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PSLE Foundation Mathematics

<p>A curve $C$ has parametric equations \[x=4t^2+1,\qquad y=2t.\]</p>
<p>The points $P$ and $Q$ on $C$ have parameters $p$ and $3p$ respectively. The point $M$ is the midpoint of $PQ$.</p>
<ol class="abc-list"><li data-list-item-id="q2-a" data-list-item-path="q2-a"><div class="list-item-paragraph">Show that the coordinates of $M$ are $(20p^2+1,4p)$, and hence find the Cartesian equation of the locus of $M$ as&nbsp;$p$ varies.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q2-b" data-list-item-path="q2-b"><div class="list-item-paragraph">The line $L$ has equation \[x=15y-39.\] It intersects the locus of $M$ at two points.</div><div class="list-item-paragraph">Find the corresponding values of $p$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="q2-c" data-list-item-path="q2-c"><div class="list-item-paragraph">For the smaller value of $p$, denote the corresponding points on $C$ by $P_1$ and $Q_1$. For the larger value, denote them by $P_2$ and $Q_2$.</div><div class="list-item-paragraph"><p>The four distinct points $P_1,Q_1,P_2,Q_2$ form a quadrilateral. Find the exact area of the quadrilateral.</p><span class="inline-marks">[5]</span></div></li></ol>

A curve $C$ has parametric equations \[x=4t^2+1,\qquad y=2t.\]

The points $P$ and $Q$ on $C$ have parameters $p$ and $3p$ respectively. The point $M$ is the midpoint of $PQ$.

Show that the coordinates of $M$ are $(20p^2+1,4p)$, and hence find the Cartesian equation of the locus of $M$ as $p$ varies.

Locus of a Midpoint

Parametric Equations

Converting Parametric Equations to Cartesian Form

Area from a parametric midpoint locus

Functions

Graphing Techniques

<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 $-2\leq t<2$.

Asymptotes and Discontinuities

Graphs of Conic Sections

Sketching Parametric Curves

2025 EJC Promo Q9

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<p>Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations</p>
<p style="text-align: center">$x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),$</p>
<p>for&nbsp;$-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi$, where&nbsp;$x$ and $y$ are measured in metres. The $x$-axis models the horizontal ground.</p>
<p>The maximum point on the curve is at $A$, and the line $AB$ is vertical. Points $C$ and $D$ are where the tangent to the curve is parallel to the $y$-axis. Point $E$ is the intersection of the lines $AB$ and $CD$.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><ol class="i-list"><li data-list-item-id="imported-item-block-3-item-1-sub-0" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-0"><div class="list-item-paragraph">Find the values of $\theta$ at points $C$ and $D$, giving your answers correct to $2$ significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-1-sub-1" data-list-item-path="imported-item-block-3-item-1/imported-item-block-3-item-1-sub-1"><div class="list-item-paragraph">Hence find the greatest width, $CD$, of the loop.<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>The $x$-coordinate of point $A$ is $28$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><ol class="i-list"><li data-list-item-id="imported-item-block-5-item-0-sub-0" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-0"><div class="list-item-paragraph">Find the value of $\theta$ at point $A$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-1" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-1"><div class="list-item-paragraph">Find the $2$ possible values of $\theta$ at point $B$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-5-item-0-sub-2" data-list-item-path="imported-item-block-5-item-0/imported-item-block-5-item-0-sub-2"><div class="list-item-paragraph">Hence find the distance $AB$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>
<p>Engineers know that, for safety reasons, the distance $AE$ should be within $10\%$ of the distance $\frac{1}{2}CD$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="imported-item-block-7-item-0" data-list-item-path="imported-item-block-7-item-0"><div class="list-item-paragraph">Determine whether the design of this section of track satisfies this condition.<span class="inline-marks">[2]</span></div></li></ol>

Engineers are designing a roller coaster. The diagram shows the side view of a section of track with a loop. This section is modelled by the curve with parametric equations

$x = 28(\theta \cos \theta + 1), \quad y = 20(2 - \theta \sin \theta),$

for $-\frac{2}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi$, where $x$ and $y$ are measured in metres. The $x$-axis models the horizontal ground.

