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PSLE

PSLE Foundation Mathematics

<p>A curve has the equation $y = x^3 - ax^2 + bx + 1$, where&nbsp;$a$ and $b$ are constants. Find, respectively, the relationship between $a$ and $b$ for which</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">$y$ is always increasing, and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">the curve has two distinct stationary points.</div></li><li data-list-item-id="item-1770378230806-b7vs3p6qd" data-list-item-path="item-1770378230806-b7vs3p6qd"><div class="list-item-paragraph">In the case where&nbsp;$a = 8$ and $b = 10$, and the rate of decrease of $x$ is $0.05$ units/s when&nbsp;$x = 1$, find the rate of change of $y$ at this instant.</div></li></ol>

A curve has the equation $y = x^3 - ax^2 + bx + 1$, where $a$ and $b$ are constants. Find, respectively, the relationship between $a$ and $b$ for which

the curve has two distinct stationary points.

Parameter Conditions for Stationary Points

Connected Rates of Change

Increasing and Decreasing Functions

Differentiation

<p>A curve has the equation $y = x^3 + 4x^2 + kx + 3$, where&nbsp;$k$ is a constant. Given that there exists at least one point on the curve at which the tangent to the curve is perpendicular to the $y$-axis, find</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">the range of values of $k$, and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">the $x$-coordinate(s), in terms of $k$, of such point(s) on the curve.</div></li></ol>

A curve has the equation $y = x^3 + 4x^2 + kx + 3$, where $k$ is a constant. Given that there exists at least one point on the curve at which the tangent to the curve is perpendicular to the $y$-axis, find

the $x$-coordinate(s), in terms of $k$, of such point(s) on the curve.

Tangents and Normals Parallel to the Axes

<p>A curve has the equation $y = \frac{ax^2 + x}{2x - 1},$ where&nbsp;$a$ is a constant and $a\notin\{0,-2\}$. Find, respectively, the range of values of $a$ such that</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">the gradient of the curve is always non-positive, and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">there exists at least one point on the curve at which the tangent to the curve is parallel to the $x$-axis.</div></li></ol>

A curve has the equation $y = \frac{ax^2 + x}{2x - 1},$ where $a$ is a constant and $a\notin\{0,-2\}$. Find, respectively, the range of values of $a$ such that

the gradient of the curve is always non-positive, and

there exists at least one point on the curve at which the tangent to the curve is parallel to the $x$-axis.

<p>A curve $C$ has equation $y = ax+b+\frac{b-2a}{x+2}$, where&nbsp;$a$ and $b$ are real constants such that&nbsp;$a &gt; 0$, $a \neq \frac{1}{2}b$ and $x \neq -2$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$C$ has stationary points, use differentiation to find the relationship between $a$ and $b$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is now given that&nbsp;$a = 1$ and $b = 3$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Prove algebraically that&nbsp;$y$ cannot lie between $-1$ and $3$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch $C$, stating the equations of any asymptotes and the coordinates of any axial intercepts and turning points.</span><span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has equation $y = ax+b+\frac{b-2a}{x+2}$, where $a$ and $b$ are real constants such that $a > 0$, $a \neq \frac{1}{2}b$ and $x \neq -2$.

Given that $C$ has stationary points, use differentiation to find the relationship between $a$ and $b$.

It is now given that $a = 1$ and $b = 3$.

Prove algebraically that $y$ cannot lie between $-1$ and $3$.

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

Sketching Rational Functions

Asymptotes and Discontinuities

Finding Possible y-values of a Curve

2025 RVHS P1 Q9

Techniques of Differentiation

Graphing Techniques

Applications of Differentiation

<p>The graph of $y = \frac{ax^2 + bx + c}{x + d}$ has a vertical asymptote with equation $x = -1$ and an oblique asymptote with equation $y = 2x + 2$. It is given that&nbsp;$c \neq 2$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Write down the value of $d$ and show that&nbsp;$a = 2$ and $b = 4$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Hence find the range of values of $c$ if the graph has no stationary points.</span><span class="inline-marks">[2]</span></div></li></ol>

The graph of $y = \frac{ax^2 + bx + c}{x + d}$ has a vertical asymptote with equation $x = -1$ and an oblique asymptote with equation $y = 2x + 2$. It is given that $c \neq 2$.

Write down the value of $d$ and show that $a = 2$ and $b = 4$.

Hence find the range of values of $c$ if the graph has no stationary points.

