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PSLE

PSLE Foundation Mathematics

<ol class="i-list"><li data-list-item-id="atan-30" data-list-item-path="atan-30"><div class="list-item-paragraph">Using a right-angled triangle, show that, for&nbsp;$a&gt;0$, \[\tan^{-1}(a)+\tan^{-1}\left(\frac1a\right)=\frac{\pi}{2}.\]<span class="inline-marks">[2]</span></div></li></ol>
<p>The curves $D_1$ and $D_2$ have equations \[D_1:\ y=\frac{2}{1+(x-1)^2},\qquad D_2:\ y=\frac{k}{12x-7},\] where&nbsp;$k&gt;0$.</p>
<ol class="i-list" start="2"><li data-list-item-id="atan-33" data-list-item-path="atan-33"><div class="list-item-paragraph">Determine the values of $k$ for which the curves have at least one point of intersection.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the remaining parts, take $k=5$.</p>
<ol class="i-list" start="3"><li data-list-item-id="atan-36" data-list-item-path="atan-36"><div class="list-item-paragraph">Sketch both curves on the same axes. Label their intersections with the coordinate axes, the maximum point of $D_1$, and all asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="atan-37" data-list-item-path="atan-37"><div class="list-item-paragraph">Calculate the exact area of the finite region enclosed by the curves, simplifying your answer using part <strong>(i)</strong>.<span class="inline-marks">[5]</span></div></li></ol>

Using a right-angled triangle, show that, for $a>0$, \[\tan^{-1}(a)+\tan^{-1}\left(\frac1a\right)=\frac{\pi}{2}.\]

The curves $D_1$ and $D_2$ have equations \[D_1:\ y=\frac{2}{1+(x-1)^2},\qquad D_2:\ y=\frac{k}{12x-7},\] where $k>0$.

Determine the values of $k$ for which the curves have at least one point of intersection.

Sketch both curves on the same axes. Label their intersections with the coordinate axes, the maximum point of $D_1$, and all asymptotes.

Calculate the exact area of the finite region enclosed by the curves, simplifying your answer using part **(i)**.

Sketching Rational Functions

Inverse Trigonometric Functions

Parameter Conditions for Graphs and Equations

Asymptotes and Discontinuities

Area bounded by 2 curves

A complementary-angle identity and a shifted-curve area

Applications of Integration

Graphing Techniques

Functions

<ol class="i-list"><li data-list-item-id="atan-21" data-list-item-path="atan-21"><div class="list-item-paragraph">By considering two perpendicular lines, explain why \[\tan^{-1}(3)-\tan^{-1}\left(-\frac13\right)=\frac{\pi}{2}.\]<span class="inline-marks">[1]</span></div></li></ol>
<p>The curves $C_1$ and $C_2$ have equations \[C_1:\ y=\frac{1}{1+x^2},\qquad C_2:\ y=\frac{k}{4x+3},\] where&nbsp;$k&gt;0$.</p>
<ol class="i-list" start="2"><li data-list-item-id="atan-24" data-list-item-path="atan-24"><div class="list-item-paragraph">Find the set of values of $k$ for which the curves have at least one point of intersection.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the remaining parts, take $k=\frac32$.</p>
<ol class="i-list" start="3"><li data-list-item-id="atan-27" data-list-item-path="atan-27"><div class="list-item-paragraph">Sketch both curves on the same axes, labelling their intersections with the coordinate axes and stating the equations of all asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="atan-28" data-list-item-path="atan-28"><div class="list-item-paragraph">Find the exact area of the finite region enclosed by the curves. Use the identity in part <strong>(i)</strong> to simplify your answer.<span class="inline-marks">[5]</span></div></li></ol>

By considering two perpendicular lines, explain why \[\tan^{-1}(3)-\tan^{-1}\left(-\frac13\right)=\frac{\pi}{2}.\]

The curves $C_1$ and $C_2$ have equations \[C_1:\ y=\frac{1}{1+x^2},\qquad C_2:\ y=\frac{k}{4x+3},\] where $k>0$.

