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PSLE

PSLE Foundation Mathematics

<p>A curve passing through the points $(1,2)$ and $(\frac{1}{4},-10)$ has a gradient which is inversely proportional to $x^2$. Find the equation of the curve.</p>

A curve passing through the points $(1,2)$ and $(\frac{1}{4},-10)$ has a gradient which is inversely proportional to $x^2$. Find the equation of the curve.

Solving Differential Equations by Direct Integration

Modelling with Differential Equations

Differential Equations

<p>The variables $w$, $x$ and $y$ are connected by the following differential equations:</p>
<p style="text-align: center">$\frac{\mathrm{d}w}{\mathrm{d}x} = -\frac{3}{2}w - 2 \quad\quad (A)$</p>
<p style="text-align: center">$\frac{\mathrm{d}y}{\mathrm{d}x} = w \qquad\qquad\quad\,\,\, (B)$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Solve equation (A) to find $w$ in terms of $x$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $y$ in terms of $x$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Different arbitrary constants found in part <strong>(ii)</strong> yields different solution curves. Some of the solution curves are straight lines. Write down the equation of one of these lines. On a single diagram, sketch your line together with a non-linear solution curve that has your line as an asymptote, indicating clearly any axes intercepts.<span class="inline-marks">[3]</span></div></li></ol>

The variables $w$, $x$ and $y$ are connected by the following differential equations:

$\frac{\mathrm{d}w}{\mathrm{d}x} = -\frac{3}{2}w - 2 \quad\quad (A)$

$\frac{\mathrm{d}y}{\mathrm{d}x} = w \qquad\qquad\quad\,\,\, (B)$

Solve equation (A) to find $w$ in terms of $x$.

Sketching Exponential Functions and Solving Equations Graphically

Solving Differential Equations by Separation of Variables

Asymptotes and Discontinuities

Sketching Solution Curves of Differential Equations

Order, General and Particular Solutions of Differential Equations

2016 SAJC P1 Q8 [Modified]

Graphing Techniques

<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;$2xy - x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = 2x^4 + y^2$. Using the substitution $wy = x^2$, show that the differential equation can be transformed to $\frac{\mathrm{d}w}{\mathrm{d}x} = 1 + 2w^2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, solve the differential equation $2xy - x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = 2x^4 + y^2$.</span><span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;$u$ satisfies $\frac{\mathrm{d}^2u}{\mathrm{d}x^2} = e^{2x}$ and has a stationary value of $\frac{e^2}{4}$ when&nbsp;$x = 1$. Find $u$ in terms of $x$.</span><span class="inline-marks">[3]</span></div></li></ol>

It is given that $u$ satisfies $\frac{\mathrm{d}^2u}{\mathrm{d}x^2} = e^{2x}$ and has a stationary value of $\frac{e^2}{4}$ when $x = 1$. Find $u$ in terms of $x$.

Second-Order Linear Differential Equations

Solving Differential Equations by Substitution

2024 EJC P1 Q5

<p>Show that the differential equation $y\left( \frac{\mathrm{d}y}{\mathrm{d}x}+2y \right)=\frac{x}{{{\mathrm{e}}^{4x+{{x}^{2}}}}}$ can be reduced by the substitution $z={{y}^{2}}{{\mathrm{e}}^{4x}}$ to $\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{2x}{{{\mathrm{e}}^{{{x}^{2}}}}}$. Hence, find the general solution in the form ${{y}^{2}}=\mathrm{f}\left( x \right)$. </p><p style="text-align: right;">[4]</p>

Show that the differential equation $y\left( \frac{\mathrm{d}y}{\mathrm{d}x}+2y \right)=\frac{x}{{{\mathrm{e}}^{4x+{{x}^{2}}}}}$ can be reduced by the substitution $z={{y}^{2}}{{\mathrm{e}}^{4x}}$ to $\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{2x}{{{\mathrm{e}}^{{{x}^{2}}}}}$. Hence, find the general solution in the form ${{y}^{2}}=\mathrm{f}\left( x \right)$. 

2025 YIJC P1 Q1

<p>By using the substitution $y=ux$, solve the differential equation $x\frac{\mathrm{d}y}{\mathrm{d}x}=y+x$, given that $x&gt;0$ and $y=2$ when $x=1$.</p>

By using the substitution $y=ux$, solve the differential equation $x\frac{\mathrm{d}y}{\mathrm{d}x}=y+x$, given that $x>0$ and $y=2$ when $x=1$.

