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N(A) E Math

PSLE

PSLE Foundation Mathematics

A curve is defined parametrically by $x=3\tan t$ and $y=2\sec t$, where $-\frac{\pi}{2}<t<\frac{\pi}{2}$.

Tangents and Normals to Parametric Curves

Converting Parametric Equations to Cartesian Form

Normal and branch restriction of a trigonometric curve

Applications of Differentiation

Graphing Techniques

A curve is defined parametrically by $x=\ln(t+1)$ and $y=\ln(5-t)$, where $t$ takes all real values for which both expressions are defined.

Normal and coordinate ranges of a logarithmic curve

<p>A curve $C$ has parametric equations \[x=t^2+1,\qquad y=t^3-2.\]</p>
<p>The points $P$ and $Q$ on $C$ have parameters $p$ and $2p$ respectively, where&nbsp;$p&gt;0$. The tangents to $C$ at $P$ and $Q$ meet at $R$.</p>
<ol class="abc-list"><li data-list-item-id="q1-a" data-list-item-path="q1-a"><div class="list-item-paragraph">Show that the tangent to $C$ at the point with parameter $t$ has equation \[3tx-2y=t^3+3t+4.\]<span class="inline-marks">[3]</span></div></li><li data-list-item-id="q1-b" data-list-item-path="q1-b"><div class="list-item-paragraph">Find the coordinates of $R$ in terms of $p$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="q1-c" data-list-item-path="q1-c"><div class="list-item-paragraph"><p>Show that the area of triangle $PQR$ is $\dfrac{5}{3}p^5$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="q1-d" data-list-item-path="q1-d"><div class="list-item-paragraph">Given that the area of triangle $PQR$ is $\dfrac{160}{3}$, find the value of $p$.<span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has parametric equations \[x=t^2+1,\qquad y=t^3-2.\]

The points $P$ and $Q$ on $C$ have parameters $p$ and $2p$ respectively, where $p>0$. The tangents to $C$ at $P$ and $Q$ meet at $R$.

Show that the tangent to $C$ at the point with parameter $t$ has equation \[3tx-2y=t^3+3t+4.\]

Find the coordinates of $R$ in terms of $p$.

Given that the area of triangle $PQR$ is $\dfrac{160}{3}$, find the value of $p$.

Differentiation of Parametric Equations

Area enclosed by two parametric tangents

Techniques of Differentiation

<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.

Volume of Revolution about the y-axis

Asymptotes and Discontinuities

Sketching Parametric Curves

2025 RVHS Promo Q11

Applications of Integration

<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>The asteroid, a curve $C$ which is used to characterize various properties of energy and magnetism, has parametric equations \[x = a \cos^3 t, y = a \sin^3 t,\] where&nbsp;$0 \le t \le \frac{1}{2}\pi$ and $a$ is a positive constant.</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 the equation of the tangent to $C$ at the point $P$ with parameter $p$.<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"><div class="list-item-paragraph">The tangent at $P$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$. Show that the length $AB$ depends only on $a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">Find a cartesian equation of $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-3" data-list-item-path="imported-item-block-3-item-3"><div class="list-item-paragraph">The region bounded by $C$ and the $x$- and $y$-axes is rotated through $360^\circ$ about the $y$-axis. Find the exact value of the volume of revolution of the solid formed.<span class="inline-marks">[4]</span></div></li></ol>

The asteroid, a curve $C$ which is used to characterize various properties of energy and magnetism, has parametric equations \[x = a \cos^3 t, y = a \sin^3 t,\] where $0 \le t \le \frac{1}{2}\pi$ and $a$ is a positive constant.

Find the equation of the tangent to $C$ at the point $P$ with parameter $p$.

The tangent at $P$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$. Show that the length $AB$ depends only on $a$.

The region bounded by $C$ and the $x$- and $y$-axes is rotated through $360^\circ$ about the $y$-axis. Find the exact value of the volume of revolution of the solid formed.

Derivatives of Trigonometric Functions

Chain Rule

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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 Parallel to the Axes

Product Rule

Parametric Equations

N2025 P1 Q11

<p>A curve has parametric equations given by $x = 2t^2 + t$, $y = t^3 - 3t$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the equation of the tangent on the curve which is parallel to the $y$-axis.</div></li><li><div class="list-item-paragraph">Find the coordinates of the point when the tangent at $t = -\frac{1}{2}$ meets the curve again.</div></li></ol>

A curve has parametric equations given by $x = 2t^2 + t$, $y = t^3 - 3t$.