The maximum point on the curve is at $A$, and the line $AB$ is vertical. Points $C$ and $D$ are where the tangent to the curve is parallel to the $y$-axis. Point $E$ is the intersection of the lines $AB$ and $CD$.

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

The $x$-coordinate of point $A$ is $28$.

Engineers know that, for safety reasons, the distance $AE$ should be within $10\%$ of the distance $\frac{1}{2}CD$.

Determine whether the design of this section of track satisfies this condition.

Tangents and Normals to Parametric Curves

Derivatives of Trigonometric Functions

Differentiation of Parametric Equations

Tangents and Normals Parallel to the Axes

Product Rule

N2025 P1 Q11

Applications of Differentiation

Techniques of Differentiation

<p>A curve $C$ has parametric equations $x = 3t^2$, $y = 2t^3$, for&nbsp;$t &lt; 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch $C$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Points $P$ and $Q$ have parameters $p$ and $q$ respectively. Given that&nbsp;$q = 2p$, find the cartesian equation of the locus of the mid-point of $PQ$ as&nbsp;$p$ varies.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>A curve $L$ has equation $x = y^2 + 1$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">It is given that the curves $C$ and $L$ touch at the point $M$. Show that&nbsp;$4t^6 - 3t^2 + 1 = 0$ at $M$.<br>Hence, or otherwise, find the exact coordinates of $M$.<span class="inline-marks">[6]</span></div></li></ol>

A curve $C$ has parametric equations $x = 3t^2$, $y = 2t^3$, for $t < 0$.

Points $P$ and $Q$ have parameters $p$ and $q$ respectively. Given that $q = 2p$, find the cartesian equation of the locus of the mid-point of $PQ$ as $p$ varies.

A curve $L$ has equation $x = y^2 + 1$.

It is given that the curves $C$ and $L$ touch at the point $M$. Show that $4t^6 - 3t^2 + 1 = 0$ at $M$.\nHence, or otherwise, find the exact coordinates of $M$.

N2013 P1 Q11 [Modified]

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the parametric curve $x = t^{3} + 1$, $y = 2 - t$ for&nbsp;$t \in \mathbb{R}$.</div></li><li><div class="list-item-paragraph">Find the coordinates of the points of intersection of the curve in <strong>(i)</strong> and the straight line $4y + x = 9$.</div></li></ol>

<ol class="i-list"><li><div class="list-item-paragraph">Sketch the parametric curve $x = 2t^{2} - 1$, $y = 3(t + 1)$ for&nbsp;$t \in \mathbb{R}$.</div></li><li><div class="list-item-paragraph">Find the coordinates of the points of intersection of the curve in <strong>(i)</strong> and the straight line $3x - 4y = 3$.</div></li></ol>

<p>Application of Vectors: Motion</p>
<p>A rocket was launched at a fireworks display. At time $t$ seconds, the position of the rockets, in metres relative to the launch location (let this be the origin $O$), was $\mathbf{r}=20t\mathbf{i}+\left[ {{(t-4)}^{3}}+64 \right]\mathbf{j}$ where $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertical unit vectors respectively. The rocket exploded when $t=4$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Write down the height of the rocket when it exploded.</div></li><li><div class="list-item-paragraph">Find the displacement of the rocket from $t=3$ to $t=4$.</div></li><li><div class="list-item-paragraph">Show that the Cartesian equation representing the path of the rocket was $y={{\left( \frac{x}{20}-4 \right)}^{3}}+64$ and sketch the path of the rocket. Hence state the direction of movement of the rocket when it exploded.</div></li></ol>

A rocket was launched at a fireworks display. At time $t$ seconds, the position of the rockets, in metres relative to the launch location (let this be the origin $O$), was $\mathbf{r}=20t\mathbf{i}+\left[ {{(t-4)}^{3}}+64 \right]\mathbf{j}$ where $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertical unit vectors respectively. The rocket exploded when $t=4$.

