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PSLE

PSLE Foundation Mathematics

<p>The curve $C$ has equation</p>
<p>\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]</p>
<p>The line $x=1$ intersects $C$ at $A(1,0)$ and at a second point $B$.</p>
<ol class="i-list"><li data-list-item-id="b818afb7-41c3-4440-814f-cebde95ce4c2" data-list-item-path="b818afb7-41c3-4440-814f-cebde95ce4c2"><div class="list-item-paragraph">Using differentiation, find the exact coordinates of the point on $C$ where&nbsp;$x$ has its greatest value.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="35fabeb6-60ee-401c-9944-333d45b502e4" data-list-item-path="35fabeb6-60ee-401c-9944-333d45b502e4"><div class="list-item-paragraph">Sketch $C$ and the line $x=1$ on the same axes. State the equation of the asymptote of $C$ and find the coordinates of $B$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92" data-list-item-path="aa9e4784-84aa-43c0-bc7a-f6b2d9a58f92"><div class="list-item-paragraph">Find the area of the finite region enclosed by $C$ and the line $x=1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="9a5b99e2-d202-43e6-9ba3-93603dc40d0d" data-list-item-path="9a5b99e2-d202-43e6-9ba3-93603dc40d0d"><div class="list-item-paragraph">Find the volume generated when this region is rotated through $2\pi$ radians about the $y$-axis.<span class="inline-marks">[2]</span></div></li></ol>

\[x=(y+1)e^{-\frac{y}{2}},\qquad y\ge0.\]

The line $x=1$ intersects $C$ at $A(1,0)$ and at a second point $B$.

Using differentiation, find the exact coordinates of the point on $C$ where $x$ has its greatest value.

Sketch $C$ and the line $x=1$ on the same axes. State the equation of the asymptote of $C$ and find the coordinates of $B$.

Sketching Exponential Functions and Solving Equations Graphically

Volume of Revolution about the y-axis

Asymptotes and Discontinuities

Area bounded by 2 curves

Stationary Points and Their Nature

An enclosed region beside a sideways exponential curve

Applications of Integration

Graphing Techniques

Applications of Differentiation

<p>The curves $C_1$ and $C_2$ have equations</p>
<p>\[C_1:\quad x=\frac{\sqrt y}{4-y^2},\qquad C_2:\quad x=1-\frac y2,\]</p>
<p>where&nbsp;$0\le y&lt;2$.</p>
<p>The curves intersect at $P$. The region $R$ is bounded by $C_1$, $C_2$ and the $x$-axis.</p>
<ol class="i-list"><li data-list-item-id="7372e40b-dbf3-4bdb-8cc4-370e31da0bdc" data-list-item-path="7372e40b-dbf3-4bdb-8cc4-370e31da0bdc"><div class="list-item-paragraph">Sketch $C_1$ and $C_2$ on the same axes, indicating their intersections with the $x$-axis. Find the coordinates of $P$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="6f926be5-df64-4a1c-8713-a15ccc14de0d" data-list-item-path="6f926be5-df64-4a1c-8713-a15ccc14de0d"><div class="list-item-paragraph">Find the area of $R$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="ec60edec-9f1d-4c19-8cd4-e7f52ea34663" data-list-item-path="ec60edec-9f1d-4c19-8cd4-e7f52ea34663"><div class="list-item-paragraph">Find the volume generated when&nbsp;$R$ is rotated through $2\pi$ radians about the $y$-axis.<span class="inline-marks">[4]</span></div></li></ol>

The curves $C_1$ and $C_2$ have equations

\[C_1:\quad x=\frac{\sqrt y}{4-y^2},\qquad C_2:\quad x=1-\frac y2,\]

The curves intersect at $P$. The region $R$ is bounded by $C_1$, $C_2$ and the $x$-axis.

Sketch $C_1$ and $C_2$ on the same axes, indicating their intersections with the $x$-axis. Find the coordinates of $P$.

Find the volume generated when $R$ is rotated through $2\pi$ radians about the $y$-axis.