Parameter Conditions for Graphs and Equations

2025 ACJC P1 Q2

<p>The curve $C$ has the equation $y=ax+\frac{b}{x-2}$, where&nbsp;$a$ and $b$ are non-zero constants.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the condition(s) required for&nbsp;$a$ and $b$ for this curve to have two turning points.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$a=b$, find the range of values of $y$ can take, in terms of $a$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The inequality</div><div class="list-item-paragraph" style="text-align: center">$ax+\frac{b}{x-2} &lt; ax+m$</div><div class="list-item-paragraph">where $a&gt;0$, $b&lt;0$ and $m$ is a positive constant, has the solution set $\{x\in \mathbb{R}:x&lt;0$ or $x&gt;2\}$. Find $m$ in terms of $b$.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has the equation $y=ax+\frac{b}{x-2}$, where $a$ and $b$ are non-zero constants.

Inequalities involving Rational Functions

2020 DHS P1 Q4

Inequalities

<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

Graphical Root Counting

2024 ACJC Promo Q10

<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 $a>0$.

Integration by the Reverse Chain Rule

Volume of Revolution about the y-axis

Stationary Points and Points of Inflection

2024 RI P1 Q9

Applications of Integration

Techniques of Integration

<ol class="abc-list"><li data-list-item-id="item-1760700624129-xks5wjpxt" data-list-item-path="item-1760700624129-xks5wjpxt"><div class="list-item-paragraph">The diagram shows the curve with equation $y=\mathrm{f}\left( x \right)$. The curve crosses the $x$-axis at $x=1$ and $x=2.5$, crosses the $y$-axis at $y=1$ and has a maximum point at $\left( 4,\ 6 \right)$. The equations of the asymptotes are $x=2$ and $y=3$. Sketch the graph of $y=\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$, giving the equations of asymptotes, coordinates of turning points and axial intercepts, where possible.<span class="inline-marks">[2]</span></div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760700646547-qe4ykbfrydl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3MDA2NDY1NDctcWU0eWtiZnJ5ZGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU1NjM0MywiZXhwIjoxNzkzNTU2NjQzfQ._VK16hejCEchONTTnwjPyuPR9YCcn6kCxRJMfgi7iOE" alt="List item image" class="aligncenter custom-width" style="max-width: min(500px, 100%); height: auto;" /></div></li><li data-list-item-id="item-1760700624766-3zt3230tl" data-list-item-path="item-1760700624766-3zt3230tl"><div class="list-item-paragraph">The curve $C$ has equation $y=\frac{{{x}^{2}}+kx-1}{x+1}$, where&nbsp;$k$ is a non-zero constant.</div><ol class="i-list"><li data-list-item-id="subitem-1760700651533-w0amyfl0v" data-list-item-path="item-1760700624766-3zt3230tl/subitem-1760700651533-w0amyfl0v"><div class="list-item-paragraph">Find the range of values of $k$ for which $C$ has no stationary points.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1760700652194-719o65g8g" data-list-item-path="item-1760700624766-3zt3230tl/subitem-1760700652194-719o65g8g"><div class="list-item-paragraph">Given that&nbsp;$y=x+3$ is an asymptote of $C$, show that&nbsp;$k=4$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760700652786-5cjnudcc2" data-list-item-path="item-1760700624766-3zt3230tl/subitem-1760700652786-5cjnudcc2"><div class="list-item-paragraph">State a sequence of transformations which transform the graph of $y=\frac{x}{4}-\frac{1}{x}$ onto the graph of $y=\frac{{{x}^{2}}+4x-1}{x+1}$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

Composite Transformations

Sketching Derivative Graphs

2024 TJC P1 Q7

Transformations of Graphs

<p>It is given that $\mathrm{f}\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx$, where $a$, $b$ and $c$ are non-zero real constants.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $\mathrm{f}\left( -x \right)=-\mathrm{f}\left( x \right)$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">It is given that $\mathrm{f}\left( x \right)=0$ has only one real root and one of the non-real roots is $p+q\mathbf{i}$, where $p$ and $q$ are non-zero real constants. Find, in terms of $p$ and $q$, all the other non-real roots of $\mathrm{f}\left( x \right)=0$, justifying your answers.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $\mathop{\displaystyle\int }_{0}^{3}\mathrm{f}\left( x \right)\mathrm{d}x=-5$, state the values of $\displaystyle\int_{-3}^{3}{\mathrm{f}\left( x \right)\mathrm{d}x}$ and $\displaystyle\int_{-3}^{3}{\mathrm{f}\left( \left| x \right| \right)\mathrm{d}x}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Let $a=1$ and $b=3$ .</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">By considering $f\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)$, find the range of values of $c$ such that the curve with equation $y=\mathrm{f}\left( x \right)$ has 2 stationary points, showing your working clearly.</div><p style="text-align: right;">[3]</p></li></ol>

It is given that $\mathrm{f}\left( x \right)=a{{x}^{5}}+b{{x}^{3}}+cx$, where $a$, $b$ and $c$ are non-zero real constants.