Vector Kinematics

AS Mathematics, Berry J., Fentem R., Francis B., Graham T., Harper Collins 2000, Ex 24. P1 Q6 [Modified]

Vectors

<p>A curve has parametric equations</p>
<p style="text-align: center">$x=\tan \theta $, $y=2\sec \theta $ for&nbsp;$0\le \theta &lt;2\pi $.</p>
<p>The equation of the tangent to the curve at the point $P$ with parameter $p$ is given by</p>
<p>$y=\left( 2\sin p \right)x+2\cos p$.</p>
<ol class="i-list"><li data-list-item-id="item-1760868732601-83n7tph3v" data-list-item-path="item-1760868732601-83n7tph3v"><div class="list-item-paragraph">The tangent at $P$ meets the $x$ and $y$ axes at the points $A$ and $B$ respectively.</div><div class="list-item-paragraph">Find the Cartesian equation of the locus of the mid-point of $AB$ as&nbsp;$p$ varies.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760868734086-ner95306k" data-list-item-path="item-1760868734086-ner95306k"><div class="list-item-paragraph">The tangent at $P$ meets the line $y=2x$ at the point $S$ and the line $y=-2x$ at the point $T$. Show that the area of the triangle $OST$ is independent of $p$, where&nbsp;$O$ is the origin.<span class="inline-marks">[4]</span></div></li></ol>

$x=\tan \theta $, $y=2\sec \theta $ for $0\le \theta <2\pi $.

The equation of the tangent to the curve at the point $P$ with parameter $p$ is given by

2018 SRJC P1 Q2

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760838133629-rtrhy1228n9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA4MzgxMzM2MjktcnRyaHkxMjI4bjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU3NDM5NSwiZXhwIjoxNzkzNTc0Njk1fQ.Gnx31DvzhV7YkGpDotxbfJ3LnOvGsBvPW6TXy4slkfo" width="400" class="aligncenter" /></figure>
<p>The figure above shows an ellipse with parametric equations</p>
<p style="text-align: center">${x = 2\cos\theta, y = 6\sin\left(\theta + \frac{\pi}{3}\right), -\pi \leq \theta < \pi}$</p>
<p>The curve meets the coordinate axes at points ${A}$, ${B}$, ${C}$ and ${D}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the coordinates of the points $A$, $B$, $C$ and $D$.</div></li></ol>
<p>The straight line $l$ is the tangent to the ellipse at the point $E$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">State the equation of $l$, given that $l$ is parallel to the $y$-axis.</div></li><li><div class="list-item-paragraph">Find the value of $\theta $ at the point $E$.</div></li></ol>
<p>The finite region bounded by the ellipse and the $x$-axis for which $y\ge 0$ is shown shaded in the figure above.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Show that the area of this region is given by the integral</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{-\frac{\pi }{3}}^{\frac{2\pi }{3}}{3-3\cos 2\theta +3\sqrt{3}\sin 2\theta \,\mathrm{d}\theta }$.</div></li><li><div class="list-item-paragraph">Hence find the area of the shaded region.</div></li></ol>

The figure above shows an ellipse with parametric equations

${x = 2\cos\theta, y = 6\sin\left(\theta + \frac{\pi}{3}\right), -\pi \leq \theta < \pi}$

The curve meets the coordinate axes at points ${A}$, ${B}$, ${C}$ and ${D}$.

The straight line $l$ is the tangent to the ellipse at the point $E$.

The finite region bounded by the ellipse and the $x$-axis for which $y\ge 0$ is shown shaded in the figure above.

Area Bounded by Curves

Parametric Equations Q2

Applications of Integration

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760837813404-tu6u9e9jqj9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA4Mzc4MTM0MDQtdHU2dTllOWpxajkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU3NDM5NSwiZXhwIjoxNzkzNTc0Njk1fQ.fUM2WslWbBHgIoPiYICrTth1a_U5X61H5DSoGNQLB90" width="400" class="aligncenter" /></figure>
<p>The figure above shows a curve known as a re-entrant cycloid, with parametric equations</p>
<p style="text-align: center">$x=\theta -4\sin \theta $, $y=1-2\cos \theta $, $0\le \theta \le 2\pi $.</p>
<p>The curve crosses the $x$-axis at the points $P$ and $Q$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $\theta $ at the points $P$ and $Q$.</div></li><li><div class="list-item-paragraph">Show that the area of the finite region bounded by the curve and the $x$-axis, shown shaded in the figure above, is given by the integral</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{1-6\cos \theta +8{{\cos }^{2}}\theta \,\mathrm{d}\theta }$,</div><div class="list-item-paragraph">where ${{\theta }_{1}}$ and ${{\theta }_{2}}$ must be stated.</div></li><li><div class="list-item-paragraph">Find an exact value for the above integral.</div></li></ol>

The figure above shows a curve known as a re-entrant cycloid, with parametric equations

$x=\theta -4\sin \theta $, $y=1-2\cos \theta $, $0\le \theta \le 2\pi $.

