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<p>The line $y=mx+\frac{3}{m}$ is a tangent to the parabola ${{y}^{2}}=kx$ for all values of $m$ where $m\ne 0$.</p>
<p>Find the constant $k$ and the coordinates of the point of contact between the two graphs, giving your answer in terms of $m$.</p><p style="text-align: right;">[4]</p>

The line $y=mx+\frac{3}{m}$ is a tangent to the parabola ${{y}^{2}}=kx$ for all values of $m$ where $m\ne 0$.

Find the constant $k$ and the coordinates of the point of contact between the two graphs, giving your answer in terms of $m$.

2014 VJC P1 Q1

Applications of Differentiation

<p>A curve $C$ has equation $3{{x}^{2}}-2xy+5{{y}^{2}}=14$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{3x-y}{x-5y}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the exact $x$-coordinates of the points on the curve $C$ at which the tangent is parallel to the $y$-axis.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">A point $P\left( x,y \right)$ moves along the curve $C$ in such a way that $y$ decreases at a constant rate of $7$ units per second. Given that $x$ increases at the instant when $y=1$, find the corresponding rate of change in $x$.</span></div><p style="text-align: right;">[3]</p></li></ol>

A curve $C$ has equation $3{{x}^{2}}-2xy+5{{y}^{2}}=14$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{3x-y}{x-5y}$.

Find the exact $x$-coordinates of the points on the curve $C$ at which the tangent is parallel to the $y$-axis.

A point $P\left( x,y \right)$ moves along the curve $C$ in such a way that $y$ decreases at a constant rate of $7$ units per second. Given that $x$ increases at the instant when $y=1$, find the corresponding rate of change in $x$.

2019 DHS P2 Q2

<p>A group of student councillors bought an 8 m long piece of tarpaulin canvas to build a tentage for a school event. The canvas would extend diagonally at an angle of $\theta $ from the ground to a height of 3 m, where it will then stretch horizontally to the school building (see diagram for the cross-sectional view).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770108489940-2nbbnq8xste.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzAxMDg0ODk5NDAtMm5iYm5xOHhzdGUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMzQ1NCwiZXhwIjoxNzg1NzEzNzU0fQ.mTaIYemCLFNjikSvFAIWKOs8xvGtBs3E8SkZl6atj9U" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">Given that the total cross-sectional area covered by the canvas is $A\mathrm{ }{{\mathrm{m}}^{2}}$, show that&nbsp;$A=24-9\operatorname{cosec}\theta +\frac{9}{2}\cot \theta $.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find, by differentiation, the largest possible value of $A$.<span class="inline-marks">[5]</span></div></li></ol>

A group of student councillors bought an 8 m long piece of tarpaulin canvas to build a tentage for a school event. The canvas would extend diagonally at an angle of $\theta $ from the ground to a height of 3 m, where it will then stretch horizontally to the school building (see diagram for the cross-sectional view).

Given that the total cross-sectional area covered by the canvas is $A\mathrm{ }{{\mathrm{m}}^{2}}$, show that $A=24-9\operatorname{cosec}\theta +\frac{9}{2}\cot \theta $.

Find, by differentiation, the largest possible value of $A$.

2015 HCI P1 Q4

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1758960497417-emz2mzg2gwh.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTg5NjA0OTc0MTctZW16Mm16ZzJnd2gucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMzQ1NCwiZXhwIjoxNzg1NzEzNzU0fQ.mEc-rThWQgHeMZo6BDBqF9kXo-4S-O4v6iGXf4mlB8g" width="400" class="aligncenter" /></figure>
<p>The diagram shows a mechanism for converting rotational motion into linear motion. The point $P$ is on the circumference of a disc of fixed radius r which can rotate about a fixed point $O$. The point $Q$ can only move on the line $OX$, and $P$ and $Q$ are connected by a rod of length $2r$. As the disc rotates, the point $Q$ is made to slide along $OX$. At time t, angle $POQ$ is $\theta$ and the distance $OQ$ is $x$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State the maximum and minimum values of $x$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that $x=r\left( \cos \theta +\sqrt{4-{{\sin }^{2}}\theta } \right)$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">At a particular instant, $\theta =\frac{\pi }{6}$ and $\frac{\mathrm{d}\theta }{\mathrm{d}t}=0.3$. Find the numerical rate at which point $Q$ is moving towards point $O$ at that instant, leaving your answer in terms of $r$.</span></div><p style="text-align: right;">[3]</p></li></ol>