Modulus Functions

Odd and Even Functions

2024 HCI P1 Q7

Functions

<p>A curve $C$ has equation</p>
<p style="text-align: center">${{x}^{3}}+{{y}^{3}}-xy=A$</p>
<p>where&nbsp;$A$ is a non-zero constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1760427585431-f69ru43cs" data-list-item-path="item-1760427585431-f69ru43cs"><div class="list-item-paragraph">Show that any stationary points on $C$ lie on</div><div class="list-item-paragraph" style="text-align: center">$y=3{{x}^{2}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760427586174-lfr6gnob6" data-list-item-path="item-1760427586174-lfr6gnob6"><div class="list-item-paragraph">Find, in terms of $A$, the $x$-coordinates of the stationary points, and hence determine the range of values of $A$ for which $C$ has two distinct stationary points.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760427586955-0oxq8snrq" data-list-item-path="item-1760427586955-0oxq8snrq"><div class="list-item-paragraph">Suppose that&nbsp;$C$ has two stationary points. Determine the nature of each of the stationary points found in part <strong>(b)</strong>.<span class="inline-marks">[3]</span></div></li></ol>

Implicit Differentiation

Stationary Points and Their Nature

2023 EJC P2 Q4

<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 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 $b$.

2009 AJC Promo Q13

<p>A curve $~C$ has equation $y=ax+b+\frac{a+2b}{x-1}$, where $a$ and $b$ are real constants such that $a>0$, $b\ne -\frac{1}{2}a$ and $x\ne 1$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that $~C$ has no stationary points, use differentiation to find the relationship between $a$ and $b$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>It is now given that $b=-2a$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Sketch $~C$ on the axes below stating the equations of any asymptotes and the coordinates of the points where$~C$ crosses the axes.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">On the same axes, sketch the graph of $y=ax-a$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Hence solve the inequality $x-2-\frac{3}{x-1}\le x-1$.</div><p style="text-align: right;">[2]</p></li></ol>

A curve $~C$ has equation $y=ax+b+\frac{a+2b}{x-1}$, where $a$ and $b$ are real constants such that $a>0$, $b\ne -\frac{1}{2}a$ and $x\ne 1$.

Graphical Method

N2022 P1 Q10

<p>The curve $C$ has equation $y=\frac{2{{x}^{2}}-a}{x+k}$ where&nbsp;$a,\ \,k&gt;0$.</p>
<ol class="i-list"><li data-list-item-id="item-1760088251155-l78b57v2k" data-list-item-path="item-1760088251155-l78b57v2k"><div class="list-item-paragraph">Given that&nbsp;$2{{k}^{2}}\ne a$, find the range of values of $k$ such that the curve $C$ has at least one tangent parallel to the $x$-axis.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760088251965-pu7xmzmpf" data-list-item-path="item-1760088251965-pu7xmzmpf"><div class="list-item-paragraph">What can be said about the graph of $y=\frac{2{{x}^{2}}-a}{x+k}$ when&nbsp;$2{{k}^{2}}=a$?<span class="inline-marks">[2]</span></div></li></ol>
<p>For the range of values of $k$ in <strong>(i)</strong>,</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760088287372-bjyr49kwa" data-list-item-path="item-1760088287372-bjyr49kwa"><div class="list-item-paragraph">Sketch the curve of $C$, clearly indicating the equations of the asymptotes and any intercepts with the axis.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760088288025-y8wme30zv" data-list-item-path="item-1760088288025-y8wme30zv"><div class="list-item-paragraph">The curve $y=\frac{2{{x}^{2}}-a}{x+k}$ does not intersect the curve ${{(x+k)}^{2}}-{{\left( \frac{y+4k}{b} \right)}^{2}}=1$ where&nbsp;$b&gt;0$. By drawing the $2$ curves on the same diagram, find the range of values of $b$.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $y=\frac{2{{x}^{2}}-a}{x+k}$ where $a,\ \,k>0$.