<p>By using the substitution $u=xe^{y}$, find $y$ in terms of $x$ given that $e^{y}\left(x\frac{\mathrm{d}y}{\mathrm{d}x}+1\right)=2x$, and that $y=0$ when $x=2$.</p>

By using the substitution $u=xe^{y}$, find $y$ in terms of $x$ given that $e^{y}\left(x\frac{\mathrm{d}y}{\mathrm{d}x}+1\right)=2x$, and that $y=0$ when $x=2$.

<ol class="i-list"><li><div class="list-item-paragraph">Solve $\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{y}{x}=1$, using the substitution $y=vx$.</div></li><li><div class="list-item-paragraph">Solve $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y-x+1}{y-x+5}$, using the substitution $z=y-x$.</div></li><li><div class="list-item-paragraph">Solve $2x^{3}\frac{\mathrm{d}y}{\mathrm{d}x}=x^{2}y+1$, using the substitution $y=u\sqrt{x}$.</div></li><li><div class="list-item-paragraph">Solve $\frac{\mathrm{d}y}{\mathrm{d}x}+\sin 2y=\sin^{2}y$, using the substitution $z=\tan y$.</div></li></ol>

<p>Find the general solution of the following differential equations:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$e^{2x^{5}-1}\frac{\mathrm{d}y}{\mathrm{d}x}=x^{4}$</div></li><li><div class="list-item-paragraph">$\sqrt{1-x^{2}}\frac{\mathrm{d}y}{\mathrm{d}x}=2x-1$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}y}{\mathrm{d}x}=x^{2}+\frac{1}{x^{2}}$</div></li></ol>

Find the general solution of the following differential equations:

<p>Solve the following differential equations:</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}x^{2}}=2$ where $x=0$, $\frac{\mathrm{d}y}{\mathrm{d}x}=2$, $y=4$</div></li><li><div class="list-item-paragraph">$\sec^{2}x\frac{{\mathrm{d}}^{2}y}{\mathrm{d}x^{2}}+2=0$</div></li><li><div class="list-item-paragraph">$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x\ln x}$</div></li></ol>

Solve the following differential equations:

<ol class="i-list"><li data-list-item-id="item-1760695858153-wqvfhqfyd" data-list-item-path="item-1760695858153-wqvfhqfyd"><div class="list-item-paragraph"><p>Find $\frac{\mathrm{d}}{\mathrm{d}x}{{\mathrm{e}}^{-{{x}^{2}}}}$. Hence find $\int x^3 \mathrm{e}^{-x^2} \mathrm{d}x$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760695858941-ekqdq7dy2" data-list-item-path="item-1760695858941-ekqdq7dy2"><div class="list-item-paragraph">By using the substitution $z={{\mathrm{e}}^{-{{x}^{2}}}}y$, find the general solution of $\frac{\mathrm{d}y}{\mathrm{d}x}-2xy={{x}^{3}}$, expressing $y$ in teams of $x$.<span class="inline-marks">[4]</span></div></li></ol>
<p style="text-align: center"></p>

Integration by Parts

Derivatives of Exponential Functions

Chain Rule

2023 ACJC P1 Q5

Techniques of Differentiation

Techniques of Integration

<ol class="abc-list"><li><div class="list-item-paragraph">Show that the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{2y}{x}={{x}^{3}}$ may be reduced by the substitution $y=u{{x}^{2}}$ to $\frac{\mathrm{d}u}{\mathrm{d}x}=x$. Hence find the general solution for $y$ in terms of $x$.</div><p style="text-align: right;">[4]</p></li></ol>

2023 TMJC P1 Q10

<p>The variables $x$ and $y$ are related by the differential equation</p>
<p style="text-align: center">$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=2{{\mathrm{e}}^{-2x}}$</p>
<p>Solve the differential equation. Hence find a possible solution curve that has an oblique asymptote, and state the equation of the oblique asymptote.</p><p style="text-align: right;">[4]</p>

The variables $x$ and $y$ are related by the differential equation

$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=2{{\mathrm{e}}^{-2x}}$

Solve the differential equation. Hence find a possible solution curve that has an oblique asymptote, and state the equation of the oblique asymptote.