Find the set of values of $k$ for which the curves have at least one point of intersection.

For the remaining parts, take $k=\frac32$.

Sketch both curves on the same axes, labelling their intersections with the coordinate axes and stating the equations of all asymptotes.

Find the exact area of the finite region enclosed by the curves. Use the identity in part **(i)** to simplify your answer.

An inverse-tangent identity and an enclosed rational-curve area

<p>The curve $C_1$ has equation $y=\frac{ax^2+bx+3}{x+2}$, where&nbsp;$a$ and $b$ are constants. It is given that&nbsp;$C_1$ has an asymptote $y=x-3$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-ri-promo-q7-ri-q7-item-block-1-item-0" data-list-item-path="h2-2025-ri-promo-q7-ri-q7-item-block-1-item-0"><div class="list-item-paragraph">State the value of $a$ and show that&nbsp;$b=-1$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q7-ri-q7-item-block-1-item-1" data-list-item-path="h2-2025-ri-promo-q7-ri-q7-item-block-1-item-1"><div class="list-item-paragraph"><p>Using an <strong>algebraic method</strong>, find the set of values that&nbsp;$y$ cannot take.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>The locus of a point is defined as the path traced out by that point as it moves.</p>
<ol class="abc-list" start="3"><li data-list-item-id="h2-2025-ri-promo-q7-ri-q7-item-block-3-item-0" data-list-item-path="h2-2025-ri-promo-q7-ri-q7-item-block-3-item-0"><div class="list-item-paragraph">Let&nbsp;$P(x, y_1)$ be a point on the curve $C_1$ and $Q(x, y_2)$ be a point on the curve $C_2$ with equation $y=\frac{3x-x^2}{x+2}$. Find the cartesian equation of the locus of the midpoint of $P$ and $Q$ as&nbsp;$x$ varies.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C_1$ has equation $y=\frac{ax^2+bx+3}{x+2}$, where $a$ and $b$ are constants. It is given that $C_1$ has an asymptote $y=x-3$.

State the value of $a$ and show that $b=-1$.

The locus of a point is defined as the path traced out by that point as it moves.

Let $P(x, y_1)$ be a point on the curve $C_1$ and $Q(x, y_2)$ be a point on the curve $C_2$ with equation $y=\frac{3x-x^2}{x+2}$. Find the cartesian equation of the locus of the midpoint of $P$ and $Q$ as $x$ varies.

Locus of a Midpoint

Finding Possible y-values of a Curve

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<p>A curve $C_1$ has equation $(ky)^2-8ky-x^2+2x=-11$, where&nbsp;$k&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-hci-promo-q4-hci-q4-item-block-1-item-0" data-list-item-path="h2-2025-hci-promo-q4-hci-q4-item-block-1-item-0"><div class="list-item-paragraph">Show that the equation of $C_1$ can be expressed as&nbsp;$\alpha\left(y-\frac{4}{k}\right)^2-\beta(x-1)^2=1$, where&nbsp;$\alpha$ is a constant in terms of $k$ and $\beta$ is a constant to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-hci-promo-q4-hci-q4-item-block-1-item-1" data-list-item-path="h2-2025-hci-promo-q4-hci-q4-item-block-1-item-1"><div class="list-item-paragraph">It is given that one of the asymptotes of $C_1$ has equation $y=\frac{1}{2}x+\frac{3}{2}$. Find the value of $k$. Hence write down the equation of the other asymptote.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-hci-promo-q4-hci-q4-item-block-1-item-2" data-list-item-path="h2-2025-hci-promo-q4-hci-q4-item-block-1-item-2"><div class="list-item-paragraph"><p>Using the value of $k$ found in part <strong>(b)</strong>, sketch $C_1$, stating clearly the equations of its asymptotes and the coordinates of any turning points.</p><span class="inline-marks">[3]</span></div></li></ol>

A curve $C_1$ has equation $(ky)^2-8ky-x^2+2x=-11$, where $k>0$.

