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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

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IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-0" data-list-item-path="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-0"><div class="list-item-paragraph">The curve $C_1$ has equation $\frac{(x-1)^2}{16}-\frac{(y+3)^2}{4}=1$. Sketch $C_1$, giving the equations of the asymptotes and the coordinates of the centre and vertices.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-1" data-list-item-path="h2-2025-vjc-promo-q2-vjc-q2-item-block-0-item-1"><div class="list-item-paragraph">The curve $C_2$ has equation $x^2-4x+y^2+6y+13=a^2$, where&nbsp;$a&gt;0$, intersects $C_1$ at exactly two points. Find the range of values of $a$.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C_1$ has equation $\frac{(x-1)^2}{16}-\frac{(y+3)^2}{4}=1$. Sketch $C_1$, giving the equations of the asymptotes and the coordinates of the centre and vertices.

The curve $C_2$ has equation $x^2-4x+y^2+6y+13=a^2$, where $a>0$, intersects $C_1$ at exactly two points. Find the range of values of $a$.

Asymptotes and Discontinuities

Graphs of Conic Sections

Circle–Curve Intersection Conditions

2025 VJC Promo Q2

Graphing Techniques

<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.

Parameter Conditions for Graphs and Equations

2025 HCI Promo Q4

<p>The curve $C_1$ has equation \[(x+3)^2-4(y+1)^2=16.\]</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-ejc-promo-q9-ejc-q9-item-block-2-item-0" data-list-item-path="h2-2025-ejc-promo-q9-ejc-q9-item-block-2-item-0"><div class="list-item-paragraph"><p>Sketch the curve $C_1$, stating the equations of any asymptote(s) and other key features clearly.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>Another curve $C_2$ has parametric equations $x=t-\dfrac{1}{t-2}$ and $y=-t+5$ where&nbsp;$-2\leq t&lt;2$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="h2-2025-ejc-promo-q9-ejc-q9-item-block-4-item-0" data-list-item-path="h2-2025-ejc-promo-q9-ejc-q9-item-block-4-item-0"><div class="list-item-paragraph"><p>Find the coordinates of any point(s) of intersection between $C_1$ and $C_2$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q9-ejc-q9-item-block-4-item-1" data-list-item-path="h2-2025-ejc-promo-q9-ejc-q9-item-block-4-item-1"><div class="list-item-paragraph"><p>Sketch the curve $C_2$ on the same diagram as part <strong>(a)</strong>, showing clearly any asymptote(s) and the coordinates of any axial intercept(s).</p><span class="inline-marks">[4]</span></div></li></ol>

The curve $C_1$ has equation \[(x+3)^2-4(y+1)^2=16.\]

Another curve $C_2$ has parametric equations $x=t-\dfrac{1}{t-2}$ and $y=-t+5$ where $-2\leq t<2$.

Sketching Parametric Curves

Parametric Equations

2025 EJC Promo Q9

<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 $C_1$ 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 $y = 2x + 3 + \frac{12}{x - 2}$, $x \neq 2$.

