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

Parametric Equations

ACJC Tutorial 2 Q8

Graphing Techniques

<p>The curve $C$ has parametric equations</p>
<p style="text-align: center">$x={{t}^{2}}+4t$, $y={{t}^{3}}+{{t}^{2}}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the curve for $-2\le t\le 1$.</span></div><p style="text-align: right;">[1]</p></li></ol>
<p>The tangent to the curve at point $P$ where $t=2$ is denoted by $l$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of $l$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">The tangent $l$ meets $C$ again at the point $Q$. Use a non-calculator method to find the coordinates of $Q$.</span></div><p style="text-align: right;">[4]</p></li></ol>

$x={{t}^{2}}+4t$, $y={{t}^{3}}+{{t}^{2}}$.

The tangent to the curve at point $P$ where $t=2$ is denoted by $l$.

The tangent $l$ meets $C$ again at the point $Q$. Use a non-calculator method to find the coordinates of $Q$.

N2009 P2 Q1

<p style="text-align: center">$ \left( x-a \right)^2 +\left( y-b \right)^2=r^2  $</p>
<div class="graph-block" data-graph-config="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" data-width="200" data-height="200" style="width:200px;height:200px;display:block;margin:0 auto;"></div>
<p>Centre: <br>Radius:</p>

$ \left( x-a \right)^2 +\left( y-b \right)^2=r^2  $

Circle 

<p>A curve $C$ has parametric equations $x=\tan \theta +1$, $y=\sec \theta $.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find a cartesian equation of $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch $C$, giving the coordinates of any points where $C$ crosses the $x$- and $y$-axes and the equations of any asymptotes.</span></div></li></ol>

A curve $C$ has parametric equations $x=\tan \theta +1$, $y=\sec \theta $.

Sketch $C$, giving the coordinates of any points where $C$ crosses the $x$- and $y$-axes and the equations of any asymptotes.

NYJC Advanced Graphing Techniques Self-Review 7

<p>The curve $C$ is defined by the parametric equations</p>
<p style="text-align: center">$x={{\sin }^{3}}\left( 2t \right)$, $y=\sqrt{3}\tan t$, for $t\ge 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the curve $C$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the exact area of the region $R$ enclosed by $C$, the $y$-axis and the lines $y=1$ and $y=\sqrt{3}$.</span></div><p style="text-align: right;">[4]</p></li></ol>

The curve $C$ is defined by the parametric equations

$x={{\sin }^{3}}\left( 2t \right)$, $y=\sqrt{3}\tan t$, for $t\ge 0$.

Find the exact area of the region $R$ enclosed by $C$, the $y$-axis and the lines $y=1$ and $y=\sqrt{3}$.

2018 BT1 VJC Q3

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">$ \frac{ \left(x-a \right)^2}{p^2}-\frac{ \left(y-b \right)^2}{q^2} = 1  $</p><div class="graph-block" data-graph-config="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" data-width="200" data-height="200" style="width:200px;height:200px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column-divider"></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p style="text-align: center">$\frac{ \left(y-b \right)^2}{q^2}- \frac{ \left(x-a \right)^2}{p^2} = 1  $</p><div class="graph-block" data-graph-config="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" data-width="200" data-height="200" style="width:200px;height:200px;max-width:100%;display:block;margin:0 auto;"></div></div></div></div>
<p>Finding the Oblique Asymptotes:<br><br><br>Gradients of Oblique Asymptotes:</p>

Finding the Oblique Asymptotes:


Gradients of Oblique Asymptotes:

Hyperbola

<p>Write down all the asymptote(s) and axial intercepts of $y=\frac{2-{{x}^{2}}}{2x+1}$ and sketch the curve.</p>

Write down all the asymptote(s) and axial intercepts of $y=\frac{2-{{x}^{2}}}{2x+1}$ and sketch the curve.

<ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$a$ is a positive constant, sketch the curve with equation $y=\frac{x+a}{x-a}$. State the equations of any asymptotes and the coordinates of the points where the curve crosses the axes.<span class="inline-marks">[3]</span></div></li><li><ol class="abc-list"><li><div class="list-item-paragraph">Solve the inequality $\left| \frac{x+a}{x-a} \right|&gt;1$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Solve the inequality $\frac{\left| x \right|+a}{\left| x \right|-a}&gt;1$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

2021 VJC P1 Q2

<p>The curve $C$ has equation $\frac{9{{\left( y+3 \right)}^{2}}}{4}-\frac{{{\left( x-2 \right)}^{2}}}{4}=1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$, stating the equations of any asymptotes and the coordinates of any turning points.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence find the set of values of $m$ such that the straight line with gradient $m$ that passes through the point $\left( 2,-3 \right)$ intersects $C$ at least once.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $\frac{9{{\left( y+3 \right)}^{2}}}{4}-\frac{{{\left( x-2 \right)}^{2}}}{4}=1$.

