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PSLE

PSLE Foundation Mathematics

<p>A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch $C$, indicating on the diagram the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts.<span class="inline-marks">[4]</span></div></li></ol>
<p>A curve $D$ has parametric equations</p>
<p style="text-align: center">\[x = 23\sin 2\theta - 1,\ y = 23 + 23\cos 2\theta\text{ for}\hspace{0.5em} -\frac{\pi}{2} &lt; \theta &lt; \frac{\pi}{2}.\]</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">Find the cartesian equation of curve $D$.<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">Hence, on the same diagram as in part <strong>(a)</strong>, sketch the curve $D$ and determine the number of intersection points between curve $C$ and curve $D$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$.

\[x = 23\sin 2\theta - 1,\ y = 23 + 23\cos 2\theta\text{ for } -\frac{\pi}{2} < \theta < \frac{\pi}{2}.\]

Sketching Rational Functions

Graphical Root Counting

Asymptotes and Discontinuities

Sketching Parametric Curves

Converting Parametric Equations to Cartesian Form

2024 EJC P1 Q4

Graphing Techniques

<p>The curve $C_1$ has equation $y = 2x + 3 + \frac{12}{x - 2}$, $x \neq 2$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Sketch $C_1$, showing clearly the equations of asymptotes, the coordinates of the axial intercepts and the turning points, if any.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">By completing the square for both&nbsp;$x$ and $y$, give a geometrical interpretation of the curve $C_2$ with equation $x^2 - 4x + y^2 + 6y - 3 = 0$.</div><div class="list-item-paragraph">Hence write down the number of real roots for the equation \[(x - 2)^2 + \left(2x + 6 + \frac{12}{x - 2}\right)^2 = 4^2.\]<span class="inline-marks">[3]</span></div></li></ol>

The curve $C_1$ has equation $y = 2x + 3 + \frac{12}{x - 2}$, $x \neq 2$.

Graphs of Conic Sections

2024 TMJC Promo Q3

<p>The curve $C$ has equation $y = \frac{x^2 + 3x}{x - 1}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact set of values of $y$ for which there are no point on $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Draw a sketch of $C$, showing clearly any axis intercepts, asymptotes and turning points.</span></div></li><li><div class="list-item-content"><span class="list-item-text">In the same diagram, draw a sketch of the curve $y = (x - 5)^2 + 1$ and hence find the number of real roots of the equation $x^3 - 12x^2 + 33x - 26 = 0$.</span></div></li></ol>

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

Find the exact set of values of $y$ for which there are no point on $C$.

Draw a sketch of $C$, showing clearly any axis intercepts, asymptotes and turning points.

In the same diagram, draw a sketch of the curve $y = (x - 5)^2 + 1$ and hence find the number of real roots of the equation $x^3 - 12x^2 + 33x - 26 = 0$.

Finding Possible y-values of a Curve

N1990 P1 Q1

<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

Parameter Conditions for Stationary Points

2024 ACJC Promo Q10

Techniques of Differentiation

Applications of Differentiation

<ol class="abc-list"><li data-list-item-id="item-1760701767667-3e17w9o5w" data-list-item-path="item-1760701767667-3e17w9o5w"><div class="list-item-paragraph">The curve $C$ has equation $y=\mathrm{f}\left( x \right)$ where&nbsp;$\mathrm{f}\left( x \right)=\frac{a{{x}^{2}}+bx+c}{x+d}$, and $a$, $b$, $c$ and $d$ are constants, and $a\ne 0$. Given that&nbsp;$C$ has asymptote $y=x+1$, find the value of $a$ and show that&nbsp;$b=d+1$.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">If&nbsp;$\mathrm{f}$ is an increasing function for all $x\in \mathbb{R}$, $x\ne -d$, show that&nbsp;$c\le d$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760701768590-fql24blu9" data-list-item-path="item-1760701768590-fql24blu9"><div class="list-item-paragraph">It is further given that&nbsp;$c=1$ and $d=2$.</div><ol class="i-list"><li data-list-item-id="subitem-1760701784340-3tm31dynp" data-list-item-path="item-1760701768590-fql24blu9/subitem-1760701784340-3tm31dynp"><div class="list-item-paragraph">Sketch $C$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760701785045-o0glkwown" data-list-item-path="item-1760701768590-fql24blu9/subitem-1760701785045-o0glkwown"><div class="list-item-paragraph">By sketching a suitable graph in the same diagram in part <strong>(b)(i)</strong>, find the number of real roots to the equation ${{\left( \frac{{{x}^{2}}+3x+1}{x+2} \right)}^{2}}+4{{x}^{2}}=16$<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Parameter Conditions for Graphs and Equations