Show that the equation of $C_1$ can be expressed as $\alpha\left(y-\frac{4}{k}\right)^2-\beta(x-1)^2=1$, where $\alpha$ is a constant in terms of $k$ and $\beta$ is a constant to be determined.

It is given that one of the asymptotes of $C_1$ has equation $y=\frac{1}{2}x+\frac{3}{2}$. Find the value of $k$. Hence write down the equation of the other asymptote.

Graphs of Conic Sections

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<ol class="abc-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0"><div class="list-item-paragraph">The graphs of $y=2|5x-p|$ and $y=qx$, where&nbsp;$p,q&gt;0$, intersect at only one point.</div><ol class="i-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-0"><div class="list-item-paragraph"><p>On a single diagram, sketch the two graphs.</p><p>State the range of values of $q$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-1" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-0-item-0-sub-1"><div class="list-item-paragraph"><p>Solve $2|5x-p|&lt;10x$. Deduce the solution set of the inequality $\left|\frac{10}{x}-2p\right|&lt;\frac{10}{x}$.</p><p>Leave both answers in terms of $p$.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>
<ol class="abc-list" start="2"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0"><div class="list-item-paragraph"><p>The diagram shows a sketch of the curve $y=2\mathrm{f}(-x-1)$.</p><p>The curve has asymptotes with equation $x=-1$, $x=1$ and $y=0$.</p><p>The curve has a stationary point of inflexion at $(0,0)$, a maximum point at $(-2,3)$ and cuts the $x$-axis at $(-1.5,0)$.</p></div><div class="graph-block" 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data-width="460" data-height="360" style="width:460px;height:360px;max-width:100%;display:block;margin:0 auto;"></div><div class="list-item-paragraph">On separate diagrams, sketch the graphs of</div><ol class="i-list"><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-0" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-0"><div class="list-item-paragraph">$y=\frac{1}{2\mathrm{f}(-x-1)}$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-1" data-list-item-path="h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0/h2-2025-dhs-promo-q11-dhs-q11-item-block-1-item-0-after-1-item-1"><div class="list-item-paragraph"><p>$y=\mathrm{f}(x)$,</p><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content">stating clearly the equation of any asymptotes, and the coordinates of any turning points and axial intercepts.</div></li></ol>

The graphs of $y=2|5x-p|$ and $y=qx$, where $p,q>0$, intersect at only one point.

Inequalities involving Substitution

Graphical Method

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

Recovering Original Graphs from Transformations

Graphs of Modulus Function

Inequalities involving Modulus Functions

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Inequalities

Transformations of Graphs

<p>The curve $C$ has equation $\frac{(y-3)^2}{5}-\frac{(x+2)^2}{9}=1$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch $C$, stating the equations of the asymptotes and their point of intersection.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence state the exact range of values of $m$ such that the straight line $y=m(x+2)+3$ intersect $C$.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $\frac{(y-3)^2}{5}-\frac{(x+2)^2}{9}=1$.

<p>The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$, where&nbsp;$a$ is a constant, $a \neq 6$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find the set of values of $a$ such that&nbsp;$C$ does not cut the $x$-axis.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">It is given that the equation of $C$ is $y = \frac{3x^2 - 6x + 3}{x + 1}$. Sketch $C$ and give the equations of any asymptotes. Also, state the coordinates of any points where&nbsp;$C$ crosses the axes and of any turning points.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Let&nbsp;$k$ be a positive constant. By sketching a suitable graph in the same diagram in part <strong>(b)</strong>, find an inequality satisfied by $k$ such that the equation</div><div class="list-item-paragraph" style="text-align: center">\[k^2(x - 1)^2 + \left(\frac{3x^2 - 6x + 3}{x + 1} + 12\right)^2 = k^2\]</div><div class="list-item-paragraph">has 2 real and distinct roots.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$, where $a$ is a constant, $a \neq 6$.