<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 $C$ is defined by the parametric equations $x=at^2$, $y=\frac{a}{t}$, $t \neq 0$, 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 the equation of the tangent to the curve at the point $P\left(ap^2, \frac{a}{p}\right)$ is $2p^3y+x=3ap^2$.<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">Find the equations of the tangents to $C$ at the points where&nbsp;$x=a$. Find also the acute angle between these tangents.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><ol class="i-list"><li data-list-item-id="item-c-i" data-list-item-path="item-c/item-c-i"><div class="list-item-paragraph">Show that&nbsp;$(q-p)^2(q+2p)=q^3-3p^2q+2p^3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c-ii" data-list-item-path="item-c/item-c-ii"><div class="list-item-paragraph">The tangent to the curve at $P$ cuts the curve again at $Q\left(aq^2, \frac{a}{q}\right)$. Find $q$ in terms of $p$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>
<p>The following diagram shows the graph of $C$ for&nbsp;$a=1$, and the Cartesian equation of the curve $C$ is $xy^2=1$. The tangent to $C$ at the point $\left(p^2, \frac{1}{p}\right)$ cuts the curve again at the point $\left(4p^2, -\frac{1}{2p}\right)$.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" data-proportional style="width:300px;height:200px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:300px;--graph-aspect:300 / 200;"></div>

<p>The region $S$ is bounded by the tangent to $C$ at the point $\left( p^2, \frac{1}{p} \right)$, the curve $C$ and the line</p>
<p>$x = p^2$</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the area of $S$ in terms of $p$.<span class="inline-marks">[3]</span></div></li></ol>

The curve $C$ is defined by the parametric equations $x=at^2$, $y=\frac{a}{t}$, $t \neq 0$, where $a$ is a positive constant.

Show that the equation of the tangent to the curve at the point $P\left(ap^2, \frac{a}{p}\right)$ is $2p^3y+x=3ap^2$.

Find the equations of the tangents to $C$ at the points where $x=a$. Find also the acute angle between these tangents.

The following diagram shows the graph of $C$ for $a=1$, and the Cartesian equation of the curve $C$ is $xy^2=1$. The tangent to $C$ at the point $\left(p^2, \frac{1}{p}\right)$ cuts the curve again at the point $\left(4p^2, -\frac{1}{2p}\right)$.

The region $S$ is bounded by the tangent to $C$ at the point $\left( p^2, \frac{1}{p} \right)$, the curve $C$ and the line

Find the area of $S$ in terms of $p$.

Area Bounded by Curves

2025 ACJC P1 Q11

<p>A curve $C$ has a parametric equations</p>
<p style="text-align: center">$x=5{{t}^{2}}$, $y=2t$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $t$ at the point on $C$ where the tangent has gradient $0.8$.</div></li><li><div class="list-item-paragraph">The normal at the point $P\left( 5{{p}^{2}},2p \right)$ on $C$ meets the $x-$axis at point $Q$. Find the Cartesian equation of the locus of the mid-point of $PQ$ as $p$ varies.</div></li></ol>

Locus of a Midpoint

Worked Example 3

<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x = 5t^{2}$, $y = 2t$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $t$ at the point on $C$ where the tangent has gradient $0.8$.</div></li><li><div class="list-item-paragraph">The normal at the point $P(5p^{2}, 2p)$ on $C$ meets the $x$-axis at point $Q$. Find the Cartesian equation of the locus of the midpoint of $PQ$ as&nbsp;$p$ varies.</div></li></ol>

<p>A curve has parametric equation $x=2t-1$ , $y=\frac{1}{{{t}^{2}}+1}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Prove that the equation of the tangent at the point with parameter $t$ is $(t^{2} + 1)^{2}y + tx = 3t^{2} - t + 1$.</div></li><li><div class="list-item-paragraph">The tangent at the point where&nbsp;$t = 3$ meets the curve again at the point where&nbsp;$t = q$. Find the value of $q$ and hence find the coordinates of the point of intersection.</div></li></ol>

A curve has parametric equation $x=2t-1$ , $y=\frac{1}{{{t}^{2}}+1}$.