Area and volume between sideways curves

<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

Differentiation of Parametric Equations

Sketching Parametric Curves

Converting Parametric Equations to Cartesian Form

2025 RVHS Promo Q11

Techniques of Differentiation

<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

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Show that&nbsp;$\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.<span class="inline-marks">[2]</span></div></li></ol>
<p>A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Using the result in part <strong>(a)</strong>, find the exact volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $x$-axis.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Find the volume generated when&nbsp;$R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.<span class="inline-marks">[3]</span></div></li></ol>

Show that $\frac{\mathrm{d}}{\mathrm{d}x}(x \ln x - x) = \ln x$.

A curve $C$ is given by the equation $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm{e}$.

Find the volume generated when $R$ is rotated $360^\circ$ about the $y$-axis, giving your answer correct to $2$ decimal places.

Derivatives of Natural Logarithmic Functions

Volume of Revolution about the x-axis

Product Rule

N2025 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

Tangents and Normals to Implicit Curves

Graphs of Conic Sections

2025 NJC P2 Q2

<p>The curve $C$ is defined by the parametric equations</p>
<p style="text-align: center">$x=6t-1$, $y=8{{t}^{2}}+2$ where&nbsp;$t\ge \frac{1}{6}$.</p>
<p>The line $N$, with equation $3y=8x-10$, is the tangent to $C$ at the point $\left( 5,10 \right)$. The region enclosed by $C$, $N$ and the $y$-axis is denoted by $R$.</p>
<ol class="abc-list"><li data-list-item-id="item-1761041124949-ho4ty3zb6" data-list-item-path="item-1761041124949-ho4ty3zb6"><div class="list-item-paragraph">Find the area of $R$.</div></li><li data-list-item-id="item-1761041125533-yv0ghak7m" data-list-item-path="item-1761041125533-yv0ghak7m"><div class="list-item-paragraph">Find the cartesian equation of $C$. Hence, find the volume generated when&nbsp;$R$ is rotated about the $y$-axis through ${{360}^{\circ }}$.</div></li></ol>

The curve $C$ is defined by the parametric equations

$x=6t-1$, $y=8{{t}^{2}}+2$ where $t\ge \frac{1}{6}$.

The line $N$, with equation $3y=8x-10$, is the tangent to $C$ at the point $\left( 5,10 \right)$. The region enclosed by $C$, $N$ and the $y$-axis is denoted by $R$.

NYJC Applications of Integration Tutorial Q1

<p>A curve $C$ has equation $y=2ax+\frac{b}{x}$ where&nbsp;$a$ and $b$ are non-zero real constants and $x\ne 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760765552949-17xocwmpx" data-list-item-path="item-1760765552949-17xocwmpx"><div class="list-item-paragraph">Using differentiation, determine whether $C$ has any stationary points if&nbsp;$ab &lt; 0$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$b=\frac{1}{2}a$ where&nbsp;$a&gt;0$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760765589976-7l9zd6yvg" data-list-item-path="item-1760765589976-7l9zd6yvg"><div class="list-item-paragraph">Sketch $C$, stating the equations of any asymptotes, and the coordinates of any stationary points and points where&nbsp;$C$ crosses the axes (if any).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760765593674-qsqu3k010" data-list-item-path="item-1760765593674-qsqu3k010"><div class="list-item-paragraph">State the range of values of $k$, in terms of $a$, for which the equation $2ax+\frac{b}{x}=kx$ has no real roots.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760765594477-wx20pfe6v" data-list-item-path="item-1760765594477-wx20pfe6v"><div class="list-item-paragraph">The region bounded by $C$, the axes, the lines $x=\frac{1}{2}$ and $y=4a$ is rotated about the $y$-axis through $2\pi $ radians. Show that the volume generated is given by $\frac{1}{2}a\pi +\frac{\pi }{8{{a}^{2}}}\displaystyle\displaystyle \displaystyle\int_{2a}^{4a}{{{y}^{2}}}-2{{a}^{2}}-y\sqrt{{{y}^{2}}-4{{a}^{2}}}\ \mathrm{d}y$.</div><div class="list-item-paragraph">Hence, find, in terms of $a$ and $\pi $, the exact volume generated.<span class="inline-marks">[6]</span></div></li></ol>

A curve $C$ has equation $y=2ax+\frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x\ne 0$.

It is now given that $b=\frac{1}{2}a$ where $a>0$.