Increasing and Decreasing Functions

2024 VJC P1 Q7

<ol class="abc-list"><li data-list-item-id="item-1760695036327-a9ckxswnk" data-list-item-path="item-1760695036327-a9ckxswnk"><div class="list-item-paragraph">State a sequence of transformations that will transform the curve with equation ${{y}^{2}}-{{x}^{2}}=1$ on to the curve with equation $9{{y}^{2}}-54y-{{x}^{2}}-2x+79=0$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760695037024-ina5nxhaw" data-list-item-path="item-1760695037024-ina5nxhaw"><div class="list-item-paragraph">A curve $C$ has equation</div><div class="list-item-paragraph" style="text-align: center">$9{{y}^{2}}-54y-{{x}^{2}}-2x+79=0$.</div><ol class="i-list"><li data-list-item-id="subitem-1760695054305-49r742fa2" data-list-item-path="item-1760695037024-ina5nxhaw/subitem-1760695054305-49r742fa2"><div class="list-item-paragraph">For real values $x$, use a non-graphical method to determine that&nbsp;$y$ cannot lie between $a$ and $b$, where&nbsp;$a$ and $b$ are exact real constants to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760695056217-r7eoweqsq" data-list-item-path="item-1760695037024-ina5nxhaw/subitem-1760695056217-r7eoweqsq"><div class="list-item-paragraph">Sketch the curve $C$, indicating clearly the equations of all asymptotes and the coordinates of the turning points.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760695057028-3gept328l" data-list-item-path="item-1760695037024-ina5nxhaw/subitem-1760695057028-3gept328l"><div class="list-item-paragraph">By adding a suitable curve, determine the number of real roots of the equation,</div><div class="list-item-paragraph" style="text-align: center">$9{{\left[ {{\left( x+1 \right)}^{2}}+3 \right]}^{2}}-54\left[ {{\left( x+1 \right)}^{2}}+3 \right]-{{x}^{2}}-2x+79=0$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Composite Transformations

2024 ASRJC P1 Q7

Transformations of Graphs

<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 $x=1$.

2023 CJC P1 Q6

<p>The function $y=\mathrm{f}\left( x \right)$ is defined by</p>
<p style="text-align: center">$y=\frac{{{x}^{2}}+x+p}{x+2}$, where&nbsp;$p$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value of $p$ if&nbsp;$\mathrm{f}$ is a linear function.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">If&nbsp;$p$ is greater than the value found in part <strong>(i)</strong>, use an algebraic method to determine the range of values that&nbsp;$y$ can take, leaving your answer in terms of $p$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Take $p=12$ for the rest of this question.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Sketch the graph of $y=\frac{{{x}^{2}}+x+12}{x+2}$. Give the equation(s) of any asymptotes, the coordinates of any turning point(s) and the points of intersection with the axes.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the number of real roots of the equation</div><div class="list-item-paragraph" style="text-align: center">${{\left( {{x}^{2}}+x+12 \right)}^{2}}-{{\left( x+2 \right)}^{2}}\left( 36-{{x}^{2}} \right)=0$</div><div class="list-item-paragraph">by adding a suitable graph on the sketch in part <strong>(iii)</strong>.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Describe a sequence of $3$ transformations which transforms the graph of $y=\frac{{{x}^{2}}+x+12}{x+2}$ onto $y=-2x+3-\frac{7}{x+1}$.<span class="inline-marks">[3]</span></div></li></ol>

The function $y=\mathrm{f}\left( x \right)$ is defined by

$y=\frac{{{x}^{2}}+x+p}{x+2}$, where $p$ is a constant.