Sketching Rational Functions

Graphical Root Counting

2024 TMJC Promo Q3

<p>The diagram shows an ellipse, centred at the origin with semi-minor axis $a$ and semi-major axis $2a$, where&nbsp;$a &gt; 0$. The rectangle $PQRS$ is inscribed in the ellipse such that its four vertices $P$, $Q$, $R$ and $S$ lie on the ellipse. The coordinates of $P$ is $(x, y)$, where&nbsp;$x$ and $y$ are positive.</p>
<div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjI1MCwiaGVpZ2h0IjoyNTAsImVsZW1lbnRzIjpbeyJpZCI6Ijg5NTU1ODRlLTE3ZDYtNDRiMi1iNGNiLTQzNDE0MDZkMDc0ZiIsInR5cGUiOiJlbGxpcHNlIiwiY29sb3IiOiIjMDAwMDAwIiwiY2VudGVyIjpbMCwwXSwicmFkaXVzWCI6MiwicmFkaXVzWSI6NCwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6ImNmNmJmNTNjLTdmY2UtNGZhZC1iMTdlLWY5MWU3ODYyYWIwNyIsInR5cGUiOiJwb2x5Z29uIiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnRzIjpbWy0xLjYsMi4zN10sWzEuNjEsMi4zN10sWzEuNjEsLTIuNF0sWy0xLjYsLTIuNF1dLCJmaWxsQ29sb3IiOiIjMDAwMDAwIiwic2hhcGVLaW5kIjoicmVjdGFuZ2xlIiwiZmlsbE9wYWNpdHkiOjAsInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6LTEuOTgsInkiOjIuNDksImlkIjoiNjJlMGRkODgtMGYyZS00MTdmLTg5MWQtZjYwNmM2MjhlNzVjIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKFNcXCkiLCJ0ZXh0Um90YXRpb24iOjB9LHsieCI6LTIuMiwieSI6LTIuNCwiaWQiOiJhMjk1MzlkYS00MzliLTRjOTItOTI2ZC1kNWQ4Y2FiNWFlNjIiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoUlxcKSIsInRleHRSb3RhdGlvbiI6MH0seyJ4IjoxLjg3LCJ5IjotMi40LCJpZCI6ImVjZDdlODZkLTI5ZjctNGI4Ni05YjdlLWQ3YzRkMGVjYjFiYiIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChRXFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOjEuODcsInkiOjIuNDksImlkIjoiM2RiZjQyMDEtNWYzOS00MjQyLWFkNDAtZDIyMTk0MzU5YTYyIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKFAoeCx5KVxcKSIsInRleHRSb3RhdGlvbiI6MH0seyJ4IjoyLjE3LCJ5IjotMC4zMSwiaWQiOiIyN2Y2MjFhMy1mZTQyLTRhYTEtOTVmZS1lZjJmNGY2ODAzOWEiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoYVxcKSIsInRleHRSb3RhdGlvbiI6MH0seyJ4IjotMi43OSwieSI6LTAuMzEsImlkIjoiODQ0YTllYWQtNmI3NS00MGMxLWI3ZGMtYjIwM2UxNjY1YWRlIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKC1hXFwpIiwidGV4dFJvdGF0aW9uIjowfSx7IngiOjAuMiwieSI6LTQuMzQsImlkIjoiMjJlZmI4MjgtZWNmYi00ZDgwLWJmYmUtZjJlMTIzNGIwYzQwIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKC0yYVxcKSIsInRleHRSb3RhdGlvbiI6MH0seyJ4IjowLjIsInkiOjQuMzIsImlkIjoiMTUyYjEzZDQtZTUxMi00YmNmLTg3MzgtMTIxMTFlODE1MGNlIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKDJhXFwpIiwidGV4dFJvdGF0aW9uIjowfV0sInNob3dYQXhpcyI6dHJ1ZSwic2hvd1lBeGlzIjp0cnVlLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInlBeGlzTGFiZWwiOiJcXCh5XFwpIiwiYm91bmRpbmdCb3giOlstNSw1LDUsLTVdLCJvcmlnaW5MYWJlbCI6IlxcKE9cXCkiLCJncmlkU2V0dGluZ3MiOnsibWFqb3JDb2xvciI6IiM5NGEzYjgiLCJtYWpvclN0ZXBYIjoxLCJtYWpvclN0ZXBZIjoxLCJtaW5vckNvbG9yIjoiI2NiZDVlMSIsIm1ham9yT3BhY2l0eSI6MC40NSwibWlub3JPcGFjaXR5IjowLjQ1LCJtaW5vckVsZW1lbnRzWCI6NCwibWlub3JFbGVtZW50c1kiOjQsIm1ham9yU3Ryb2tlV2lkdGgiOjAuOCwibWlub3JTdHJva2VXaWR0aCI6MC41fSwic2hvd0F4aXNMYWJlbHMiOmZhbHNlLCJzaG93TmF2aWdhdGlvbiI6ZmFsc2UsImtlZXBBc3BlY3RSYXRpbyI6dHJ1ZSwieEF4aXNMYWJlbE9mZnNldCI6WzIsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0xNCw1XSwib3JpZ2luTGFiZWxPZmZzZXQiOlstNSwtMl19" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Write down the cartesian equation of the ellipse in terms of $a$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">By considering the coordinates of $P$, show that&nbsp;$A$, the area of rectangle $PQRS$, is $8x\sqrt{a^2 - x^2}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using differentiation, find, in terms of $a$, the value of $x$ when&nbsp;$A$ is a maximum. You do not need to prove that&nbsp;$A$ is a maximum.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Find the value of $a$ if the maximum value of $A$ is $100$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an ellipse, centred at the origin with semi-minor axis $a$ and semi-major axis $2a$, where $a > 0$. The rectangle $PQRS$ is inscribed in the ellipse such that its four vertices $P$, $Q$, $R$ and $S$ lie on the ellipse. The coordinates of $P$ is $(x, y)$, where $x$ and $y$ are positive.