Sketching Rational Functions

Parameter Conditions for Graphs and Equations

Parameter Conditions for Stationary Points

Integration by the Reverse Chain Rule

Stationary Points and Points of Inflection

2024 RI P1 Q9

Techniques of Integration

<p>The region bounded by the curve with equation $x=\frac{y}{\sqrt{2y-{{y}^{2}}}}$, the lines $y=1$, $y=1.6$ and the $y$-axis is rotated through $2\pi $ radians about the $y$-axis to form a solid ornament.</p>
<ol class="abc-list"><li data-list-item-id="item-1760765492274-msblxptdn" data-list-item-path="item-1760765492274-msblxptdn"><div class="list-item-paragraph">Find the exact volume ${{V}_{1}}$ of the ornament, giving your answer in terms of $\pi $.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760765492769-m3fqs86cx" data-list-item-path="item-1760765492769-m3fqs86cx"><div class="list-item-paragraph">An ornament designer designs a different ornament by rotating the region bounded by another curve with equation $x=\frac{by}{\sqrt{2y-{{y}^{2}}}}$, where&nbsp;$b&gt;0$, the lines $y=1$, $y=1.6$ and the $y$-axis. The region is rotated through $2\pi $ radians about the $y$-axis. The volume generated is now ${{V}_{2}}$. State the ratio of ${{V}_{1}}$ to ${{V}_{2}}$.<span class="inline-marks">[1]</span></div></li></ol>

The region bounded by the curve with equation $x=\frac{y}{\sqrt{2y-{{y}^{2}}}}$, the lines $y=1$, $y=1.6$ and the $y$-axis is rotated through $2\pi $ radians about the $y$-axis to form a solid ornament.

Integrals Involving Long Division

2024 HCI P1 Q3

<p>The diagram shows the region ${R}$, which is bounded by the curve ${(y-2)^{2}=2x-3}$ and the line ${x=3.5}$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the</p><ol class="i-list"><li data-list-item-id="item-1790411500021-19e5lk948" data-list-item-path="item-1790411500021-19e5lk948"><div class="list-item-paragraph"><p>area of region ${R}$.</p></div></li><li data-list-item-id="item-1790411500793-i7g0t6tg3" data-list-item-path="item-1790411500793-i7g0t6tg3"><div class="list-item-paragraph"><p>volume when&nbsp;${R}$ is rotated through four right angles about the ${x}$-axis.</p></div></li><li data-list-item-id="item-1790411501589-99mij59u7" data-list-item-path="item-1790411501589-99mij59u7"><div class="list-item-paragraph"><p>volume when&nbsp;${R}$ is rotated through ${2\pi}$ about the ${y}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="256" data-height="223" data-proportional style="width:256px;height:223px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:256px;--graph-aspect:256 / 223;"></div></div></div></div>

The diagram shows the region ${R}$, which is bounded by the curve ${(y-2)^{2}=2x-3}$ and the line ${x=3.5}$.

Area Bounded by Curves

Volume of Revolution: Roots of Equations

<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><p>Find the volumes of solid formed when region ${R}$ is rotated through ${2\pi}$ about</p><ol class="abc-list"><li data-list-item-id="item-1790409768941-4f23ijx0s" data-list-item-path="item-1790409768941-4f23ijx0s"><div class="list-item-paragraph"><p>${y}$-axis,</p></div></li><li data-list-item-id="item-1790409769593-b62yblex9" data-list-item-path="item-1790409769593-b62yblex9"><div class="list-item-paragraph"><p>${x}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="244" data-height="187" data-proportional style="width:244px;height:187px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:244px;--graph-aspect:244 / 187;"></div></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the volumes of solid formed when region ${R}$ is rotated through ${2\pi}$ about</p><ol class="abc-list"><li data-list-item-id="item-1790405400625-njfim6qjc" data-list-item-path="item-1790405400625-njfim6qjc"><div class="list-item-paragraph"><p>${y}$-axis,</p></div></li><li data-list-item-id="item-1790405401525-9ne1comxv" data-list-item-path="item-1790405401525-9ne1comxv"><div class="list-item-paragraph"><p>${x}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="351" data-height="333" data-proportional style="width:351px;height:333px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:351px;--graph-aspect:351 / 333;"></div></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the volumes of solid formed when region ${R}$ is rotated through ${2\pi}$ about</p><ol class="abc-list"><li data-list-item-id="item-1790411074084-ml5uwwzi8" data-list-item-path="item-1790411074084-ml5uwwzi8"><div class="list-item-paragraph"><p>${x}$-axis,</p></div></li><li data-list-item-id="item-1790411074774-22zuzkks6" data-list-item-path="item-1790411074774-22zuzkks6"><div class="list-item-paragraph"><p>${y}$-axis.</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="334" data-height="307" data-proportional style="width:334px;height:307px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:334px;--graph-aspect:334 / 307;"></div></div></div></div>