Take $p=12$ for the rest of this question.

2022 HCI J1 BT Q8

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+4}{x+b}$.</p>
<p>It is given that $C$ has vertical asymptote $x=-1$ and a stationary point at $x=2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Determine the values of $a$ and $b$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the equation of the other asymptote of $C$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Prove, using an algebraic method, that $C$ cannot lie between two values (to be determined).</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Draw a sketch of $C$, showing clearly any axial intercepts, asymptotes and stationary points.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Deduce the number of real roots of the equation $\left( 4-{{x}^{2}} \right){{\left( x+1 \right)}^{2}}={{\left( {{x}^{2}}-4x+4 \right)}^{2}}$.</div><p style="text-align: right;">[2]</p></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+4}{x+b}$.

It is given that $C$ has vertical asymptote $x=-1$ and a stationary point at $x=2$.

Stationary Points and Their Nature

2007 HCI P1 Q13

<p>The curve $C$ has equation $y=\frac{3{{x}^{2}}-px-1}{x-p}$ where $p$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">If $C$ has no stationary points, use an alegbraic method to find the range of possible values of $p$.</div><p style="text-align: right;">[4]</p></li></ol>
<p>It is given that the equation of the oblique asymptote of $C$ is $y=3x-1$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the value of $p$.</div><p style="text-align: right;">[1]</p></li></ol>
<p>For the rest of the question, use the value of $p$ found in part <strong>(ii)</strong>.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Sketch the curve $C$, stating clearly the equations of the asymptotes and the coordinates of any stationary point(s) and the coordinates of any axial intercept(s).</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By sketching another suitable curve on the same diagram in part <strong>(iii)</strong>, deduce the number of real roots of the equation $3{{x}^{3}}-p{{x}^{2}}-x=2x-2p$.</div><p style="text-align: right;">[3]</p></li></ol>

The curve $C$ has equation $y=\frac{3{{x}^{2}}-px-1}{x-p}$ where $p$ is a constant.

It is given that the equation of the oblique asymptote of $C$ is $y=3x-1$.

For the rest of the question, use the value of $p$ found in part **(ii)**.

2022 CJC J1 MYE Q9

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+ax-4}{x+b}$, where $a$ and $b$ are constants, and $x\ne -b$. It is given that the asymptotes of $C$ are $y=x-2$ and $x=1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the values of $a$ and $b$. Hence sketch $C$, stating the coordinates of the point(s) where the curve crosses the axes, and the equations of the asymptotes.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">By drawing a suitable curve on the same diagram in part <strong>(i)</strong>, find the number of real distinct roots of the equation ${{x}^{2}}-2x-20+21{{\left( \frac{{{x}^{2}}+ax-4}{x+b} \right)}^{2}}=0$.</div><p style="text-align: right;">[3]</p></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+ax-4}{x+b}$, where $a$ and $b$ are constants, and $x\ne -b$. It is given that the asymptotes of $C$ are $y=x-2$ and $x=1$.

2018 EJC Promo 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.

Stationary Points and Points of Inflection

FM 395 P1 Q1

<p>A curve $C$ has equation $y=\frac{2{{x}^{2}}+5x-1}{x+3}$.</p>
<ol class="i-list"><li data-list-item-id="item-1759820094000-2x33o95od" data-list-item-path="item-1759820094000-2x33o95od"><div class="list-item-paragraph">Using an algebraic method, determine the range of values that&nbsp;$y$ can take.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759820094601-qplwxb5ek" data-list-item-path="item-1759820094601-qplwxb5ek"><div class="list-item-paragraph">Sketch $C$, labelling any turning points and equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759820095365-avxwagj7e" data-list-item-path="item-1759820095365-avxwagj7e"><div class="list-item-paragraph">On a separate diagram, sketch the graph of ${{\left( x-3 \right)}^{2}}+k{{y}^{2}}=k$ where&nbsp;$1 &lt; k\le 25$. Label all critical points clearly. Hence, determine all possible numbers of solutions to ${{\left( x-3 \right)}^{2}}+k{{\left( \frac{2{{x}^{2}}+5x-1}{x+3} \right)}^{2}}=k$.<span class="inline-marks">[3]</span></div></li></ol>

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

2022 EJC J1 MYE Q8

<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 $y=\frac{2{{x}^{2}}+3}{x-2},x\in \mathbb{R},x\ne 2$.