2023 VJC Promo Q10

<p>The curve $C_1$ is given by the equation</p>
<p style="text-align: center">\[y = \frac{ax^2 + bx}{x + c}\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The curve $C_1$ has a vertical asymptote at $x = 2$. The linear oblique asymptote of $C_1$ passes through the points with coordinates $(0, 10)$ and $(-2, 0)$. Determine the values of the unknown constants $a$, $b$ and $c$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence sketch $C_1$, stating clearly the equation(s) of any asymptote(s) and the coordinates of any stationary point(s).</span></div></li></ol>

The curve $C_1$ is given by the equation

The curve $C_1$ has a vertical asymptote at $x = 2$. The linear oblique asymptote of $C_1$ passes through the points with coordinates $(0, 10)$ and $(-2, 0)$. Determine the values of the unknown constants $a$, $b$ and $c$.

Hence sketch $C_1$, stating clearly the equation(s) of any asymptote(s) and the coordinates of any stationary point(s).

2014 NJC MYE Q7

<p>A sketch of the curve $y = \frac{ax^2 + bx + c}{x + d}$ where&nbsp;$a$, $b$, $c$ and $d$ are constants, is shown, not drawn to scale, in the diagram.</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/1769493570220-vaw3i8da8w.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk0OTM1NzAyMjAtdmF3M2k4ZGE4dy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTY5NTAxLCJleHAiOjE3OTM1Njk4MDF9.h0Eg6ZNYU1vlk-VlpcjTvmlzp4wAEG8-5DydNOL7dmU" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The upper branch of the curve cuts the $y$-axis at the point $A\left(0, \frac{3}{2}\right)$ and the equation of the asymptotes, also shown in the diagram, are $x = -2$ and $y = x - 3$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-content"><span class="list-item-text">Write down the value of $d$ and determine the values of $a$, $b$ and $c$.</span></div></li><li data-list-item-id="item-ii"><div class="list-item-content"><span class="list-item-text">Prove that&nbsp;$y$ cannot take any values in the interval $(p, q)$ where&nbsp;$p$ and $q$ are constants to be determined.</span></div></li><li data-list-item-id="item-iii"><div class="list-item-content"><span class="list-item-text">Using the graph given and by considering the line that passes through the point of intersection of the $2$ asymptotes, determine the range of values of $m$ if the equation $ax^2 + bx + c = (x + d)[m(x + 2) - 5]$ has&nbsp;$2$ real roots.</span></div></li></ol>

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

The upper branch of the curve cuts the $y$-axis at the point $A\left(0, \frac{3}{2}\right)$ and the equation of the asymptotes, also shown in the diagram, are $x = -2$ and $y = x - 3$.

Write down the value of $d$ and determine the values of $a$, $b$ and $c$.

Prove that $y$ cannot take any values in the interval $(p, q)$ where $p$ and $q$ are constants to be determined.

Using the graph given and by considering the line that passes through the point of intersection of the $2$ asymptotes, determine the range of values of $m$ if the equation $ax^2 + bx + c = (x + d)[m(x + 2) - 5]$ has $2$ real roots.

<p>The function $\mathrm{f}$ is given by</p>
<p>\[\mathrm{f}(x) = \begin{cases} 2 + \sqrt{2^2 - (x-2)^2}, & 0 &lt; x \leq 4, \\ 2 - \sqrt{2^2 - (x-6)^2}, & 4 &lt; x \leq 8. \end{cases}\]</p>
<p>It is given that&nbsp;$\mathrm{f}(x) = \mathrm{f}(x+8)$ for all real values of $x$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">On the diagram in the Printed Answer Book, sketch the graph of $y = \mathrm{f}(x)$ for&nbsp;$-6 \leq x \leq 7$, indicating clearly the coordinates of the end points and the points where the graph cuts the axes.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Without integrating, write down the exact area of the region bounded by $y = \mathrm{f}(x)$, the line $x = 4$, the $x$-axis and the $y$-axis.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The curve $C$ has equation $\frac{(y-3)^2}{a^2} - (x+2)^2 = 1$, where&nbsp;$a$ is a positive real constant.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">State the equations of the asymptotes of $C$ in terms of $a$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Determine the range of values of $a$ if there is at most one intersection between $C$ and the graph of $y = \mathrm{f}(x)$.</span><span class="inline-marks">[2]</span></div></li></ol>

\[\mathrm{f}(x) = \begin{cases} 2 + \sqrt{2^2 - (x-2)^2}, & 0 < x \leq 4, \\ 2 - \sqrt{2^2 - (x-6)^2}, & 4 < x \leq 8. \end{cases}\]

It is given that $\mathrm{f}(x) = \mathrm{f}(x+8)$ for all real values of $x$.

