( -1,-3 \right)$. The graph has a turning point at $x=1$, and $\displaystyle\int_{0}^{1}{\mathrm{f}\left( x \right)}\mathrm{d}x=6$.</p>
<p>Find the values of $a$, $b$, $c$ and $d$.<span class="inline-marks">[5]</span></p>

A function $\mathrm{f}$ is defined by $\mathrm{f}\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d$. The graph of $y=\mathrm{f}\left( x \right)$ passes through the points $\left( 1,5 \right)$ and $\left( -1,-3 \right)$. The graph has a turning point at $x=1$, and $\displaystyle\int_{0}^{1}{\mathrm{f}\left( x \right)}\mathrm{d}x=6$.

Find the values of $a$, $b$, $c$ and $d$.

Determining Model Coefficients Using Linear Systems

Definite Integral

N2021 P1 Q1

<p>It is given that $\mathrm{f}\left( x \right)=\frac{ax+b}{cx+d}$, for non-zero constants $a$, $b$, $c$, and $d$?</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that $ad-bc\ne 0$, show by differentiation that the graph of $y=\mathrm{f}\left( x \right)$ has no turning points.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">What can be said about the graph of $y=\mathrm{f}\left( x \right)$ when $ad-bc=0$?</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Deduce from part (i) that the graph of $y=\frac{3x-7}{2x+1}$ has a positive gradient at all points of the graph.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Sketch the graph of $y=\frac{3x-7}{2x+1}$, including the coordinates of the points where the graphs cross the axes and the equations of any asymptotes.</div><p style="text-align: right;">[2]</p></li></ol>

It is given that $\mathrm{f}\left( x \right)=\frac{ax+b}{cx+d}$, for non-zero constants $a$, $b$, $c$, and $d$?

Given that $ad-bc\ne 0$, show by differentiation that the graph of $y=\mathrm{f}\left( x \right)$ has no turning points.

What can be said about the graph of $y=\mathrm{f}\left( x \right)$ when $ad-bc=0$?

Deduce from part (i) that the graph of $y=\frac{3x-7}{2x+1}$ has a positive gradient at all points of the graph.

Sketch the graph of $y=\frac{3x-7}{2x+1}$, including the coordinates of the points where the graphs cross the axes and the equations of any asymptotes.

Math Question Bank