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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>A curve $C$ has parametric equations \[x=3t-\frac{3}{t},\ y=t+\frac{1}{t},\quad \text{where}\hspace{0.5em} t&lt;0.\]</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-0" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-0"><div class="list-item-paragraph"><p>Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{t^2-1}{3(t^2+1)}$.</p></div><div class="list-item-after-content">Hence, show the equation of the tangent to $C$ at the point $P$ with parameter $p$ is</div><div class="list-item-after-content"><p>\[3\left(p^2+1\right)y=\left(p^2-1\right)x+12p.\]</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-1" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-1"><div class="list-item-paragraph">Show that the cartesian equation of $C$ may be written as&nbsp;$9y^2-x^2=36$, stating the restriction on the values of $y$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-2" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-2"><div class="list-item-paragraph">Sketch the graph of $C$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-3" data-list-item-path="h2-2025-rvhs-promo-q11-rvhs-q11-item-block-2-item-3"><div class="list-item-paragraph"><p>The region bounded by $C$, the $y$-axis, the $x$-axis and the line $x=4.5$ is rotated through $2\pi$ radians about the $y$-axis. Find the volume of revolution of the solid formed, giving your answer correct to $2$ decimal places.</p><span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has parametric equations \[x=3t-\frac{3}{t},\ y=t+\frac{1}{t},\quad \text{where } t<0.\]

Show that the cartesian equation of $C$ may be written as $9y^2-x^2=36$, stating the restriction on the values of $y$.

Sketch the graph of $C$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes.

Tangents and Normals to Parametric Curves

Volume of Revolution about the y-axis

Differentiation of Parametric Equations

Asymptotes and Discontinuities

Sketching Parametric Curves

Converting Parametric Equations to Cartesian Form

2025 RVHS Promo Q11

Graphing Techniques

Applications of Integration

Techniques of Differentiation

Applications of Differentiation

<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 $-2\leq t<2$.

Graphs of Conic Sections

Parametric Equations

2025 EJC Promo Q9

<p>A curve $C$ is defined parametrically by $x=6t,\ y=t^2$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-0" data-list-item-path="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-0"><div class="list-item-paragraph">Show that the equation of the normal at the point $P(6p, p^2)$ is $p^3+(18-y)p-3x=0$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-1" data-list-item-path="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-1"><div class="list-item-paragraph">The normal to the curve at the point $P$ meets the curve again at the point $Q(-54,81)$. Find the possible values of $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-2" data-list-item-path="h2-2025-ejc-promo-q6-ejc-q6-item-block-1-item-2"><div class="list-item-paragraph">Sketch the graph of $C$, indicating clearly the coordinates of $Q$ and the possible points of $P$. Sketch also the normals at each of these points $P$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ is defined parametrically by $x=6t,\ y=t^2$.

Show that the equation of the normal at the point $P(6p, p^2)$ is $p^3+(18-y)p-3x=0$.

The normal to the curve at the point $P$ meets the curve again at the point $Q(-54,81)$. Find the possible values of $p$.

Sketch the graph of $C$, indicating clearly the coordinates of $Q$ and the possible points of $P$. Sketch also the normals at each of these points $P$.

2025 EJC Promo Q6

<p>A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch $C$, indicating on the diagram the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts.<span class="inline-marks">[4]</span></div></li></ol>
<p>A curve $D$ has parametric equations</p>
<p style="text-align: center">\[x = 23\sin 2\theta - 1,\ y = 23 + 23\cos 2\theta\text{ for}\hspace{0.5em} -\frac{\pi}{2} &lt; \theta &lt; \frac{\pi}{2}.\]</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the cartesian equation of curve $D$.<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">Hence, on the same diagram as in part <strong>(a)</strong>, sketch the curve $D$ and determine the number of intersection points between curve $C$ and curve $D$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$.

\[x = 23\sin 2\theta - 1,\ y = 23 + 23\cos 2\theta\text{ for } -\frac{\pi}{2} < \theta < \frac{\pi}{2}.\]

Sketching Rational Functions

Graphical Root Counting

2024 EJC P1 Q4

<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>

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

<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 $M$.