<ol class="abc-list" start="2"><li data-list-item-id="item-1760764282779-cx3i3qqa3" data-list-item-path="item-1760764282779-cx3i3qqa3"><div class="list-item-paragraph">A chef plans to create an ornament for his master dish. The ornament is made by rotating the shaded 	region as shown in the diagram completely about the $y$-axis. The region is bounded by the parabola 	$y={{\left( x-3 \right)}^{2}}$, the curve $C$ and the lines $y=9+d$ and $y=d$, where&nbsp;$0 &lt; d &lt; 9$. The curve $C$ is the 	reflection of the parabola along the line $y=9$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760764327490-iag1bxr6cy8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3NjQzMjc0OTAtaWFnMWJ4cjZjeTgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mzg1MzgxMywiZXhwIjoxNzkzODU0MTEzfQ.cIIwvkF_PouvwUZlJuqIk3uOsflAONkBKLI3dQep8-4" alt="List item image" class="aligncenter custom-width" style="max-width: min(500px, 100%); height: auto;" /></div><ol class="i-list"><li data-list-item-id="subitem-1760764326707-hrn2bk8ug" data-list-item-path="item-1760764282779-cx3i3qqa3/subitem-1760764326707-hrn2bk8ug"><div class="list-item-paragraph">Find the equation of the curve $C$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760764337087-6c71ytip6" data-list-item-path="item-1760764282779-cx3i3qqa3/subitem-1760764337087-6c71ytip6"><div class="list-item-paragraph">Find the volume of the ornament in terms of ${d}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="subitem-1760764342127-tmgdrosg0" data-list-item-path="item-1760764282779-cx3i3qqa3/subitem-1760764342127-tmgdrosg0"><div class="list-item-paragraph">By varying the value of ${d}$, find the maximum volume for the ornament correct to the nearest integer. State the value of ${d}$ corresponding to this maximum volume.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Problems Involving Maximisation and Minimisation