ACJC Tutorial 2 Q8

<p>The diagram below shows the graph of the function $y=\mathrm{f}\left( x \right)$, where</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)=\frac{a{{x}^{2}}+bx+c+2}{x+d}$ and $a$, $b$, $c$ and $d$ are integer constants.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759222408207-jc1q7i9b48.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkyMjI0MDgyMDctamMxcTdpOWI0OC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTc3MDA1LCJleHAiOjE3OTM1NzczMDV9.s3DfAlxhg_gDmQIbXrE8rTHM_L6vK2Y4ZB9egxMfSPE" width="400" class="aligncenter" /></figure>
<p>The curve has a vertical asymptote at the $y$-axis and an oblique asymptote which intersects the axes at $x=1$ and $y=-1$. Find the values of $a$, $b$ and $d$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">By considering$\frac{\mathrm{d}y}{\mathrm{d}x}$, show that $c<-2$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using the values of $a$, $b$ and $d$ found in (i), sketch ${{C}_{1}}$, the graph of $y=\mathrm{f}\left( x \right)$ if $c=-1$, labelling clearly the equation of asymptotes, coordinates of the turning points and axial intercepts (if any).</span></div></li></ol>
<p>On the same diagram, sketch the curve ${{C}_{2}}$, defined by the parametric equations</p>
<p style="text-align: center">$x=t-\cos t$, $y=2\sin t$, $0\le t\le \pi $.</p>
<p>Deduce the number of intersection points between ${{C}_{1}}$ and ${{C}_{2}}$.</p>

The diagram below shows the graph of the function $y=\mathrm{f}\left( x \right)$, where

$\mathrm{f}\left( x \right)=\frac{a{{x}^{2}}+bx+c+2}{x+d}$ and $a$, $b$, $c$ and $d$ are integer constants.

The curve has a vertical asymptote at the $y$-axis and an oblique asymptote which intersects the axes at $x=1$ and $y=-1$. Find the values of $a$, $b$ and $d$.

By considering$\frac{\mathrm{d}y}{\mathrm{d}x}$, show that $c<-2$.

Using the values of $a$, $b$ and $d$ found in (i), sketch ${{C}_{1}}$, the graph of $y=\mathrm{f}\left( x \right)$ if $c=-1$, labelling clearly the equation of asymptotes, coordinates of the turning points and axial intercepts (if any).

On the same diagram, sketch the curve ${{C}_{2}}$, defined by the parametric equations

$x=t-\cos t$, $y=2\sin t$, $0\le t\le \pi $.

Deduce the number of intersection points between ${{C}_{1}}$ and ${{C}_{2}}$.

NJC Graphing Techniques Tutorial Q4

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+2x+q}{x-p}$, $x\ne p$, where $p$ and $q$ are constants. It is given that the line $y=x+3$ is an asymptote of $C$.</p>
<p>Show that $p=1$.<span class="inline-marks">[2]</span></p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">If $C$ has $2$ stationary points, show that $q>-3$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch $C$ for $q=1$. By adding another graph, deduce that for all positive $\beta $, the equation $\left( \beta +1 \right){{x}^{2}}+2x+1+\beta {{x}^{3}}$ has exactly one real root.<span class="inline-marks">[5]</span></div></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+2x+q}{x-p}$, $x\ne p$, where $p$ and $q$ are constants. It is given that the line $y=x+3$ is an asymptote of $C$.

If $C$ has $2$ stationary points, show that $q>-3$.

Sketch $C$ for $q=1$. By adding another graph, deduce that for all positive $\beta $, the equation $\left( \beta +1 \right){{x}^{2}}+2x+1+\beta {{x}^{3}}$ has exactly one real root.