The diagram shows a mechanism for converting rotational motion into linear motion. The point $P$ is on the circumference of a disc of fixed radius r which can rotate about a fixed point $O$. The point $Q$ can only move on the line $OX$, and $P$ and $Q$ are connected by a rod of length $2r$. As the disc rotates, the point $Q$ is made to slide along $OX$. At time t, angle $POQ$ is $\theta$ and the distance $OQ$ is $x$.

State the maximum and minimum values of $x$.

Show that $x=r\left( \cos \theta +\sqrt{4-{{\sin }^{2}}\theta } \right)$.

At a particular instant, $\theta =\frac{\pi }{6}$ and $\frac{\mathrm{d}\theta }{\mathrm{d}t}=0.3$. Find the numerical rate at which point $Q$ is moving towards point $O$ at that instant, leaving your answer in terms of $r$.

2019 EJC P2 Q2

<p>A curve $C$ has equation ${{\left( \ln y \right)}^{2}}-3xy+{{x}^{2}}-4=0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\left( 3y-2x \right)y}{2\ln y-3xy}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">It is given that $C$ contains the point $A$ with coordinates $\left( \alpha ,1 \right)$ where $\alpha <0$.</span></div><div class="list-item-paragraph">Find the value of $\alpha $ and hence the equation of the normal to $C$ at the point $A$.</div><p style="text-align: right;">[3]</p></li></ol>

A curve $C$ has equation ${{\left( \ln y \right)}^{2}}-3xy+{{x}^{2}}-4=0$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\left( 3y-2x \right)y}{2\ln y-3xy}$.

It is given that $C$ contains the point $A$ with coordinates $\left( \alpha ,1 \right)$ where $\alpha <0$.

2019 ASRJC Promo Q3

<p>A circular cylinder is inscribed in a right circular cone of base radius $2c$ and vertical height $c$. One of the circular ends of the cylinder lies on the base of the cone and the other end is in contact with the inner surface of the cone. If the cylinder has base radius $x$, find the expressions for the volume $V$ and the total surface area $S$ of the cylinder in terms of $c$ and $x$. If $c$is fixed and $x$ varies, find the maximum value of $V$.</p><p style="text-align: right;">[8]</p>
<p>Prove that $S$ cannot exceed $8\pi {{c}^{2}}$.</p><p style="text-align: right;">[2]</p>

A circular cylinder is inscribed in a right circular cone of base radius $2c$ and vertical height $c$. One of the circular ends of the cylinder lies on the base of the cone and the other end is in contact with the inner surface of the cone. If the cylinder has base radius $x$, find the expressions for the volume $V$ and the total surface area $S$ of the cylinder in terms of $c$ and $x$. If $c$is fixed and $x$ varies, find the maximum value of $V$.

Prove that $S$ cannot exceed $8\pi {{c}^{2}}$.