Sketch $C$, stating the equations of any asymptotes and the coordinates of any turning points.

Hence find the set of values of $m$ such that the straight line with gradient $m$ that passes through the point $\left( 2,-3 \right)$ intersects $C$ at least once.

2022 VJC Promo Q3

<p>Write down all the asymptote(s) and axial intercepts of $y=\frac{5}{{{x}^{2}}+2x+5}$ and sketch the curve.</p>

Write down all the asymptote(s) and axial intercepts of $y=\frac{5}{{{x}^{2}}+2x+5}$ and sketch the curve.

<p>The parametric equations of a curve are $x=\frac{1}{2}{{\mathrm{e}}^{t}}-2{{\mathrm{e}}^{-t}}$ and $y={{\mathrm{e}}^{t}}+2{{\mathrm{e}}^{-t}}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Using calculus, find the exact gradient of the normal to the curve at the point where&nbsp;$t=\ln 4$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Show that the equation of the normal is $5x+7y=39$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the exact value of $t$ where the normal meets the curve again.<span class="inline-marks">[4]</span></div></li></ol>

The parametric equations of a curve are $x=\frac{1}{2}{{\mathrm{e}}^{t}}-2{{\mathrm{e}}^{-t}}$ and $y={{\mathrm{e}}^{t}}+2{{\mathrm{e}}^{-t}}$.

2023 NYJC Promo Q7

<p>The curve $C$ has equation ${\frac{{{\left( x-2 \right)}^{2}}}{2}+{{\left( y-3 \right)}^{2}}=1}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760868495354-qu7i0wz74" data-list-item-path="item-1760868495354-qu7i0wz74"><div class="list-item-paragraph">Sketch $C$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760868495916-nthzlmc19" data-list-item-path="item-1760868495916-nthzlmc19"><div class="list-item-paragraph">Hence state the range of positive values of $k$ such that&nbsp;${{\left( x-2 \right)}^{2}}+{{\left( y+3 \right)}^{2}}={{k}^{2}}$ intersects $C$ at twice.<span class="inline-marks">[2]</span></div></li></ol>

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

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}-3x+3}{1-x}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760426225255-w8z8w80qh" data-list-item-path="item-1760426225255-w8z8w80qh"><div class="list-item-paragraph">Sketch $C$, stating the coordinates of any turning points and points of intersection with the axes, and the equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760426225968-2ct1ijudn" data-list-item-path="item-1760426225968-2ct1ijudn"><div class="list-item-paragraph">Determine the set of exact values of $k$ such that&nbsp;$C$ and the line $y=kx$ has two points of intersection for all real values of $x$.<span class="inline-marks">[3]</span></div></li></ol>

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

2023 DHS P1 Q2

Summary Notes : Graphing Techniques

<p>A curve has parametric equations $x={{\cos }^{2}}t$, $y={{\sin }^{3}}t$, for $0\le t\le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the curve.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The tangent to the curve at the point $\left( {{\cos }^{2}}\theta ,\,\,{{\sin }^{3}}\theta \right)$ where $0<\theta <\frac{\pi }{2}$, meets the $x$- and $y$-axes at $Q$ and $R$ respectively. The origin is denoted by $O$. Show that the area of triangle $OQR$ is $\frac{1}{12}\sin \theta {{\left( 3{{\cos }^{2}}\theta +2{{\sin }^{2}}\theta \right)}^{2}}$.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Show that the area under the curve for $0\le t\le \frac{\pi }{2}$ is $2\displaystyle\int\nolimits_{0}^{\frac{\pi }{2}}{\cos t{{\sin }^{4}}\mathrm{t}\,\,\mathrm{d}t}$, and use the substituition $\sin t=u$ to find this area.</div><p style="text-align: right;">[5]</p></li></ol>

A curve has parametric equations $x={{\cos }^{2}}t$, $y={{\sin }^{3}}t$, for $0\le t\le \frac{\pi }{2}$.