2018 DHS Promo 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>The curve ${{C}_{1}}$ has equation ${{x}^{2}}-{{y}^{2}}+6y+16=0$</p>
<p>The curve ${{C}_{2}}$ has equation ${{\left( \frac{x-p}{5} \right)}^{2}}-{{\left( \frac{y-q}{5} \right)}^{2}}=1$, where&nbsp;$p$ and $q$ are constants.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch ${{C}_{1}}$, stating the coordinates of any points of intersection with the $x$- and $y$-axes and the equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The line $y=mx+c$ intersects ${{C}_{1}}$ twice. State the range of values of $m$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;${{C}_{1}}$ and ${{C}_{2}}$ have the same asymptotes, determine the values of $p$ and $q$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">On the same diagram in part <strong>(i)</strong>, sketch ${{C}_{2}}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Let&nbsp;$n$ be the number of points of intersection of the curve ${{x}^{2}}+{{(y-3)}^{2}}={{r}^{2}}$ where&nbsp;$r&gt;0$, with the curve ${{C}_{2}}$. Determine the possible values of $n$.<span class="inline-marks">[1]</span></div></li></ol>

The curve ${{C}_{1}}$ has equation ${{x}^{2}}-{{y}^{2}}+6y+16=0$

The curve ${{C}_{2}}$ has equation ${{\left( \frac{x-p}{5} \right)}^{2}}-{{\left( \frac{y-q}{5} \right)}^{2}}=1$, where $p$ and $q$ are constants.

Sketch ${{C}_{1}}$, stating the coordinates of any points of intersection with the $x$- and $y$-axes and the equations of any asymptotes.

The line $y=mx+c$ intersects ${{C}_{1}}$ twice. State the range of values of $m$.

Given that ${{C}_{1}}$ and ${{C}_{2}}$ have the same asymptotes, determine the values of $p$ and $q$.

Let $n$ be the number of points of intersection of the curve ${{x}^{2}}+{{(y-3)}^{2}}={{r}^{2}}$ where $r>0$, with the curve ${{C}_{2}}$. Determine the possible values of $n$.

2014 JJC Promo Q6

<ol class="i-list"><li data-list-item-id="item-1758939496608-73imtj679" data-list-item-path="item-1758939496608-73imtj679"><div class="list-item-paragraph">Sketch the graph of $y=\frac{{{x}^{2}}+2}{2x+1}$, stating the equation(s) of the asymptote(s), the coordinates of the turning point(s) and intersection(s) with the axes.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1758939497920-gh4aday59" data-list-item-path="item-1758939497920-gh4aday59"><div class="list-item-paragraph">Let&nbsp;$P$ be the point of intersection of the oblique asymptote of the graph $y=\frac{{{x}^{2}}+2}{2x+1}$ and the $y$-axis. Show that&nbsp;$P$ lies on the line $4y=mx-1$ for all real values of $m$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1758939499135-yvioz70ng" data-list-item-path="item-1758939499135-yvioz70ng"><div class="list-item-paragraph">Deduce the number of real root(s) for the equation $4{{x}^{2}}+8=\left( mx-1 \right)\left( 2x+1 \right)$, where&nbsp;$0 &lt; m &lt; 2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1758939499690-r8hh29sot" data-list-item-path="item-1758939499690-r8hh29sot"><div class="list-item-paragraph">Hence, what can be said about the two roots of the equation $4{{x}^{2}}+8=\left( mx-1 \right)\left( 2x+1 \right)$, where&nbsp;$0 &lt; m &lt; 2$?<span class="inline-marks">[1]</span></div></li></ol>

Sketch the graph of $y=\frac{{{x}^{2}}+2}{2x+1}$, stating the equation(s) of the asymptote(s), the coordinates of the turning point(s) and intersection(s) with the axes.

Let $P$ be the point of intersection of the oblique asymptote of the graph $y=\frac{{{x}^{2}}+2}{2x+1}$ and the $y$-axis. Show that $P$ lies on the line $4y=mx-1$ for all real values of $m$.

Deduce the number of real root(s) for the equation $4{{x}^{2}}+8=\left( mx-1 \right)\left( 2x+1 \right)$, where $0 < m < 2$.

Hence, what can be said about the two roots of the equation $4{{x}^{2}}+8=\left( mx-1 \right)\left( 2x+1 \right)$, where $0 < m < 2$?

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