2010 NYJC P2 Q2

<p>An isosceles triangle $PQR$, where $PQ=PR$, is inscribed in a circle with centre $O$ and radius $1$ unit. It is given that the angle $QOR=4\theta $ radians, where $\theta $ is an acute angle.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759133352019-sj4dxue1ap.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkxMzMzNTIwMTktc2o0ZHh1ZTFhcC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEzNDU0LCJleHAiOjE3ODU3MTM3NTR9.29SKSO2NIfeOlXY0FkXgjO-630SfUpRT-oJOvwCJaQM" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $A$, the area of triangle $PQR$, can be expressed as $\left( \sin 2\theta \right)\left( 1+\cos 2\theta \right)$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence use differentiation to prove that $\theta $ varies, $A$ is a maximum when triangle $PQR$ is an equilateral triangle.</span></div><p style="text-align: right;">[5]</p></li></ol>

An isosceles triangle $PQR$, where $PQ=PR$, is inscribed in a circle with centre $O$ and radius $1$ unit. It is given that the angle $QOR=4\theta $ radians, where $\theta $ is an acute angle.

Show that $A$, the area of triangle $PQR$, can be expressed as $\left( \sin 2\theta \right)\left( 1+\cos 2\theta \right)$.

Hence use differentiation to prove that $\theta $ varies, $A$ is a maximum when triangle $PQR$ is an equilateral triangle.

2022 ACJC J2 MYE Q3

<p>Given that $A=\sqrt{1+\frac{1}{{{x}^{2}}}}$, and that the rate of decrease of $A$ is $k$ times the rate of increase of $x$ when $x=k$, find $k$, where $k$ is a positive real number.</p><p style="text-align: right;">[4]</p>

Given that $A=\sqrt{1+\frac{1}{{{x}^{2}}}}$, and that the rate of decrease of $A$ is $k$ times the rate of increase of $x$ when $x=k$, find $k$, where $k$ is a positive real number.

2020 TJC P1 Q1

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759142565719-2v584p7biwg.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkxNDI1NjU3MTktMnY1ODRwN2Jpd2cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMzQ1NCwiZXhwIjoxNzg1NzEzNzU0fQ.4fJ-CxZK-iuLGkO2ezmxKk5N9CXuhj3aolxCAbKw3Rc" width="400" class="aligncenter" /></figure>
<p>Water is poured at a rate of $8$cm$^{3}$ per second into a conical container with base radius $8$cm. The semi-vertical angle of the cone is $\theta $, where $\tan \theta =0.4$. At time $t$ seconds after the start, the radius of the water surface is $x$cm (see diagram). Find the rate of increase of the depth of water when the depth is $10$cm.</p>
<p>[The volume of a cone of base radius $r$ and height $h$ is given by $V=\frac{1}{3}\pi {{r}^{2}}h$].</p><p style="text-align: right;">[7]</p>

Water is poured at a rate of $8$cm$^{3}$ per second into a conical container with base radius $8$cm. The semi-vertical angle of the cone is $\theta $, where $\tan \theta =0.4$. At time $t$ seconds after the start, the radius of the water surface is $x$cm (see diagram). Find the rate of increase of the depth of water when the depth is $10$cm.

[The volume of a cone of base radius $r$ and height $h$ is given by $V=\frac{1}{3}\pi {{r}^{2}}h$].

2021 VJC P1 Q4

<p>The equation of a curve is given by</p>
<p style="text-align: center">$xy-2{{y}^{2}}+4{{x}^{2}}=66$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact coordinates of the points on the curve where the tangent is parallel to the $y$-axis.</span></div><p style="text-align: right;">[4]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that every line parallel to the $x$ -axis cuts the curve at two distinct points.</span></div><p style="text-align: right;">[3]</p></li></ol>

Find the exact coordinates of the points on the curve where the tangent is parallel to the $y$-axis.

Show that every line parallel to the $x$ -axis cuts the curve at two distinct points.