The curve crosses the $x$-axis at the points $P$ and $Q$.

Integrating Powers of Sine and Cosine

Parametric Equations Q1

Techniques of Integration

<p>A curve $C$ has parametric equations ${x={{\mathrm{e}}^{t}}+\sin t}$, ${y={{\mathrm{e}}^{t}}-\cos t}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Describe the shape of $C$ as $t\to \infty $.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the Cartesian equation of the normal to $C$ at the point $P\left( {{\mathrm{e}}^{\theta }}+\sin \theta ,\ {{\mathrm{e}}^{\theta }}-\cos \theta \right)$, where $\theta >0$, giving your answer in the form $y=mx+c$.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The normal to $C$ at $P$ meets the $y$-axis at the point $D$, and the curve $C$ meets the positive $x$-axis at the point $E$ that has integral coordinates.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the coordinates of $D$ and $E$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the parametric equations of the mid-point of $DE$ as ${\theta }$ varies.</div><p style="text-align: right;">[2]</p></li></ol>

A curve $C$ has parametric equations ${x={{\mathrm{e}}^{t}}+\sin t}$, ${y={{\mathrm{e}}^{t}}-\cos t}$.

The normal to $C$ at $P$ meets the $y$-axis at the point $D$, and the curve $C$ meets the positive $x$-axis at the point $E$ that has integral coordinates.

Derivatives of Exponential Functions

2015 HCI P1 Q6

<p>A $C$ curve has parametric equations</p>
<p style="text-align: center">$x={{\mathrm{e}}^{\theta }}\sin \theta $, $y={{\mathrm{e}}^{\theta }}\cos \theta $,</p>
<p>where ${-\frac{\pi}{2} < \theta \le \frac{\pi}{2}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$, indicating clearly the axial intercepts.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">$C$ cuts the $y$- and $x$-axes at points $A$ and $B$ respectively. A particle moves along $C$ from $A$ to $B$, with its $x$-coordinate increasing at a constant rate of $0.1$ units per second. Find the exact rate of change of its $y$-coordinate when $x=\frac{1}{2}{{\mathrm{e}}^{\frac{\pi }{6}}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The tangent at the point $P$ on $C$ is parallel to the $y$-axis. Find the equation of this tangent.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The point $Q$ on $C$ is such that the angle ${POQ=\frac{\pi }{2}}$. Find the area of triangle $OPQ$.</div><p style="text-align: right;">[5]</p></li></ol>

$x={{\mathrm{e}}^{\theta }}\sin \theta $, $y={{\mathrm{e}}^{\theta }}\cos \theta $,

where ${-\frac{\pi}{2} < \theta \le \frac{\pi}{2}}$.

Connected Rates of Change

2015 VJC P1 Q9

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=3\cos \theta -2\cos 3\theta $,</p>
<p style="text-align: center">$y=9\sin \theta -\sin 3\theta $,</p>
<p>for $0\le \theta \le 2\pi $.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$ and state the cartesian equations of its lines of symmetry.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $\cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta $, find the exact values of $\theta $ at the points where $C$ meets the $y$-axis.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the area enclosed by the axes and the part of $C$ in the first quadrant is given by</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{\left( 27{{\cos }^{2}}\theta -27\cos \theta \cos 3\theta +6{{\cos }^{2}}3\theta \right)\,\mathrm{d}\theta }$,</div><div class="list-item-paragraph">where the values of ${{\theta }_{1}}$ and ${{\theta }_{2}}$ should be stated.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find the exact total area enclosed by $C$.</div><p style="text-align: right;">[5]</p></li></ol>