N2007 P1 Q11

<p>A curve is given by the parametric equations $x={{t}^{2}}-3,\,\,y=t({{t}^{2}}-3)$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the Cartesian equation.</div></li><li><div class="list-item-paragraph">Prove that the curve is symmetrical about the $x$-axis.</div></li><li><div class="list-item-paragraph">Show that there are no points on the curve for which $x < -3$.</div></li></ol>

A curve is given by the parametric equations $x={{t}^{2}}-3,\,\,y=t({{t}^{2}}-3)$.

ACJC Tutorial Q2

<p>The curve $C$ is given by the equation</p>
<p style="text-align: center">$x(x+8)=4{{(y-2)}^{2}}-12$</p>
<p>Sketch $C$, indicating clearly the key features of the curve.</p>
<p>[You are not required to give the coordinates of the axial intercepts.]</p><p style="text-align: right;">[4]</p>
<p>State the range of values of $m$ such that there is no intersection between $C$ and the line $y=mx+4m+2$.</p><p style="text-align: right;">[2]</p>

Sketch $C$, indicating clearly the key features of the curve.

[You are not required to give the coordinates of the axial intercepts.]

State the range of values of $m$ such that there is no intersection between $C$ and the line $y=mx+4m+2$.

2020 HCI Promo Q4

<p>Write down all the asymptote(s) and axial intercepts of $y=\frac{2{{x}^{2}}+1}{{{x}^{2}}+3}$ and sketch the curve.</p>

Write down all the asymptote(s) and axial intercepts of $y=\frac{2{{x}^{2}}+1}{{{x}^{2}}+3}$ and sketch the curve.

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=t-\frac{10}{t}$, $y=6t-{{t}^{2}}$, for $\sqrt{10}\le t\le 6$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the curve $C$, labelling the coordinates of any points of intersection with the axes.</div><p style="text-align: right;">[2]</p></li></ol>
<p>It is given that a plot of flat land is represented by the region bounded by $C$ and both axes, where units are measured in kilometres.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the area of the plot of land.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The landowner considers using a portion of this land to rear cattle. This portion is a rectangular piece of land, $OPRQ$, where $O$ is the origin, $R$ is a point on the curve, $P$ and $Q$ are the points on the $x$-axis and $y$-axis respectively.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">By using differentiation, find the value of $t$ that maximises the area of rectangle $OPRQ$. (You need not show that your answer gives a maximum.)</div><p style="text-align: right;">[3]</p></li></ol>
<p>It is given instead that the landowner decides to divide the plot of flat land equally into two by building a straight fence through the land, parallel to the $y$-axis. It is assumed that the thickness of the fence is negligible.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Using the result from part <strong>(ii)</strong>, find the equation of the fence that divides the plot of land equally into two parts.</div><p style="text-align: right;">[4]</p></li></ol>

$x=t-\frac{10}{t}$, $y=6t-{{t}^{2}}$, for $\sqrt{10}\le t\le 6$.

It is given that a plot of flat land is represented by the region bounded by $C$ and both axes, where units are measured in kilometres.

The landowner considers using a portion of this land to rear cattle. This portion is a rectangular piece of land, $OPRQ$, where $O$ is the origin, $R$ is a point on the curve, $P$ and $Q$ are the points on the $x$-axis and $y$-axis respectively.

It is given instead that the landowner decides to divide the plot of flat land equally into two by building a straight fence through the land, parallel to the $y$-axis. It is assumed that the thickness of the fence is negligible.

2023 MI P1 Q12

<p>The equation of a curve $C$ is $x^3 + xy + 2y^3 = k$, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a-9"><div class="list-item-content"><span class="list-item-text">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-9"><div class="list-item-paragraph">It is given that&nbsp;$C$ has a tangent which is parallel to the $y$-axis. Show that the $y$-coordinate of the points of contact of the tangent with $C$ must satisfy $216y^6 + 4y^3 + k = 0$. Hence show that&nbsp;$k \leq \frac{1}{54}$.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-c-9"><div class="list-item-content"><span class="list-item-text">Find the possible values of $k$ in the case where the line $x = -6$ is a tangent to $C$.</span><span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve $C$ is $x^3 + xy + 2y^3 = k$, where $k$ is a constant.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $x$ and $y$.

It is given that $C$ has a tangent which is parallel to the $y$-axis.\n\nShow that the $y$-coordinate of the points of contact of the tangent with $C$ must satisfy $216y^6 + 4y^3 + k = 0$. Hence show that $k \leq \frac{1}{54}$.

Find the possible values of $k$ in the case where the line $x = -6$ is a tangent to $C$.

Math Question Bank