2012 IJC P1 Q4

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759219209120-0f9vostx9zid.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkyMTkyMDkxMjAtMGY5dm9zdHg5emlkLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3ODU3MTM0NTQsImV4cCI6MTc4NTcxMzc1NH0.z5fqo1olIeoSroDG6O1oBeSD8Rd5BEg4hjK3PWVm4FA" width="400" class="aligncenter" /></figure>
<p>The diagram shows a cardboard $ABC$ in the shape of an equilateral triangle of side $y$ cm. A smaller equilateral triangle $ADE$ of side $x$cm is cut from the cardboard. The perimeter of the remaining cardboard $BCED$ is $10$cm.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the area of the remaining cardboard, $A$cm$^{2}$, is given by</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{\sqrt{3}}{9}(25+5x-2{{x}^{2}})$</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text">Without the use of a calculator, find the maximum value of $A$ as $x$ varies, justifying that this value is a maximum.</span></div><p style="text-align: right;">[3]</p></li></ol>

The diagram shows a cardboard $ABC$ in the shape of an equilateral triangle of side $y$ cm. A smaller equilateral triangle $ADE$ of side $x$cm is cut from the cardboard. The perimeter of the remaining cardboard $BCED$ is $10$cm.

Show that the area of the remaining cardboard, $A$cm$^{2}$, is given by

Without the use of a calculator, find the maximum value of $A$ as $x$ varies, justifying that this value is a maximum.

2016 SAJC H1 Q4

<p>A solid is made up of a cone and a cylinder with the same radius, $r$ cm (see diagram below). </p>
<p>It expands in such a way that the height of the cone is always equal to the radius of the cone. Furthermore, the height of the cylinder, $h$ cm, varies with $r$, such that $h=k\sqrt{r}$, where $k$ is a real constant. The total volume of the solid is denoted by $V$ $cm^3$ and the area of the base of the cylinder is denoted by $A$ $cm^2$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759140041726-6xum3zj9tt7.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkxNDAwNDE3MjYtNnh1bTN6ajl0dDcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMzQ1NCwiZXhwIjoxNzg1NzEzNzU0fQ.cn6O5bkv5vkuP1As_MfNgY1huoCEn0kfJfNk7Z2wDX8" width="400" class="aligncenter" /></figure>
<p>Given that when $r=4$ cm, $A$ increases at a rate of $0.008\pi $ $cm^2$/min and $V$ increases at a rate of $0.026\pi $ $cm^3$/min, determine the value of $k$.</p><p style="text-align: right;">[6]</p>

A solid is made up of a cone and a cylinder with the same radius, $r$ cm (see diagram below). 

It expands in such a way that the height of the cone is always equal to the radius of the cone. Furthermore, the height of the cylinder, $h$ cm, varies with $r$, such that $h=k\sqrt{r}$, where $k$ is a real constant. The total volume of the solid is denoted by $V$ $cm^3$ and the area of the base of the cylinder is denoted by $A$ $cm^2$.

Given that when $r=4$ cm, $A$ increases at a rate of $0.008\pi $ $cm^2$/min and $V$ increases at a rate of $0.026\pi $ $cm^3$/min, determine the value of $k$.

2011 NJC P1 Q4

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770108398300-52mkaab7463.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzAxMDgzOTgzMDAtNTJta2FhYjc0NjMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMzQ1NCwiZXhwIjoxNzg1NzEzNzU0fQ.TrY1UjTvZwvUfK3wOiRxpVpp11CZSjrWJP0OqFToxug" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Fig. 1 shows a rectangular piece of cardboard $ABCD$ of sides $10$cm and $20$cm. A trapezium shape is cut out from each corner, to give the shape shown in Fig. 2. This shape consists of $2$ isosceles triangles and $3$ rectangles of different sizes. The remaining cardboard shown in Fig. 2 is folded along the dotted lines, to form a closed triangular prism shown in Fig. 3.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the volume $V$cm$^{3}$ of the closed triangular prism is given by $V=\frac{1}{5}\left( {{h}^{4}}-10{{h}^{3}}-25{{h}^{2}}+250h \right)$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Use differentiation to find the maximum value of $V$, proving that it is a maximum.<span class="inline-marks">[5]</span></div></li></ol>

Fig. 1 shows a rectangular piece of cardboard $ABCD$ of sides $10$cm and $20$cm. A trapezium shape is cut out from each corner, to give the shape shown in Fig. 2. This shape consists of $2$ isosceles triangles and $3$ rectangles of different sizes. The remaining cardboard shown in Fig. 2 is folded along the dotted lines, to form a closed triangular prism shown in Fig. 3.