Integrable Forms of Trigonometric Functions

Trigonometric Integration Using Product-to-Sum Formulae

2020 YIJC P2 Q3

<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>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=1-2{{t}^{2}}$,  $\qquad$  $y={{t}^{3}}+1$,  $\qquad$  for&nbsp;$t\in \mathbb{R}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760681791783-9nqqmho35" data-list-item-path="item-1760681791783-9nqqmho35"><div class="list-item-paragraph">Sketch $C$, indicating the coordinates of the points of intersection with the axes.<span class="inline-marks">[3]</span></div></li></ol>
<p>The line $x+y=2$ cuts $C$ at the points $A$ and $B$ where the $x$-coordinate of $A$ is greater than the $x$-coordinate of $B$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760681807825-h63tgux9l" data-list-item-path="item-1760681807825-h63tgux9l"><div class="list-item-paragraph">Find the exact length of $AB$ in the form $k\sqrt{2}$ where&nbsp;$k$ is an integer to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760681808411-iuddxwjsc" data-list-item-path="item-1760681808411-iuddxwjsc"><div class="list-item-paragraph">The point $M$ is the midpoint of $AP$ where&nbsp;$P$ is the point $\left( 1-2{{p}^{2}},{{p}^{3}}+1 \right)$. Find a cartesian equation of the curve traced by $M$ as&nbsp;$p$ varies. Leave your answer in the form ${{\left( ax+b \right)}^{3}}={{\left( cy+d \right)}^{2}}$, where&nbsp;$a$, $b$, $c$ and $d$ are integers to be determined.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph" style="text-align: center">$y=\left[ \mathrm{f}\left( p \right) \right]x+{{p}^{3}}+\frac{8p}{3}+1-\frac{4}{3p}$</div><div class="list-item-paragraph">Where&nbsp;$\mathrm{f}\left( p \right)$ is an expression in $p$ to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760681832531-8glu06df6" data-list-item-path="item-1760681832531-8glu06df6"><div class="list-item-paragraph">Show that the equation of the normal to $C$ at the point $P$ is</div></li><li data-list-item-id="item-1760681833378-u96ozaz64" data-list-item-path="item-1760681833378-u96ozaz64"><div class="list-item-paragraph">Find the acute angle between the normals to the curve at $x=-7$.<span class="inline-marks">[3]</span></div></li></ol>

$x=1-2{{t}^{2}}$,  $\qquad$  $y={{t}^{3}}+1$,  $\qquad$  for $t\in \mathbb{R}$.

The line $x+y=2$ cuts $C$ at the points $A$ and $B$ where the $x$-coordinate of $A$ is greater than the $x$-coordinate of $B$.

2023 CJC Promo Q11

<p>The parametric equations of a curve $C$ are</p>
<p style="text-align: center">$x=2a\cos \left( t+\frac{\pi }{6} \right)$, $y=a\sin t$,</p>
<p>where $0\le t < 2\pi $ and $a$ is a positive constant.</p>
<p>A sketch of curve $C$ is shown below, with $C$ intersecting the positive $x$-axis at $P$ and the negative $y$-axis at $Q$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760346058998-ebi81lhp3i.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzNDYwNTg5OTgtZWJpODFsaHAzaS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTc0Mzk1LCJleHAiOjE3OTM1NzQ2OTV9.cLz68eddBErUBwMI50wAGKQKLbI3FpofBI0TxSFHKYc" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the normal to a point on the curve $C$ with parameter $\theta $ has gradient $1+\sqrt{3}\tan \theta $.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Show that $t=\frac{4\pi }{3}$ at the point $Q$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Hence, find the equation of the normal to the curve $C$ at $Q$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The normal to the curve $C$ at $Q$ intersects the $x$-axis at $T$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Show that the ratio $\frac{OT}{OP}$ is independent of $a$ and find its value exactly.</div><p style="text-align: right;">[3]</p></li></ol>

The parametric equations of a curve $C$ are

$x=2a\cos \left( t+\frac{\pi }{6} \right)$, $y=a\sin t$,

where $0\le t < 2\pi $ and $a$ is a positive constant.

A sketch of curve $C$ is shown below, with $C$ intersecting the positive $x$-axis at $P$ and the negative $y$-axis at $Q$.