On the diagram in the Printed Answer Book, sketch the graph of $y = \mathrm{f}(x)$ for $-6 \leq x \leq 7$, indicating clearly the coordinates of the end points and the points where the graph cuts the axes.

Without integrating, write down the exact area of the region bounded by $y = \mathrm{f}(x)$, the line $x = 4$, the $x$-axis and the $y$-axis.

The curve $C$ has equation $\frac{(y-3)^2}{a^2} - (x+2)^2 = 1$, where $a$ is a positive real constant.

State the equations of the asymptotes of $C$ in terms of $a$.

Determine the range of values of $a$ if there is at most one intersection between $C$ and the graph of $y = \mathrm{f}(x)$.

Piecewise-Defined Functions

Periodic Functions

2025 VJC P1 Q5

<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 $C$.

Implicit Differentiation

Tangents and Normals Parallel to the Axes

Tangents and Normals to Implicit Curves

2025 NYJC P2 Q4

Techniques of Differentiation

Applications of Differentiation

<p>The curve $C$ has equation \[y = \frac{2x^2 + kx + 8}{x + p},\]where&nbsp;$k$ and $p$ are constants. It is given that&nbsp;$C$ has a vertical asymptote $x = 2$ and a stationary point at $x = -4$.</p>
<ol class="i-list"><li data-list-item-id="item-4-i" data-list-item-path="item-4-i"><div class="list-item-content"><span class="list-item-text">Find the equation of the oblique asymptote of $C$.</span><span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-4-ii" data-list-item-path="item-4-ii"><div class="list-item-content"><span class="list-item-text">Sketch $C$, clearly labelling the equations of asymptotes and the coordinates of stationary points.</span><span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has equation \[y = \frac{2x^2 + kx + 8}{x + p},\]where $k$ and $p$ are constants. It is given that $C$ has a vertical asymptote $x = 2$ and a stationary point at $x = -4$.

Find the equation of the oblique asymptote of $C$.

Sketch $C$, clearly labelling the equations of asymptotes and the coordinates of stationary points.

Stationary Points and Their Nature

2025 NJC P1 Q5

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MDIzNzc3MDYtMGZuNmZnenYxMWo3LnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTM1Njk1MDEsImV4cCI6MTc5MzU2OTgwMX0.Fsr2FZAb-2OHmg5xEK14otZ6FdFMf4ODbyKb-TAAOuU" 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

Increasing and Decreasing Functions

Stationary Points and Points of Inflection

2025 DHS P1 Q5 

<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/1768637712755-vc9bhdd9c69.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg2Mzc3MTI3NTUtdmM5YmhkZDljNjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU2OTUwMSwiZXhwIjoxNzkzNTY5ODAxfQ.QpadHfJS0A4E18w_QCUjFm7COuBHohJdX6PvZsgVbaE" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram above shows the curve $C$ with equation $\frac{(x+p)^2}{r} - \frac{(y+q)^2}{9} = 1$, where&nbsp;$p$, $q$ and $r$ are constants. The vertices are $(-5,3)$ and $(-1,3)$ and the asymptotes are $y = \frac{3}{2}x + \frac{15}{2}$ and $y = -\frac{3}{2}x - \frac{3}{2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-c-a"><div class="list-item-content"><span class="list-item-text">Find the values of $p$, $q$ and $r$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c-b"><div class="list-item-content"><span class="list-item-text">The curve $D$ has equation $(x+3)^2 + m(y-3)^2 = m$, where&nbsp;$m$ is a positive constant. Find the range of values of $m$ for which curves $C$ and $D$ do not intersect.</span><span class="inline-marks">[2]</span></div></li></ol>

The diagram above shows the curve $C$ with equation $\frac{(x+p)^2}{r} - \frac{(y+q)^2}{9} = 1$, where $p$, $q$ and $r$ are constants. The vertices are $(-5,3)$ and $(-1,3)$ and the asymptotes are $y = \frac{3}{2}x + \frac{15}{2}$ and $y = -\frac{3}{2}x - \frac{3}{2}$.