For the range of values of $k$ in **(i)**,

Graphs of Conic Sections

2012 AJC P1 Q9

<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

<ol class="abc-list"><li data-list-item-id="item-1759755636860-1t66gb1nz" data-list-item-path="item-1759755636860-1t66gb1nz"><div class="list-item-paragraph">The graph of $y=\mathrm{f}(x)$ is shown in the diagram. It passes through the origin and has a minimum point at $\left( 2,\mathrm{ }0 \right)$. The lines $x=1$ and $y=2$ are the asymptotes of the graph.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759755650243-f34pm5sc6li.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk3NTU2NTAyNDMtZjM0cG01c2M2bGkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU1NjM0MywiZXhwIjoxNzkzNTU2NjQzfQ.62fWv1kCv-JkaSmYDtPTVFhezNJFKkXp57OK042FZx8" alt="List item image" class="aligncenter custom-width" style="max-width: min(380px, 100%); height: auto;" /></div><div class="list-item-paragraph">Sketch the graph of $y=2-\mathrm{f}(x)$, stating the coordinates of the point where the graph crosses the $y$-axis, the coordinates of any stationary point(s) and the equations of any asymptotes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759755661017-2haot0l5v" data-list-item-path="item-1759755661017-2haot0l5v"><div class="list-item-paragraph">The curve $C$ has equation $y=\frac{2{{x}^{2}}+kx-2}{x+2}$, where&nbsp;$k$ is a constant.</div><ol class="i-list"><li data-list-item-id="subitem-1759755674420-dsx8v77re" data-list-item-path="item-1759755661017-2haot0l5v/subitem-1759755674420-dsx8v77re"><div class="list-item-paragraph">Find the set of values of $k$ for which $C$ has&nbsp;$2$ stationary points.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">For the rest of the question, take $k=-3$.</div></li><li data-list-item-id="subitem-1759755707029-cr9hjtsz6" data-list-item-path="item-1759755661017-2haot0l5v/subitem-1759755707029-cr9hjtsz6"><div class="list-item-paragraph">By expressing the equation $y=\frac{2{{x}^{2}}-3x-2}{x+2}$ in the form $y=ax+b+\frac{c}{x+2}$, where&nbsp;$a$, $b$ and $c$ are constants, write down the equations of the asymptotes of $C$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1759755710801-gy422hmex" data-list-item-path="item-1759755661017-2haot0l5v/subitem-1759755710801-gy422hmex"><div class="list-item-paragraph">Hence sketch $C$, giving the equations of any asymptotes, the coordinates of any stationary points and of the points where&nbsp;$C$ crosses the axes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1759755711619-gcd0ai1bo" data-list-item-path="item-1759755661017-2haot0l5v/subitem-1759755711619-gcd0ai1bo"><div class="list-item-paragraph">Describe a pair of transformations which transform the graph of $y=x+\frac{6}{x+2}$ to $C$.<span class="inline-marks">[2]</span></div></li></ol><div class="list-item-after-content">For the rest of the question, take $k=-3$.</div></li></ol>

2020 VJC Promo Q8

<p>The curve $C$ has equation $y=\frac{{{x}^{2}}-4x+{{k}^{2}}}{x-k}$ , $x\ne k$ , and $k$ is a constant such that&nbsp;$k\ne 0$ and $k\ne 2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the equations of the asymptotes of $C$.</div></li><li><div class="list-item-paragraph">Show that if&nbsp;$C$ has&nbsp;$2$ stationary points, then&nbsp;$k \lt 0$ or $k \gt 2$ .</div></li><li><div class="list-item-paragraph">Given that&nbsp;$y=x$ is an asymptote of $C$, find the value of $k$. With this value of $k$, sketch $C$, showing clearly the asymptotes and the stationary points.</div></li><li><div class="list-item-paragraph">By adding a suitable graph which passes through the point (4,4) on the sketch of $C$, find the range of values of $p$ for which the equation $({{x}^{2}}-4k+{{k}^{2}})-(x-k)(px+4-4p)=0$ has exactly 2 real roots for the value of $k$ found in <strong>(iii)</strong>.</div></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}-4x+{{k}^{2}}}{x-k}$ , $x\ne k$ , and $k$ is a constant such that $k\ne 0$ and $k\ne 2$.

<p>The curve $C$ has equation $y=\frac{a{{x}^{2}}+2}{x-1}$ where $x\ne 1$ and $a$ is a non-zero constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that if the curve $C$ has no stationary points, then $-2 < a < 0$.</div></li><li><div class="list-item-paragraph">Sketch the curve for $a=1$, showing clearly the asymptotes and coordinates of any intersections with the coordinate axes.</div></li><li><div class="list-item-paragraph">Verify that $y=k(x-1)+2$ passes through $(1,2)$ for all real values of $k$.</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 which the equation ${{x}^{2}}+2=k{{(x-1)}^{2}}+2(x-1)$ has no real roots.</div></li></ol>

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

2007 SAJC P1 Q7 [Modified]

<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.

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