Show that the volume $V$cm$^{3}$ of the closed triangular prism is given by $V=\frac{1}{5}\left( {{h}^{4}}-10{{h}^{3}}-25{{h}^{2}}+250h \right)$.

Use differentiation to find the maximum value of $V$, proving that it is a maximum.

2015 MJC P2 Q2

<p>There are two particles $A$ and $B$ with particle $A$ at $\left( -13,\ 0 \right)$ and particle $B$ at $\left( 0,\ -9 \right)$ with respect to origin $O$. $A$ moves towards $O$ at a constant speed of $u\ \mathrm{cm}\,{{\mathrm{s}}^{-1}}$ in the positive $x$ direction and $B$ moves towards $O$ at a constant speed of $\frac{4}{3}u\ \mathrm{cm}\,{{\mathrm{s}}^{-1}}$ in the positive $y$-direction (see diagram).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770108243653-q0p0vx3clgo.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzAxMDgyNDM2NTMtcTBwMHZ4M2NsZ28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMzQ1NCwiZXhwIjoxNzg1NzEzNzU0fQ.DBJXfTUCZx1M1gnD8E_w57hOpxyi9CiX4-ByElL5meA" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Show that, after a time of $t$ seconds, the distance between $A$ and $B$ is given by</p>
<p style="text-align: center">$\sqrt{\left( \frac{25}{9}{{u}^{2}}{{t}^{2}}-50ut+250 \right)}$cm.<span class="inline-marks">[2]</span></p>
<p>Find the coordinates of $A$ and $B$ at the instant when the two particles are closest.<span class="inline-marks">[4]</span></p>

There are two particles $A$ and $B$ with particle $A$ at $\left( -13,\ 0 \right)$ and particle $B$ at $\left( 0,\ -9 \right)$ with respect to origin $O$. $A$ moves towards $O$ at a constant speed of $u\ \mathrm{cm}\,{{\mathrm{s}}^{-1}}$ in the positive $x$ direction and $B$ moves towards $O$ at a constant speed of $\frac{4}{3}u\ \mathrm{cm}\,{{\mathrm{s}}^{-1}}$ in the positive $y$-direction (see diagram).

Show that, after a time of $t$ seconds, the distance between $A$ and $B$ is given by

$\sqrt{\left( \frac{25}{9}{{u}^{2}}{{t}^{2}}-50ut+250 \right)}$cm.

Find the coordinates of $A$ and $B$ at the instant when the two particles are closest.

2013 HCI P1 Q3

<p>A curve $C$ has equation $\frac{{{x}^{3}}-2{{y}^{2}}}{{{x}^{2}}+3xy}=1$.</p>
<ol class="i-list"><li data-list-item-id="item-1758951174832-hv2cq267d" data-list-item-path="item-1758951174832-hv2cq267d"><div class="list-item-paragraph">The points $P$ and $Q$ on $C$ each have&nbsp;$x$-coordinate $1$. Find the exact gradients of the tangents to $C$ at the points $P$ and $Q$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1758951175529-yol61u4u6" data-list-item-path="item-1758951175529-yol61u4u6"><div class="list-item-paragraph">Find the acute angle between the tangents to $C$ at the points $P$ and $Q$.<span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has equation $\frac{{{x}^{3}}-2{{y}^{2}}}{{{x}^{2}}+3xy}=1$.

The points $P$ and $Q$ on $C$ each have $x$-coordinate $1$. Find the exact gradients of the tangents to $C$ at the points $P$ and $Q$.

Find the acute angle between the tangents to $C$ at the points $P$ and $Q$.