The normal to the curve $C$ at $Q$ intersects the $x$-axis at $T$.

2024 ACJC Promo Q7

<p><strong>Modelling Movement of Wheel Using Parametric Equations</strong></p>
<p>A particle of negligible size is lodged onto the circumference of a circular wheel of radius $1$. Initially, the wheel is at rest with its lowest point at the origin. The initial position of the particle between the wheel and the floor (represented by the $x$-axis) is at the origin as well, as shown in the diagram below.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760336799886-llo4omqghbt.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzMzY3OTk4ODYtbGxvNG9tcWdoYnQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU3NDM5NSwiZXhwIjoxNzkzNTc0Njk1fQ.C3DJ8VyQdqGCHgC3-uTyjcQz0dBfAkfAL18mzf96FmU" width="400" class="aligncenter" /></figure>
<p>The wheel begins to move from rest, rotating towards the positive $x-$direction in a clockwise manner. The position of the particle when the wheel has rotated by $\theta $ radians is denoted by the point $P$, as shown in the diagram below.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760336814060-es6cih4uvzv.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzMzY4MTQwNjAtZXM2Y2loNHV2enYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU3NDM5NSwiZXhwIjoxNzkzNTc0Njk1fQ.SLgGmmMSZCsBlUHAAn4PxxArqAAEFg5JWvAvoj_inrY" width="400" class="aligncenter" /></figure>
<p>Given that the curve $C$ is the path of $P$ on the $x-y$ plane,</p>
<ol class="i-list"><li><div class="list-item-paragraph">show that $C$ has the parametric equations</div><div class="list-item-paragraph" style="text-align: center">$x=\theta -\sin \theta $, $y=1-\cos \theta $.</div></li><li><div class="list-item-paragraph">find the range of values of $y$,</div></li><li><div class="list-item-paragraph">sketch $C$, for $0\le \theta \le 2\pi $, indicating clearly the exact coordinates of the endpoints and the maximum point.</div></li></ol>

**Modelling Movement of Wheel Using Parametric Equations**

A particle of negligible size is lodged onto the circumference of a circular wheel of radius $1$. Initially, the wheel is at rest with its lowest point at the origin. The initial position of the particle between the wheel and the floor (represented by the $x$-axis) is at the origin as well, as shown in the diagram below.

The wheel begins to move from rest, rotating towards the positive $x-$direction in a clockwise manner. The position of the particle when the wheel has rotated by $\theta $ radians is denoted by the point $P$, as shown in the diagram below.

Given that the curve $C$ is the path of $P$ on the $x-y$ plane,

Finding Possible y-values of a Curve

NJC Parametric Equations Tutorial Q10

<p>A curve is defined parametrically by $x=\frac{5}{2}\cos \theta ,\,\,y=\frac{3}{2}\sin \theta \,\,,$for $0\le \theta \le 2\pi $.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the coordinates of the axial intercepts algebraically.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the Cartesian equation of the curve is $\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{9}=\frac{1}{4}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the distance of the point $P$ on the curve with parameter $p$ to the point $(2,0)$ is $\frac{5}{2}-2\cos \,p$. Find a similar expression for the distance between $P$ and $(-2,0)$.</div><p style="text-align: right;">[3]</p></li></ol>

A curve is defined parametrically by $x=\frac{5}{2}\cos \theta ,\,\,y=\frac{3}{2}\sin \theta \,\,,$for $0\le \theta \le 2\pi $.

N1995 P2 Q12 [Modified]