Problems Involving Maximisation and Minimisation

2024 JPJC Promo Q6

Applications of Differentiation

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">\[x = \operatorname{cosec} \theta + 2, \quad y = 3 \cot \theta, \quad \text{where}\hspace{0.5em} 0 &lt; \theta &lt; \pi.\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find a cartesian equation of $C$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Sketch $C$, indicating all the necessary features, including the equations of any asymptotes.<span class="inline-marks">[3]</span></div></li></ol>

\[x = \operatorname{cosec} \theta + 2, \quad y = 3 \cot \theta, \quad \text{where } 0 < \theta < \pi.\]

Converting Parametric Equations to Cartesian Form

<p>Sketch $x^2+6x-y^2+3y=25$ indicating any turning points and axial intercept(s).</p>

Sketch $x^2+6x-y^2+3y=25$ indicating any turning points and axial intercept(s).

<p>Sketch $x^2+3x-y^2+12y=5$ indicating any turning points and axial intercept(s).</p>

Sketch $x^2+3x-y^2+12y=5$ indicating any turning points and axial intercept(s).

<p>Sketch $ (3y-1)^2+(x+3)^2=18 $ indicating any turning points and axial intercept(s).</p>

Sketch $ (3y-1)^2+(x+3)^2=18 $ indicating any turning points and axial intercept(s).

<p>Sketch $ (y+5)^2-(x+2)^2=9  $ indicating any turning points and axial intercept(s).</p>

Sketch $ (y+5)^2-(x+2)^2=9  $ indicating any turning points and axial intercept(s).

<p>Sketch the following curve, labelling the coordinates of any axial intercepts and turning points:</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$6y-y^2=2x $</div></li><li><div class="list-item-paragraph">$ (y-3)^2=4x+6 $</div></li></ol>

Sketch the following curve, labelling the coordinates of any axial intercepts and turning points:

<p>Sketch ${{(2x-3)}^{2}}+{{(y+3)}^{2}}=15$ indicating any turning points and axial intercept(s).</p>

Sketch ${{(2x-3)}^{2}}+{{(y+3)}^{2}}=15$ indicating any turning points and axial intercept(s).

<p>The curve $C$ has equation $4{{x}^{2}}+{{y}^{2}}+8mx-4y+4=0$ , where&nbsp;$m\in \mathbb{R},m&gt;0$.<br>Show that&nbsp;$C$ can be expressed in the form  ${{\left( \frac{x+a}{b} \right)}^{2}}+{{\left( \frac{y+c}{d} \right)}^{2}}=1$, where&nbsp;$a$, $b$, $c$ and $d$ are constants to be determined in terms of $m$. <br>Hence describe a sequence of transformations, in terms of $m$, which transforms the graph of ${{x}^{2}}+{{\left( y-\frac{1}{m} \right)}^{2}}=1$ to $C$. </p>

The curve $C$ has equation $4{{x}^{2}}+{{y}^{2}}+8mx-4y+4=0$ , where $m\in \mathbb{R},m>0$.
Show that $C$ can be expressed in the form  ${{\left( \frac{x+a}{b} \right)}^{2}}+{{\left( \frac{y+c}{d} \right)}^{2}}=1$, where $a$, $b$, $c$ and $d$ are constants to be determined in terms of $m$. 
Hence describe a sequence of transformations, in terms of $m$, which transforms the graph of ${{x}^{2}}+{{\left( y-\frac{1}{m} \right)}^{2}}=1$ to $C$. 

Composite Transformations

2015 HCI Promo Q12

Transformations of Graphs

<p><strong>A graphing calculator is not allowed for this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the square roots of $2\mathrm{i}$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation $w^2 = 3 - 4\mathrm{i}$. Hence find the roots of the equation $z^2 - 4\mathrm{i}z + 4\mathrm{i} - 7 = 0$, giving your answers in Cartesian form.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$w = \frac{1}{2 + \mathrm{i}\alpha}$ where&nbsp;$\alpha \in \mathbb{R}$, and $w^*$ denotes the complex conjugate of $w$, find $ww^*$ and hence prove that&nbsp;$w + w^* = 4ww^*$ ---(1).</div><div class="list-item-paragraph">The complex number $w$ is represented in an Argand diagram by the point $A$. By using (1) and considering $w = x + y\mathrm{i}$, where&nbsp;$x, y \in \mathbb{R}$, show that, $A$ lies on a circle as&nbsp;$\alpha$ varies. Find the Cartesian equation of this circle, stating clearly its centre and radius.<span class="inline-marks">[6]</span></div></li></ol>

**A graphing calculator is not allowed for this question.**

Find the square roots of $2\mathrm{i}$.