2023 HCI P1 Q1

<ol class="i-list"><li><div class="list-item-paragraph">Write down $\displaystyle\int{\frac{1}{{{a}^{2}}-{{v}^{2}}}\,\mathrm{d}v}$, where $a$ is a positive constant.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">In the motion of an object through a certain medium, the medium furnishes a resisting force proportional to the square of the velocity of the moving object. Suppose that a body falls vertically through the medium, the model used to describe the velocity, $v$ms$^{-1}$ of the body at time t seconds after release from rest is given by the differential equation</div><div class="list-item-paragraph" style="text-align: center">$\frac{{{a}^{2}}}{10}\frac{\mathrm{d}v}{\mathrm{d}t}={{a}^{2}}-{{v}^{2}}$,</div><div class="list-item-paragraph">where $a$ is a positive constant.</div><ol class="abc-list"><li><div class="list-item-paragraph">Show that $v=a\left( \frac{{{e}^{\frac{20t}{a}}}-1}{{{e}^{\frac{20t}{a}}}+1} \right)$.</div><p style="text-align: right;">[8]</p></li><li><div class="list-item-paragraph">The rate of change of the displacement, $x$ metres, of the body from the point of release is the velocity of the body. Given that $a=2$, find the value of $x$ when $t=1$, giving your answer correct to $3$ decimal places.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Finding the Equation of Displacement from Velocity

Standard Integrals with Quadratic Denominators

2019 RI P1 Q9

Applications of Integration

<p>Two scientists are investigating the change of a certain population of size $n$ thousand at time $t$ years.</p>
<ol class="i-list"><li><div class="list-item-paragraph">One scientist suggests that $n$ and $t$ are related by the differential equation $\frac{{{\mathrm{d}}^{2}}n}{\mathrm{d}{{t}^{2}}}=10-6t$. Find the general solution of this differential equation. Sketch three members of the family of solution curves, given that $n=100$ when $t=0$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">The other scientist suggests that $n$ and $t$ are related by the differential equation $\frac{\mathrm{d}n}{\mathrm{d}t}=3-0.02n$. Find $n$ in terms of $t$, given again that $n=100$ when $t=0$. Explain in simple terms what will eventually happen to the population using this model.<span class="inline-marks">[7]</span></div></li></ol>

Two scientists are investigating the change of a certain population of size $n$ thousand at time $t$ years.

Equilibrium Solutions and Long-Term Behaviour

N2009 P2 Q4

<p>The variables $x$ and $y$ are connected by the differential equation</p>
<p style="text-align: center">$\left( 2-x \right)\frac{\mathrm{d}y}{\mathrm{d}x}=x-y$, $x < 2$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the substitution $y=\left( 2-x \right)u$ reduces the differential equation to $\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{x}{{{\left( 2-x \right)}^{2}}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence obtain the particular solution for which the solution curve passes through the origin.</div><p style="text-align: right;">[4]</p></li></ol>

The variables $x$ and $y$ are connected by the differential equation

$\left( 2-x \right)\frac{\mathrm{d}y}{\mathrm{d}x}=x-y$, $x < 2$.

2022 SAJC J2 BT Q1 [Modified]

<p>Sir Isaac Newton was a famous scientist renowned for his work on the laws of motion. One law states that, for an object falling vertically in a vacuum, the rate of change of velocity, $v$ ms$^{-1}$, with respect to time, $t$ seconds, is a constant, $c$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="abc-list"><li><div class="list-item-paragraph">Write down a differential equation relating $v$, $t$ and $c$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Initially the velocity of the object is $4$ ms$^{-1}$ and, after a further $2.5$ s, the velocity of the object is $29$ ms$^{-1}$. Find $v$ in terms of $t$ and state the value of $c$.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>
<p>For an object falling vertically through the atmosphere, the rate of change of velocity is less than that for an object falling in a vacuum. The new rate of change of $v$ is modelled as the difference between the value of $c$ found in part <strong>(i)(b)</strong> and an amount proportional to the velocity $v$, with a constant of proportionality $k$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Given that in this case the initial velocity is zero, find $v$ in terms of $t$ and $k$.</div><p style="text-align: right;">[5]</p></li></ol>
<p>For an object falling through the atmosphere, the ‘terminal velocity’ is the value approached by the velocity after a long time.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">A falling object has initial velocity zero and terminal velocity $40$ ms$^{-1}$. Find how long it takes the object to reach $90\%$ of its terminal velocity.</div><p style="text-align: right;">[4]</p></li></ol>

Sir Isaac Newton was a famous scientist renowned for his work on the laws of motion. One law states that, for an object falling vertically in a vacuum, the rate of change of velocity, $v$ ms$^{-1}$, with respect to time, $t$ seconds, is a constant, $c$.

For an object falling vertically through the atmosphere, the rate of change of velocity is less than that for an object falling in a vacuum. The new rate of change of $v$ is modelled as the difference between the value of $c$ found in part **(i)(b)** and an amount proportional to the velocity $v$, with a constant of proportionality $k$.