2024 DHS P1 Q10

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about</p><ol class="abc-list"><li data-list-item-id="item-1790400866275-lzq08pgwt" data-list-item-path="item-1790400866275-lzq08pgwt"><div class="list-item-paragraph"><p>$y$-axis,</p></div></li><li data-list-item-id="item-1790400867129-xfndc7i5s" data-list-item-path="item-1790400867129-xfndc7i5s"><div class="list-item-paragraph"><p>$x$-axis,</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjI3OCwiaGVpZ2h0IjoyNDYsImVsZW1lbnRzIjpbeyJpZCI6ImY1ZDgyNDRkLWMwOTEtNDMyYS04ZDRmLTNkYTFhZjQ2ZGU2MyIsInR5cGUiOiJwb2x5Z29uIiwiY29sb3IiOiIjMDAwMDAwIiwibGF5ZXIiOjEsInBvaW50cyI6W1swLDBdLFswLDJdLFsxLDRdLFsxLjA1LDMuODk3NV0sWzEuMSwzLjc5XSxbMS4xNSwzLjY3NzVdLFsxLjIsMy41Nl0sWzEuMjUsMy40Mzc1XSxbMS4zLDMuMzFdLFsxLjM1LDMuMTc3NV0sWzEuNCwzLjA0XSxbMS40NSwyLjg5NzVdLFsxLjUsMi43NV0sWzEuNTUsMi41OTc1XSxbMS42LDIuNDRdLFsxLjY1LDIuMjc3NV0sWzEuNywyLjExXSxbMS43NSwxLjkzNzVdLFsxLjgsMS43Nl0sWzEuODUsMS41Nzc1XSxbMS45LDEuMzldLFsxLjk1LDEuMTk3NV0sWzIsMV0sWzIuMDUsMC43OTc1XSxbMi4xLDAuNTldLFsyLjE1LDAuMzc3NV0sWzIuMiwwLjE2XSxbMi4yMzYwNjc5Nzc1LDBdXSwiZmlsbENvbG9yIjoiIzFFNkZCQSIsImZpbGxPcGFjaXR5IjowLjIsInN0cm9rZVdpZHRoIjowfSx7ImlkIjoiYTk4Yzk0MzAtNTFlMC00MjAwLTk4YmQtNTljYjEwNGQxZTY5IiwidHlwZSI6ImZ1bmN0aW9uZ3JhcGgiLCJ4TWF4IjoyLjM4LCJ4TWluIjotMC41MywiY29sb3IiOiIjMUU2RkJBIiwibGF5ZXIiOjUsImZpbGxDb2xvciI6Im5vbmUiLCJleHByZXNzaW9uIjoiNS14XjIiLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiZGNkZjhhYWQtMzM0ZC00NjEwLWI4ZTAtZmY5Y2ViNWU4NWNjIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiM3MDQ3QzYiLCJsYXllciI6NSwicG9pbnQxIjpbLTAuNTMsMC45NF0sInBvaW50MiI6WzEuNzIsNS40NF0sInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6MSwieSI6NCwiaWQiOiIzYjU4NTJhYy1hN2QwLTRjMjEtYTVhZC01OTE0MDQ4YTZkN2IiLCJ0eXBlIjoicG9pbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IiIsImxheWVyIjo3LCJmaWxsQ29sb3IiOiIjMDAwMDAwIiwicG9pbnRTaXplIjoyLCJwb2ludFN0eWxlIjoiY2lyY2xlIiwiZmlsbE9wYWNpdHkiOjEsInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6MS4xLCJ5Ijo0LCJpZCI6IjRmM2RjMGYzLTFhOTgtNGEwOS05ZWUzLTg5NDgyODkzMGNjOSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCgoMSw0KVxcKSIsInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6MC44MiwieSI6MS41OCwiaWQiOiIyYmMyMzFmYy0wZGI1LTQ3Y2MtOTBjNi0xZWFmODBhOTU1Y2EiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoUlxcKSIsInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6MS40NSwieSI6NS42OSwiaWQiOiJjZWE1ODZhNC05YjAxLTQ2MzEtYWI4Mi1lMDhkMjdmYjNlOGUiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoeT0yeCsyXFwpIiwic3Ryb2tlV2lkdGgiOjEuNX0seyJ4IjoxLjg2LCJ5IjoyLjE0LCJpZCI6Ijk2OTNkNjIwLTRjZWMtNDAwMS04MjRiLWRkOTVmYzUyODIxYiIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCh5PTUteF4yXFwpIiwic3Ryb2tlV2lkdGgiOjEuNX1dLCJzaG93WEF4aXMiOnRydWUsInNob3dZQXhpcyI6dHJ1ZSwieEF4aXNMYWJlbCI6IlxcKHhcXCkiLCJ4QXhpc1JhbmdlIjpbLTAuMzczNywyLjMzNjhdLCJ5QXhpc0xhYmVsIjoiXFwoeVxcKSIsInlBeGlzUmFuZ2UiOlswLjA3MjgsNC45NjYzXSwiYm91bmRpbmdCb3giOlstMC43Njk2LDUuOTI0LDIuODEwMSwtMC41NTU1XSwib3JpZ2luTGFiZWwiOiJcXChPXFwpIiwiZ3JpZFNldHRpbmdzIjp7Im1ham9yQ29sb3IiOiIjOTRhM2I4IiwibWFqb3JTdGVwWCI6MSwibWFqb3JTdGVwWSI6MSwibWlub3JDb2xvciI6IiNjYmQ1ZTEiLCJtYWpvck9wYWNpdHkiOjAuNDUsIm1pbm9yT3BhY2l0eSI6MC40NSwibWlub3JFbGVtZW50c1giOjQsIm1pbm9yRWxlbWVudHNZIjo0LCJtYWpvclN0cm9rZVdpZHRoIjowLjgsIm1pbm9yU3Ryb2tlV2lkdGgiOjAuNSwic2hvd01ham9yR3JpZExpbmVzIjp0cnVlLCJzaG93TWlub3JHcmlkTGluZXMiOnRydWV9LCJzaG93WmVyb1RpY2siOmZhbHNlLCJzaG93QXhpc0xhYmVscyI6dHJ1ZSwic2hvd05hdmlnYXRpb24iOmZhbHNlLCJrZWVwQXNwZWN0UmF0aW8iOmZhbHNlLCJ4VGlja0xhYmVsRXZlcnkiOjEwMDAsInlUaWNrTGFiZWxFdmVyeSI6MTAwMCwieEF4aXNMYWJlbE9mZnNldCI6Wy0zLjMsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0xNCw1XSwib3JpZ2luTGFiZWxPZmZzZXQiOlstMTUsMC42XX0=" data-width="278" data-height="246" data-proportional style="width:278px;height:246px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:278px;--graph-aspect:278 / 246;"></div></div></div></div>