<p>An architect designs and constructs a hump such that when a square wheel of dimensions 2 cm by 2 cm is rolled over the hump, the height of its centre above the horizontal ground (represented by the $x$-axis) is always constant. To test his invention, he places a square wheel against the hump, as shown in the diagram below. The points $R$, $S$ and $Q$ represent the centre of the square, the vertex of the square that is initially on the hump (at the origin $O$), and another vertex of the square which has a common side as $S$, respectively.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760256329216-hwqcu63lcde.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNTYzMjkyMTYtaHdxY3U2M2xjZGUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU3NDM5NSwiZXhwIjoxNzkzNTc0Njk1fQ.ofrnRZSbCIv-fJklRCq8aT5KArcztA8Yw5Z7fehtZUk" width="400" class="aligncenter" /></figure>
<p>The architect begins pushing the square wheel over the hump in the clockwise direction. He discovers that the centre of the square is always vertically above the variable point of contact between the square and the hump, $P$, as shown in the diagram below. It is further given that the angle $\theta$ is the angle between the side $QS$ of the square and the horizontal line passing through $S$ , measured in the positive $x$-direction. ($\theta$ is taken to be positive if the vertex $Q$ lies above the horizontal line passing through $S$, and negative if otherwise.)</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760256344668-ewta89bvx9r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNTYzNDQ2NjgtZXd0YTg5YnZ4OXIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU3NDM5NSwiZXhwIjoxNzkzNTc0Njk1fQ.WkvfZPIAJnhzsmasTvlqmq6hGArSLjx5jgO8SFSLCio" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">By considering the length of $RP$ or otherwise, show that the $y$-coordinate of $P$ can be expressed as $\sqrt{2}-\sec \theta $, for $-\frac{\pi }{4}\le \theta \le \frac{\pi }{4}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">State the gradient of $QS$ in terms of $\theta$ for  $-\frac{\pi }{4}\le \theta \le \frac{\pi }{4}$. Deduce that $\frac{\mathrm{d}x}{\mathrm{d}\theta }=-\sec \theta $, where $x$ is the $x$-coordinate of $P$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the area bounded between the hump and the $x$-axes, correct to 4 decimal places.</div><p style="text-align: right;">[3]</p></li></ol>

An architect designs and constructs a hump such that when a square wheel of dimensions 2 cm by 2 cm is rolled over the hump, the height of its centre above the horizontal ground (represented by the $x$-axis) is always constant. To test his invention, he places a square wheel against the hump, as shown in the diagram below. The points $R$, $S$ and $Q$ represent the centre of the square, the vertex of the square that is initially on the hump (at the origin $O$), and another vertex of the square which has a common side as $S$, respectively.

The architect begins pushing the square wheel over the hump in the clockwise direction. He discovers that the centre of the square is always vertically above the variable point of contact between the square and the hump, $P$, as shown in the diagram below. It is further given that the angle $\theta$ is the angle between the side $QS$ of the square and the horizontal line passing through $S$ , measured in the positive $x$-direction. ($\theta$ is taken to be positive if the vertex $Q$ lies above the horizontal line passing through $S$, and negative if otherwise.)

Derivatives of Reciprocal Trigonometric Functions

2017 NJC PROMO Q12

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=3{{t}^{2}}$, $y=2{{t}^{3}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point with parameter $t$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Points $P$ and $Q$ on $C$ have parameters $p$ and $q$ respectively. The tangent at $P$ meets the tangent at $Q$ and some point $R$. Show that the $x$-coordinates of $R$ is ${{p}^{2}}+pq+{{q}^{2}}$, and find the $y$-coordinates of $R$ in terms of $p$ and $q$. Given that $pq=-1$, show that $R$ lies on the curve with equation $x={{y}^{2}}+1$.</div><p style="text-align: right;">[5]</p></li></ol>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760242144093-gsk7u74l81p.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNDIxNDQwOTMtZ3NrN3U3NGw4MXAucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU3NDM5NSwiZXhwIjoxNzkzNTc0Njk1fQ.NrbRLHN7NNYPfXRobV2XkM2wZaHL1hNduxKHsVwiH18" width="400" class="aligncenter" /></figure>
<p>A curve $L$ has equation $x={{y}^{2}}+1$. The diagram shows parts of $C$ and $L$ for which $y\ge 0$. The curves $C$ and $L$ touch at the point $M$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that $4{{t}^{6}}-3{{t}^{2}}+1=0$ at $M$. Hence, or otherwise, find the exact coordinates of $M$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the exact value of the area of the shaded region bounded by $C$ and $L$ for which $y\ge 0$.</div><p style="text-align: right;">[6]</p></li></ol>

A curve $L$ has equation $x={{y}^{2}}+1$. The diagram shows parts of $C$ and $L$ for which $y\ge 0$. The curves $C$ and $L$ touch at the point $M$.

Math Question Bank