Locus of a Midpoint

N2013 P1 Q11 [Modified]

<p>Sketch the graph of the following pairs of parametric equations.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$x = e^t + 1, y = t + 1$, where&nbsp;$t &gt; 1$</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$x = \cos^3 t, y = \sin^3 t$, where&nbsp;$0 &lt; t \leq \pi$</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$x = 2\theta + \sin(2\theta), y = 3\sin\theta$, where&nbsp;$-\frac{\pi}{2} \leq \theta &lt; \frac{\pi}{2}$</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">$x = 2 + 2\cos u, y = \tan u$, for&nbsp;$0 \leq u &lt; \frac{\pi}{2}$</span><span class="inline-marks">[2]</span></div></li></ol>

Sketch the graph of the following pairs of parametric equations.

$x = e^t + 1, y = t + 1$, where $t > 1$

$x = \cos^3 t, y = \sin^3 t$, where $0 < t \leq \pi$

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

$x = 2 + 2\cos u, y = \tan u$, for $0 \leq u < \frac{\pi}{2}$

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">\[x = 1 - 2 \cos 2\theta, \quad y = \sin 2\theta - 1, \quad \text{for}\hspace{0.5em} 0 \le \theta \le \frac{3\pi}{8}.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the coordinates of the point where&nbsp;$C$ cuts the $y$-axis, giving your answer in exact form.<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">Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2} \cot 2\theta$. Find the coordinates of the points and the gradient of $C$ where&nbsp;$\theta = \frac{\pi}{4}$. What can be said about the tangent to $C$ as&nbsp;$\theta \to 0$?<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Hence sketch $C$, labelling the exact coordinates of axial intercepts and endpoints, and showing clearly the features of the curve at the points where&nbsp;$\theta = 0$ and $\theta = \frac{\pi}{4}$.<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">Find a cartesian equation of $C$.<span class="inline-marks">[2]</span></div></li></ol>

\[x = 1 - 2 \cos 2\theta, \quad y = \sin 2\theta - 1, \quad \text{for } 0 \le \theta \le \frac{3\pi}{8}.\]

Find the coordinates of the point where $C$ cuts the $y$-axis, giving your answer in exact form.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2} \cot 2\theta$. Find the coordinates of the points and the gradient of $C$ where $\theta = \frac{\pi}{4}$. What can be said about the tangent to $C$ as $\theta \to 0$?

Hence sketch $C$, labelling the exact coordinates of axial intercepts and endpoints, and showing clearly the features of the curve at the points where $\theta = 0$ and $\theta = \frac{\pi}{4}$.

Derivative as the Gradient of a Curve

2024 ASRJC Promo Q8

<p>A curve $C$ has parametric equations $x = t^2 + t$, $y = t^2 - t$, $t \in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch the graph of $C$, indicating the coordinates of the points where the curve crosses either axis.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the coordinates of the point on $C$ where the tangent to $C$ is parallel to the line $5y = 4x - 20$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point $(0, 2)$ and show algebraically that this tangent does not cut $C$ again.<span class="inline-marks">[5]</span></div></li></ol>

A curve $C$ has parametric equations $x = t^2 + t$, $y = t^2 - t$, $t \in \mathbb{R}$.

Sketch the graph of $C$, indicating the coordinates of the points where the curve crosses either axis.

Find the coordinates of the point on $C$ where the tangent to $C$ is parallel to the line $5y = 4x - 20$.

Find the equation of the tangent to $C$ at the point $(0, 2)$ and show algebraically that this tangent does not cut $C$ again.

2024 JPJC Promo Q10

<p>The curve $C$ has parametric equations \[x = -\frac{2}{t - 1}, \quad y = \frac{4}{t + 1}, \quad t &lt; 1, \quad t \neq \pm 1.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch a clearly labelled diagram of $C$, indicating any axial intercepts and asymptotes (if any) of this curve.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find also its cartesian equation, stating any restrictions where applicable.</span><span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ has parametric equations \[x = -\frac{2}{t - 1}, \quad y = \frac{4}{t + 1}, \quad t < 1, \quad t \neq \pm 1.\]

Sketch a clearly labelled diagram of $C$, indicating any axial intercepts and asymptotes (if any) of this curve.

Find also its cartesian equation, stating any restrictions where applicable.