Find the values of $p$, $q$ and $r$.

The curve $D$ has equation $(x+3)^2 + m(y-3)^2 = m$, where $m$ is a positive constant. Find the range of values of $m$ for which curves $C$ and $D$ do not intersect.

2025 CJC P1 Q1 

<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 Stationary Points

2025 ACJC P1 Q2

<p>The curve $C$ has equation $\dfrac{(x-1)^2}{2} - (y+2)^2 = 1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$, stating the equations of the asymptotes and their point of intersection.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence state the exact range of values of $m$ such that the straight line $y = m(x-1) - 2$ does not intersect $C$.</div><p style="text-align: right;">[2]</p></li></ol>

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

Worked Example 3

<p>The diagram below shows the graph of $y=\frac{a{{x}^{2}}+bx+c}{x+d}$. $y=x-1$ and $x=-2$ are asymptotes, and the curve also has a minimum point on the $y$-axis. Find the values of $a$, $b$, $c$ and $d$.</p>
<div class="graph-block" data-graph-config="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" data-width="432" data-height="301" style="width:432px;height:301px;max-width:100%;display:block;margin:0 auto;"></div>

The diagram below shows the graph of $y=\frac{a{{x}^{2}}+bx+c}{x+d}$. $y=x-1$ and $x=-2$ are asymptotes, and the curve also has a minimum point on the $y$-axis. Find the values of $a$, $b$, $c$ and $d$.

<p>The curve $C$ with equation $y=\frac{a{{x}^{2}}+bx+c}{x+d}$ is shown below. It is known that&nbsp;$C$, which has asymptotes at $x=4$ and $y=x$, cuts the $y$-axis at $-\frac{1}{4}$. Find the value of $a$, $b$, $c$ and $d$.</p>
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The curve $C$ with equation $y=\frac{a{{x}^{2}}+bx+c}{x+d}$ is shown below. It is known that $C$, which has asymptotes at $x=4$ and $y=x$, cuts the $y$-axis at $-\frac{1}{4}$. Find the value of $a$, $b$, $c$ and $d$.

<ol class="abc-list"><li data-list-item-id="item-1761383465097-lhs8n3c5h" data-list-item-path="item-1761383465097-lhs8n3c5h"><div class="list-item-paragraph">On the same diagram, sketch the graphs $y=\left| 2x-1 \right|$ and $y={{x}^{2}}-4x-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383465907-xaflj7mr2" data-list-item-path="item-1761383465907-xaflj7mr2"><div class="list-item-paragraph">Hence find the set of values of $x$ for which $\left| 2x-1 \right|&lt;{{x}^{2}}-4x-2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383466491-iqg4ujtg8" data-list-item-path="item-1761383466491-iqg4ujtg8"><div class="list-item-paragraph">Without using a calculator, find the range of values of $k$ for which $\left| 2x-1 \right|={{x}^{2}}-4x-2+k$ has only positive real roots.<span class="inline-marks">[4]</span></div></li></ol>

Polynomial Functions and Key Features

2025 YIJC P1 Q7

<p>It is given that</p>
<p style="text-align: center">$y=\frac{{{x}^{2}}+ax+1}{x-1}$, $x\in \mathbb{R}$, $x\ne 1$,</p>
<p>where $a$ is a constant.</p>
<p>Without using a calculator, find the range of values of $a$ such that $y$ can take any real values.</p>

$y=\frac{{{x}^{2}}+ax+1}{x-1}$, $x\in \mathbb{R}$, $x\ne 1$,

Without using a calculator, find the range of values of $a$ such that $y$ can take any real values.

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