Solve the equation $w^2 = 3 - 4\mathrm{i}$. Hence find the roots of the equation $z^2 - 4\mathrm{i}z + 4\mathrm{i} - 7 = 0$, giving your answers in Cartesian form.

Given that $w = \frac{1}{2 + \mathrm{i}\alpha}$ where $\alpha \in \mathbb{R}$, and $w^*$ denotes the complex conjugate of $w$, find $ww^*$ and hence prove that $w + w^* = 4ww^*$ ---(1).

Applying Complex Conjugate Identities

Finding Square Roots of Complex Numbers Algebraically

Finding Roots of Equations with Complex Coefficients

Loci in the Argand Diagram

2020 ASRJC P1 Q5(a) | 2024 SAJC P1 Q8b

Complex Numbers

<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 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

Functions

Applications of Integration

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">On a single diagram, sketch the graphs of $y = \frac{3x}{x - 1}$ and $y = 3x$, stating the equation(s) of any asymptote(s) and the coordinates of points of intersection.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Hence find the area bounded by the curve $y = \frac{3x}{x - 1}$ and the lines $y = 3x$ and $y = 8$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A curve has the equation $y = \sqrt{x^2 + 1}$ where&nbsp;$x$ is non-negative.</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">With the aid of a sketch of the above curve, explain why the value of $S = \lim_{n \to \infty} \frac{1}{n} \left[ 1 + \sqrt{\frac{1}{n^2} + 1} + \sqrt{\frac{4}{n^2} + 1} + \dots + \sqrt{\frac{(n - 1)^2}{n^2} + 1} \right]$ is $\displaystyle \displaystyle\int_0^1 \sqrt{x^2 + 1} \, \mathrm{d}x$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Hence find the value of $S$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

A curve has the equation $y = \sqrt{x^2 + 1}$ where $x$ is non-negative.

Area Bounded by Curves

Riemann Sums

2025 SAJC P1 Q7

<p>The curve $E$ has equation</p>
<p style="text-align: center">\[(x-a)^2 + 4y^2 = 4, \text{ where}\hspace{0.5em} 0 &lt; a &lt; 2.\]</p>
<ol class="i-list"><li data-list-item-id="item-i-3" data-list-item-path="item-i-3"><div class="list-item-paragraph">Sketch $E$.<span class="inline-marks">[2]</span></div></li></ol>
<p>You are now given that&nbsp;$a=1$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii-3" data-list-item-path="item-ii-3"><div class="list-item-paragraph">Show that the equation of a normal to $E$ at $x=0$ is $y=2\sqrt{3}x-\frac{\sqrt{3}}{2}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii-3" data-list-item-path="item-iii-3"><div class="list-item-paragraph">The region $R$ is bounded by $E$, the normal in part <strong>(ii)</strong> and the $y$-axis. Find the volume of the solid generated when&nbsp;$R$ is rotated through $2\pi$ radians about the $y$-axis.<span class="inline-marks">[5]</span></div></li></ol>

\[(x-a)^2 + 4y^2 = 4, \text{ where } 0 < a < 2.\]

Show that the equation of a normal to $E$ at $x=0$ is $y=2\sqrt{3}x-\frac{\sqrt{3}}{2}$.

Tangents and Normals at a Point

Volume of Revolution about the y-axis

Tangents and Normals to Implicit Curves

2025 NJC P2 Q2

<p>A curve has equation $y = \mathrm{f}(x)$, where&nbsp;$\mathrm{f}(x) = 1 - \sqrt{q^2 - x^2}$ for&nbsp;$q &gt; 1$. State the shape of $y = \mathrm{f}(x)$.<span class="inline-marks">[1]</span></p>
<ol class="i-list"><li data-list-item-id="item-3-i" data-list-item-path="item-3-i"><div class="list-item-paragraph">Sketch the curve $y = \frac{1}{\mathrm{f}(x)}$, giving the equations of any asymptotes and the coordinates of the end-points.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-3-ii" data-list-item-path="item-3-ii"><div class="list-item-paragraph">Describe the transformations that map the graph of $y = \mathrm{f}(x)$ to $y = -\sqrt{1-x^2}$.<span class="inline-marks">[3]</span></div></li></ol>

A curve has equation $y = \mathrm{f}(x)$, where $\mathrm{f}(x) = 1 - \sqrt{q^2 - x^2}$ for $q > 1$. State the shape of $y = \mathrm{f}(x)$.

Sketch the curve $y = \frac{1}{\mathrm{f}(x)}$, giving the equations of any asymptotes and the coordinates of the end-points.

Describe the transformations that map the graph of $y = \mathrm{f}(x)$ to $y = -\sqrt{1-x^2}$.

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