2024 DHS Promo Q4

<p>Find the Cartesian equation of the curve with the parametric coordinates $\left( 2t,4-{{t}^{2}} \right)$ where&nbsp;$t\in \mathbb{R}$.<br>Sketch the curve, indicating clearly the equation of any asymptote(s) and coordinates of any axial intercept(s).<span class="inline-marks">[3]</span></p>

Find the Cartesian equation of the curve with the parametric coordinates $\left( 2t,4-{{t}^{2}} \right)$ where  $t\in \mathbb{R}$.
Sketch the curve, indicating clearly the equation of any asymptote(s) and coordinates of any axial intercept(s).

<p>A curve $C$ is defined parametrically by</p>
<p style="text-align: center">\[x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} &lt; t &lt; \frac{\pi}{4},\]</p>
<p>where&nbsp;$a$ is a positive constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\frac{\tan t}{\sqrt{1 + \tan^2 t}} = \sin t$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">By using part <strong>(a)</strong> or otherwise, find the cartesian equation of $C$ in the form $y = \mathrm{f}(x)$, simplifying your answer.<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">Show that&nbsp;$\mathrm{f}(-x) = -\mathrm{f}(x)$. Hence sketch $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the exact area of the region bounded by $C$, the $x$-axis and the lines $x = -A$ and $x = A$, where&nbsp;$0 &lt; A &lt; a$, leaving your answer in terms of $A$ and $a$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ is defined parametrically by

\[x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},\]

Show that $\frac{\tan t}{\sqrt{1 + \tan^2 t}} = \sin t$.

Show that $\mathrm{f}(-x) = -\mathrm{f}(x)$. Hence sketch $C$.

Find the exact area of the region bounded by $C$, the $x$-axis and the lines $x = -A$ and $x = A$, where $0 < A < a$, leaving your answer in terms of $A$ and $a$.

Area Bounded by Curves

Odd and Even Functions

2025 DHS P1 Q6

Functions

<p>A curve $C$ has parametric equations<br>$x = 1 - \sin \theta$, $y = \theta + \cos \theta$, where&nbsp;$-\pi \le \theta \le \dfrac{\pi}{2}$.<br>Sketch the graph of $C$, stating the exact coordinates of the end points.<span class="inline-marks">[2]</span></p>

A curve $C$ has parametric equations
$x = 1 - \sin \theta$, $y = \theta + \cos \theta$, where $-\pi \le \theta \le \dfrac{\pi}{2}$.
Sketch the graph of $C$, stating the exact coordinates of the end points.

Worked Example 7

<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 $C$ has parametric equations</p>
<p style="text-align: center">$x=a\left( \frac{4}{\pi }t+\sin 2t \right)+2a$, </p>
<p style="text-align: center">$y=a\cos t$,</p>
<p>where $-\frac{\pi }{2}\le t\le \frac{\pi }{2}$ and $a$ is a positive constant.</p>
<ol class="abc-list"></ol>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$. State clearly the coordinates of any points where $C$ meets the $x$-axis.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the area enclosed by $C$ and the $x$-axis is given by</div><div class="list-item-paragraph" style="text-align: center">${{a}^{2}}\displaystyle\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{\left( \frac{4}{\pi }\cos t+2\cos t\cos 2t \right)\mathrm{d}t}$</div><div class="list-item-paragraph">where ${{\theta }_{1}}$ and ${{\theta }_{2}}$ are values to be stated.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find, in terms of $a$, the exact area enclosed by $C$ and the $x$-axis.</div><p style="text-align: right;">[3]</p></li></ol>

$x=a\left( \frac{4}{\pi }t+\sin 2t \right)+2a$, 

where $-\frac{\pi }{2}\le t\le \frac{\pi }{2}$ and $a$ is a positive constant.