For an object falling through the atmosphere, the ‘terminal velocity’ is the value approached by the velocity after a long time.

Modelling with Multiple Rates of Change

N2017 P1 Q11

<ol class="abc-list"><li data-list-item-id="item-1760259796821-ejp888es2" data-list-item-path="item-1760259796821-ejp888es2"><ol class="i-list"><li data-list-item-id="subitem-1760259797624-da8ing43y" data-list-item-path="item-1760259796821-ejp888es2/subitem-1760259797624-da8ing43y"><div class="list-item-paragraph">Find the general solution of the differential equation</div><div class="list-item-paragraph" style="text-align: center">$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\sec x\tan x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760259798517-vqywvpha0" data-list-item-path="item-1760259796821-ejp888es2/subitem-1760259798517-vqywvpha0"><div class="list-item-paragraph">Find the particular solution of the differential equation for which $y=2$ and $\frac{\mathrm{d}y}{\mathrm{d}x}=1$ when&nbsp;$x=0$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1760259818947-xiotpsi84" data-list-item-path="item-1760259818947-xiotpsi84"><div class="list-item-paragraph">The rate of change in the population of a colony of insects of size $x$ thousands at time $t$ days after it is first discovered, satisfies the differential equation</div><div class="list-item-paragraph" style="text-align: center">$\frac{\mathrm{d}x}{\mathrm{d}t}=40-0.1x$.</div><div class="list-item-paragraph">Find $x$ in terms of $t$, given that&nbsp;$x=300$ when&nbsp;$t=0$.<span class="inline-marks">[5]</span></div><div class="list-item-paragraph">Sketch the part of the curve with is equation which is relevant in this context, and state what happens to the population of the colony of insects eventually.<span class="inline-marks">[2]</span></div></li></ol>

2011 RVHS Q12 [Modified]

<p>In a chemical reaction, two substances $A$ and $B$ are combined to form a new substance $C$. The initial masses of $A$, $B$and $C$ are $8$, $6$ and $0$ units respectively. After time $t$ seconds, the masses of $A$ and $B$ are each reduced by $x$ units, and the mass of $C$ increases by $2x$ units. The rate of change of the mass of $C$ with respect to $t$ is proportional to the product of the masses of $A$ and $B$ at any time $t$.</p>
<ol class="i-list"><li data-list-item-id="item-1760253665607-n8a4tunsy" data-list-item-path="item-1760253665607-n8a4tunsy"><div class="list-item-paragraph">Write down a differential equation relating $x$ and $t$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760253666071-wemd8og4y" data-list-item-path="item-1760253666071-wemd8og4y"><div class="list-item-paragraph">It is observed that when&nbsp;$t=5$, $x=5$. By solving the differential equation in part <strong>(i)</strong>, show that</div><div class="list-item-paragraph" style="text-align: center">$x=\frac{24{{\left( \frac{9}{4} \right)}^{-\frac{t}{5}}}-24}{3{{\left( \frac{9}{4} \right)}^{-\frac{t}{5}}}-4}$.<span class="inline-marks">[7]</span></div></li><li data-list-item-id="item-1760253666783-dnqxmnci9" data-list-item-path="item-1760253666783-dnqxmnci9"><div class="list-item-paragraph">Sketch the graph of $x$ against $t$.<span class="inline-marks">[2]</span></div></li></ol>
<p>In another experiment with the same initial masses of substances $A$ and $B$ as before, the increase in the mass of $C$ is modelled by the differential equation</p>
<p style="text-align: center">$\frac{{{\mathrm{d}}^{2}}x}{\mathrm{d}{{t}^{2}}}=-8{{\mathrm{e}}^{-2t}}-\frac{16}{9}{{\left( t+1 \right)}^{-\frac{7}{3}}}$</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1760253725748-wscdubg73" data-list-item-path="item-1760253725748-wscdubg73"><div class="list-item-paragraph">Given that&nbsp;$x$ eventually stabilises at $6$ units after a long time, find $x$ in terms of $t$.<span class="inline-marks">[4]</span></div></li></ol>

In a chemical reaction, two substances $A$ and $B$ are combined to form a new substance $C$. The initial masses of $A$, $B$and $C$ are $8$, $6$ and $0$ units respectively. After time $t$ seconds, the masses of $A$ and $B$ are each reduced by $x$ units, and the mass of $C$ increases by $2x$ units. The rate of change of the mass of $C$ with respect to $t$ is proportional to the product of the masses of $A$ and $B$ at any time $t$.