<p>A curve is defined by the parametric equations $x = 2t - 1$, $y = t^{2} + 2$. Show that the equation of the tangent to the curve at the point $P(2p - 1, p^{2} + 2)$ is $y = px - p^{2} + p + 2$. Hence, find the equations of the tangents that pass through the point $(-1, -2)$.</p>

A curve is defined by the parametric equations $x = 2t - 1$, $y = t^{2} + 2$. Show that the equation of the tangent to the curve at the point $P(2p - 1, p^{2} + 2)$ is $y = px - p^{2} + p + 2$. Hence, find the equations of the tangents that pass through the point $(-1, -2)$.

<p>A curve $C$ is given parametrically by $x = \frac{1}{1 - t^{2}}$, $y = \frac{t}{1 - t^{2}}$. Show that the equation of the tangent at the point $\left(-\frac{1}{3}, -\frac{2}{3}\right)$ is $4y - 5x = -1$.</p>

A curve $C$ is given parametrically by $x = \frac{1}{1 - t^{2}}$, $y = \frac{t}{1 - t^{2}}$. Show that the equation of the tangent at the point $\left(-\frac{1}{3}, -\frac{2}{3}\right)$ is $4y - 5x = -1$.

Quotient Rule

<p>The parametric equations of a curve are $x = \frac{t}{1 - t^{3}}$ and $y = \frac{1}{1 - t^{3}}$, where&nbsp;$t \ne 1$. Find the equation of the normal to the curve at the point where&nbsp;$t = 2$.</p>

The parametric equations of a curve are $x = \frac{t}{1 - t^{3}}$ and $y = \frac{1}{1 - t^{3}}$, where $t \ne 1$. Find the equation of the normal to the curve at the point where $t = 2$.

<p>A curve is defined parametrically by the equations $x = \frac{2}{t}$ and $y = 1 - 2t$, where&nbsp;$t$ is a real parameter. Find the equation of the tangent to the curve at $t = -2$ and determine if it cuts the curve again.</p>

A curve is defined parametrically by the equations $x = \frac{2}{t}$ and $y = 1 - 2t$, where $t$ is a real parameter. Find the equation of the tangent to the curve at $t = -2$ and determine if it cuts the curve again.

<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">$x=3{{u}^{2}}$, $y=6u$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the equation of the normal to the curve at the point $P\left( 3{{p}^{2}},\ 6p \right)$, where $p$ is a non-zero constant.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The normal meets the $x$-axis at $Q$ and the $y$-axis at $R$. Find the coordinates of $Q$ and of $R$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find two possible expressions for the area of the triangle bounded by the normal and the axes in terms of $p$, stating the range of values of $p$ in each case.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $p$ is positive and increasing at a rate of $2$ units/s, find the rate of change of the area of the triangle in terms of $p$.</div><p style="text-align: right;">[2]</p></li></ol>

Connected Rates of Change

2024 DHS Promo Q7

<p>The parametric equations of the curve $C$ are</p>
<p style="text-align: center">$x=\tan \theta -1$, $y=2\sec \theta $, where&nbsp;$-\frac{\pi }{2}&lt;\theta &lt;\frac{\pi }{2}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760961275279-njpvri7j9" data-list-item-path="item-1760961275279-njpvri7j9"><div class="list-item-paragraph">Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $\theta $, leaving your answer in a single trigonometric function.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760961275936-uic3edge1" data-list-item-path="item-1760961275936-uic3edge1"><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point where it cuts the $y$-axis, leaving your answer in exact form.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760961276913-61m5k7kii" data-list-item-path="item-1760961276913-61m5k7kii"><div class="list-item-paragraph">Find the Cartesian equation of $C$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760961277949-yh1yw236v" data-list-item-path="item-1760961277949-yh1yw236v"><div class="list-item-paragraph">Find the equations of the asymptotes and state the coordinates of any turning point(s).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760961279831-43bl3h9nr" data-list-item-path="item-1760961279831-43bl3h9nr"><div class="list-item-paragraph">The region bounded by the tangent found in part <strong>(b)</strong>, the curve $C$ and the line $x=-1$ is rotated completely about the $x$-axis. Calculate the volume obtained.<span class="inline-marks">[3]</span></div></li></ol>

The parametric equations of the curve $C$ are

$x=\tan \theta -1$, $y=2\sec \theta $, where $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$.

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