2021 ACJC P1 Q2

<p>A kite $50$m above ground is being blown away from the person holding its string in a direction parallel to the ground at a rate $5$m per second. Assuming that the string is taut, at the instant when the length of the string already let out is $100$m, find, leaving your answers in exact form,</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the rate of change of the angle between the string and the ground,</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">the rate at which the string of the kite should be let out.</span></div><p style="text-align: right;">[4]</p></li></ol>

A kite $50$m above ground is being blown away from the person holding its string in a direction parallel to the ground at a rate $5$m per second. Assuming that the string is taut, at the instant when the length of the string already let out is $100$m, find, leaving your answers in exact form,

the rate of change of the angle between the string and the ground,

the rate at which the string of the kite should be let out.

2019 ACJC P1 Q4

<p>A solid cylinder has radius $r$ cm, height $h$ cm and total surface area $900$ cm$^{2}$. Find the exact value of the maximum possible volume of the cylinder. Find also the ratio $r:h$ that gives this maximum volume.</p><p style="text-align: right;">[7]</p>

A solid cylinder has radius $r$ cm, height $h$ cm and total surface area $900$ cm$^{2}$. Find the exact value of the maximum possible volume of the cylinder. Find also the ratio $r:h$ that gives this maximum volume.

N2019 P2 Q3

<p>The diagram shows the curve with equation ${{x}^{3}}{{y}^{2}}+{{x}^{2}}{{y}^{3}}=1$. The curve is symmetrical about the line $y=x$ and has a stationary point $A$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759142999358-mhwhzpe4zm9.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkxNDI5OTkzNTgtbWh3aHpwZTR6bTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMzQ1NCwiZXhwIjoxNzg1NzEzNzU0fQ.tJ7ybICeoaKSXpbQqwSasO5FEUPis-jJ6n4gAIK7rAQ" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact coordinates of $A$.</span></div><p style="text-align: right;">[5]</p></li><li><div class="list-item-content"><span class="list-item-text">The point $B$ is the point of the curve at which the tangent is parallel to the $y$-axis. Hence, or otherwise, state the exact coordinates of $B$.</span></div><p style="text-align: right;">[1]</p></li></ol>

The diagram shows the curve with equation ${{x}^{3}}{{y}^{2}}+{{x}^{2}}{{y}^{3}}=1$. The curve is symmetrical about the line $y=x$ and has a stationary point $A$.

The point $B$ is the point of the curve at which the tangent is parallel to the $y$-axis. Hence, or otherwise, state the exact coordinates of $B$.

2015 SAJC P1 Q4

<p>The curve $C$ has equation ${{a}^{2}}{{x}^{2}}-{{y}^{2}}-2\left( ax-2y+2 \right)=0$ where&nbsp;$a$ is a constant such $a\in \mathbb{R}\backslash \left\{ 0 \right\}$.</p>
<p>Show that&nbsp;$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a\left( ax-1 \right)}{y-2}$.</p>
<p>Hence, find the coordinates of the points on $C$ at which the tangent is parallel to $y$-axis.<span class="inline-marks">[6]</span></p>

The curve $C$ has equation ${{a}^{2}}{{x}^{2}}-{{y}^{2}}-2\left( ax-2y+2 \right)=0$ where $a$ is a constant such $a\in \mathbb{R}\backslash \left\{ 0 \right\}$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a\left( ax-1 \right)}{y-2}$.

Hence, find the coordinates of the points on $C$ at which the tangent is parallel to $y$-axis.

2019 EJC Promo Q3

<p>A closed cylinder with a circular base has radius $r$ cm and fixed height $h$ cm. Given that its volume $V$ is increasing at a rate twice as fast as its surface area $A$, with respect to time $t$, show that $r=\frac{2h}{h-4}$. Hence show that $h>4$.</p><p style="text-align: right;">[6]</p>

A closed cylinder with a circular base has radius $r$ cm and fixed height $h$ cm. Given that its volume $V$ is increasing at a rate twice as fast as its surface area $A$, with respect to time $t$, show that $r=\frac{2h}{h-4}$. Hence show that $h>4$.

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