Integrable Forms of Trigonometric Functions

Trigonometric Integration Using Product-to-Sum Formulae

2021 HCI P2 Q1

Techniques of Integration

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x={{\mathrm{e}}^{u}}$, $y={{u}^{2}}$, for&nbsp;$u\in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760961546488-3mertv1nm" data-list-item-path="item-1760961546488-3mertv1nm"><div class="list-item-paragraph">Sketch $C$, stating the equation of the asymptote and the coordinates of any points where&nbsp;$C$ cuts the axes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760961547471-p6du25dj9" data-list-item-path="item-1760961547471-p6du25dj9"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point where&nbsp;$u=-1$. Leave your answer in the form of $y=mx+c$ where&nbsp;$m$ and $c$ are exact values to be determined.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the rest of the question, all lengths are measured in kilometres (km).</p>
<p>The region bounded by $C$ and the line $y=5$ represents a man-made lake. A circle with radius $0.5$ and centre $P\left( 1,2 \right)$ represents a man-made island on the lake. A bridge is built from a point $Q$ on $C$, where&nbsp;$u\le 0$, to a point $R$ on the edge of the circular island such that&nbsp;$QR$ produced passes through point $P$. It is given that the distance between $Q$ and $P$ is denoted by $l$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760961587054-ajrf9irlu" data-list-item-path="item-1760961587054-ajrf9irlu"><div class="list-item-paragraph">Show that&nbsp;${{l}^{2}}={{\mathrm{e}}^{2u}}-2{{\mathrm{e}}^{u}}+{{u}^{4}}-4{{u}^{2}}+5$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760961588570-iaq6ljw6m" data-list-item-path="item-1760961588570-iaq6ljw6m"><div class="list-item-paragraph">Use differentiation to find the values of $u$ which give stationary values of ${{l}^{2}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760961589238-4s92dab5v" data-list-item-path="item-1760961589238-4s92dab5v"><div class="list-item-paragraph">Show that only one of the values of $u$ found in part <strong>(iv)</strong> gives a minimum value of ${{l}^{2}}$. Hence find the shortest length of the bridge $QR$.<span class="inline-marks">[4]</span></div></li></ol>

$x={{\mathrm{e}}^{u}}$, $y={{u}^{2}}$, for $u\in \mathbb{R}$.

For the rest of the question, all lengths are measured in kilometres (km).

The region bounded by $C$ and the line $y=5$ represents a man-made lake. A circle with radius $0.5$ and centre $P\left( 1,2 \right)$ represents a man-made island on the lake. A bridge is built from a point $Q$ on $C$, where $u\le 0$, to a point $R$ on the edge of the circular island such that $QR$ produced passes through point $P$. It is given that the distance between $Q$ and $P$ is denoted by $l$.

Problems Involving Maximisation and Minimisation

2022 VJC Promo Q12

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=a\left( {{t}^{2}}-t \right)$, $y=a\left( 1-2{{t}^{2}} \right)$, $-3 &lt; t\le 2$,</p>
<p>where&nbsp;$a$ is a positive constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1760865685598-5b3lnkqc8" data-list-item-path="item-1760865685598-5b3lnkqc8"><div class="list-item-paragraph">Sketch the graph of $C$, stating the coordinates of the end-points.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760865686138-3nhbtyoyo" data-list-item-path="item-1760865686138-3nhbtyoyo"><div class="list-item-paragraph">Find, in terms of $a$,</div><ol class="i-list"><li data-list-item-id="subitem-1760865711939-ij2o8bddy" data-list-item-path="item-1760865686138-3nhbtyoyo/subitem-1760865711939-ij2o8bddy"><div class="list-item-paragraph">the equation of the normal at the point where&nbsp;$t=-1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1760865712508-966l0bede" data-list-item-path="item-1760865686138-3nhbtyoyo/subitem-1760865712508-966l0bede"><div class="list-item-paragraph">the exact coordinates of the point where the normal cuts $C$ again.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

$x=a\left( {{t}^{2}}-t \right)$, $y=a\left( 1-2{{t}^{2}} \right)$, $-3 < t\le 2$,

2024 YIJC Promo Q6

<p>A curve $C$ has parametric equations $x=1-\sin \theta $, $y=\theta +\cos \theta $, where $-\pi \le \theta \le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $C$, stating the exact coordinates of the end points.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta $. What can be said about the tangent to $C$ as $\theta \to -\frac{\pi }{2}$?</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">A point $P$ on $C$ has parameter $p$, where $0 < p < \frac{\pi }{2}$. Show that the normal to $C$ at $P$ crosses the $y$-axis at point $Q$ with coordinates $\left( 0,p \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Show that the area of region bounded by $C$, the normal to $C$ at point $P$ and the $y$-axis is given by $a\pi +bp+c\cos p$, where $a$, $b$ and $c$ are to be determined.</div><p style="text-align: right;">[6]</p></li></ol>

A curve $C$ has parametric equations $x=1-\sin \theta $, $y=\theta +\cos \theta $, where $-\pi \le \theta \le \frac{\pi }{2}$.

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