In another experiment with the same initial masses of substances $A$ and $B$ as before, the increase in the mass of $C$ is modelled by the differential equation

$\frac{{{\mathrm{d}}^{2}}x}{\mathrm{d}{{t}^{2}}}=-8{{\mathrm{e}}^{-2t}}-\frac{16}{9}{{\left( t+1 \right)}^{-\frac{7}{3}}}$

2022 NYJC P1 Q11

<p>Given that $z={{e}^{2x}}\frac{\mathrm{d}y}{\mathrm{d}x}$ , express $\frac{\mathrm{d}z}{\mathrm{d}x}$ in terms of $x$, $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}$.</p>
<p>Hence show that the differential equation $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2\frac{\mathrm{d}y}{\mathrm{d}x}={{e}^{1-4x}}$$\left( * \right)$, can be reduced to the differential equation $\frac{\mathrm{d}z}{\mathrm{d}x}={{\mathrm{e}}^{1-2x}}$.</p>
<p>Hence solve the differential equation $\left( * \right)$, expressing $y$ in terms of $x$.</p>

Given that $z={{e}^{2x}}\frac{\mathrm{d}y}{\mathrm{d}x}$ , express $\frac{\mathrm{d}z}{\mathrm{d}x}$ in terms of $x$, $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}$.

Hence show that the differential equation $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2\frac{\mathrm{d}y}{\mathrm{d}x}={{e}^{1-4x}}$$\left( * \right)$, can be reduced to the differential equation $\frac{\mathrm{d}z}{\mathrm{d}x}={{\mathrm{e}}^{1-2x}}$.

Hence solve the differential equation $\left( * \right)$, expressing $y$ in terms of $x$.

Higher Derivatives by Implicit Differentiation

Product Rule

2012 DHS P1 Q11(a) [Modified]

<p>A spherical container of radius $5$ m is formed by rotating the following circle $C$ about the $y$-axis.</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/converted/1760169456333-9vxe3gvjjbg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNjk0NTYzMzMtOXZ4ZTNndmpqYmcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTQzNSwiZXhwIjoxNzkyNzMxNzM1fQ.fz1NNaxZrlUz7s-aAzl707Dx6t1LkMMFoEz69TLXKbM" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The container has negligible thickness, and the circle $C$ passes through the origin $O$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State a cartesian equation of $C$.<span class="inline-marks">[1]</span></div></li></ol>
<p>Initially the spherical container is completely filled with water. Two engineers are calculating the time needed for the container to be completely drained from a small circular hole at the bottom. The volume of water in the container at time $t$ seconds is denoted by $V$ m$^{3}$.</p>
<p>[The volume of a sphere of radius $r$ is $\frac{4}{3}\pi {{r}^{3}}$.]</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The first engineer proposes that the rate of change of $V$ with respect to $t$ is a constant $k$.<span class="inline-marks">[1]</span></div><ol class="i-list"><li><div class="list-item-paragraph">Write down a differential equation relating $V$, $t$ and $k$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Determine, with justification, the sign of $k$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find $V$ in terms of $t$ and $k$, leaving your answer in exact form.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The second engineer argues that the rate at which water flows out from the hole will be at its greatest in the beginning, and decreases as the depth of water in the container decreases. He suggests using Torricelli’s law, which says that</div><div class="list-item-paragraph" style="text-align: center">$\frac{\mathrm{d}V}{\mathrm{d}t}=-\alpha \sqrt{20h}$,</div><div class="list-item-paragraph">where&nbsp;$\alpha $ m$^{2}$ is the area of the circular hole at the bottom of the container, and $h$m is the depth of water in the container at time $t$ seconds. The radius of the hole at the bottom of the container is found to be constant at $1$ cm.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$\left( 10{{h}^{\frac{1}{2}}}-{{h}^{\frac{3}{2}}} \right)\frac{\mathrm{d}h}{\mathrm{d}t}=-\frac{\sqrt{5}}{5000}$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence find the numerical value of $t$ when the container is completely drained.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

A spherical container of radius $5$ m is formed by rotating the following circle $C$ about the $y$-axis.

The container has negligible thickness, and the circle $C$ passes through the origin $O$.

Initially the spherical container is completely filled with water. Two engineers are calculating the time needed for the container to be completely drained from a small circular hole at the bottom. The volume of water in the container at time $t$ seconds is denoted by $V$ m$^{3}$.

[The volume of a sphere of radius $r$ is $\frac{4}{3}\pi {{r}^{3}}$.]

Math Question Bank