Volume of Revolution

<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><p>Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about</p><ol class="abc-list"><li data-list-item-id="item-1790400458572-bja0be2ze" data-list-item-path="item-1790400458572-bja0be2ze"><div class="list-item-paragraph"><p>$y$-axis,</p></div></li><li data-list-item-id="item-1790400459696-k2kdjmpha" data-list-item-path="item-1790400459696-k2kdjmpha"><div class="list-item-paragraph"><p>$x$-axis,</p></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="244" data-height="241" data-proportional style="width:244px;height:241px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:244px;--graph-aspect:244 / 241;"></div></div></div></div>

<p>Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about</p>
<ol class="abc-list"><li data-list-item-id="item-1760758152540-uhcpq8kt7" data-list-item-path="item-1760758152540-uhcpq8kt7"><div class="list-item-paragraph">$y$-axis,</div></li><li data-list-item-id="item-1760758153042-beywfvtim" data-list-item-path="item-1760758153042-beywfvtim"><div class="list-item-paragraph">$x$-axis,</div></li></ol>
<div class="graph-block" data-graph-config="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" data-width="188" data-height="177" data-proportional style="width:188px;height:177px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:188px;--graph-aspect:188 / 177;"></div>

Find the volumes of solid formed when region $R$ is rotated through $2\pi $ about

<p>Find the total volume of solids formed when regions $P$ and $Q$ are rotated through $2\pi $ about $y$-axis.</p>
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Find the total volume of solids formed when regions $P$ and $Q$ are rotated through $2\pi $ about $y$-axis.

<p>Find the volume of solid formed when region $R$ is rotated through $2\pi $ about $y$-axis.</p>
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Find the volume of solid formed when region $R$ is rotated through $2\pi $ about $y$-axis.

<ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760753909486-xr3u9j011k.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3NTM5MDk0ODYteHIzdTlqMDExay5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzODUzODEzLCJleHAiOjE3OTM4NTQxMTN9.7HjDnKFTJ3aW_jwZ90cJnOm1wnvGG-cT5JiJoF3vgQE" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><div class="list-item-paragraph">The diagram shows part of the graph of $y={{x}^{2}}+2$ from $x=1$ to $x=2$. The area under the curve in this interval may be approximated by the total area of $n$ rectangles, $A$, as shown. The width of each rectangle is $\frac{1}{n}$.</div><div class="list-item-paragraph">Given that&nbsp;$\underset{r=1}{\overset{n}{\mathop \sum }}\,{{r}^{2}}=\frac{n}{6}\left( n+1 \right)\left( 2n+1 \right)$, show that&nbsp;$A=\frac{13}{3}+\frac{3}{2n}+\frac{1}{6{{n}^{2}}}$.</div><div class="list-item-paragraph">Deduce the exact value of $\mathop{\displaystyle\displaystyle \displaystyle\int }_{1}^{2}\left( 2+{{x}^{2}} \right)\,\mathrm{d}x$, justifying your answer. <span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Region $R$ is bounded by the curve $y=\ln x$, $x=1$, and $y=\ln 5$.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the area of $R$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the exact volume of the solid generated when&nbsp;$R$ is rotated through $4$ right angles about the $y$-axis.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

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