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PSLE

PSLE Foundation Mathematics

A rectangle has length 4 cm more than its width and area 96 cm2. Find its dimensions.

A rectangle has length $4$ cm more than its width and area $96\text{ cm}^2$. Find its dimensions.

Applications of Quadratic Functions and Equations

Rectangle model

Quadratic Functions, Equations and Inequalities

<p>The profit, $\$y$ of a company can be modelled by the equation $y = a(x - h)^2 + k$, where&nbsp;$x$ is the number of goods sold and $a$, $h$ and $k$ are constants.<br>The company obtained the maximum profit of $\$24\,500$ when&nbsp;$800$ goods were sold.<br>The company incurred a loss of $\$7\,500$ when no goods were sold.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">State the value of $h$ and of $k$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph"><p>Using the values of $h$ and $k$ found in part <strong>(a)</strong>, find the value of $a$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Find the range of the number of goods the company needed to sell to earn a profit.<span class="inline-marks">[2]</span></div></li></ol>

The profit, $\$y$ of a company can be modelled by the equation $y = a(x - h)^2 + k$, where $x$ is the number of goods sold and $a$, $h$ and $k$ are constants.
The company obtained the maximum profit of $\$24\,500$ when $800$ goods were sold.
The company incurred a loss of $\$7\,500$ when no goods were sold.

Quadratic Inequalities

Maximum and Minimum Values of General Quadratic Functions

2024 CGSS P1 Q2

<p>The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where&nbsp;$x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.</p>
<p>The graph of the function models the curved surface of the half-pipe.</p>
<ol class="i-list"><li data-list-item-id="item-1779863970301-8i7w7kccu" data-list-item-path="item-1779863970301-8i7w7kccu"><div class="list-item-paragraph"><p>Show that the equation of the curve can be written in the form $y = \frac{1}{2}(x - 4)^2 + 1.$</p><p>Hence, state the coordinates of the turning point.</p></div></li><li data-list-item-id="item-1779864013798-i6efiat6a" data-list-item-path="item-1779864013798-i6efiat6a"><div class="list-item-paragraph">Explain what the turning point represents in the context of the half-pipe.</div></li><li data-list-item-id="item-1779864053174-je0mqtrit" data-list-item-path="item-1779864053174-je0mqtrit"><div class="list-item-paragraph">Sketch the graph of the quadratic function clearly showing:</div><ul class="bullet-list"><li data-list-item-id="subitem-1779864068279-6voek6d7l" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864068279-6voek6d7l"><div class="list-item-paragraph">the turning point,</div></li><li data-list-item-id="subitem-1779864074574-hmuf5w09s" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864074574-hmuf5w09s"><div class="list-item-paragraph"><p>the intercept with the $y$-axis,</p></div></li><li data-list-item-id="subitem-1779864074994-t5ucj8kvb" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864074994-t5ucj8kvb"><div class="list-item-paragraph">the axis of symmetry.</div></li></ul></li><li data-list-item-id="item-1779864093967-t3qlykxyu" data-list-item-path="item-1779864093967-t3qlykxyu"><div class="list-item-paragraph">If the right end of the half pipe has a height of $13.5$ m from the ground, find the width of the half pipe from one end to the other.</div></li></ol>

The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where $x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.

The graph of the function models the curved surface of the half-pipe.

Completing the Square and Vertex Form

Sketching Quadratic Graphs

Solving Quadratic Equations

OPSS 2EM MYE 2018 Q13

<p>Aqil's father is $x^{2}$ years old while Aqil is $x$ years old now. In $5x$ years' time, Aqil's father is twice as old as him.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Form an equation in $x$ and show that it reduces to $x^{2}-7x=0$ .</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve the equation $x^{2}-7x=0$ .</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the age of Aqil's father now.</span><span class="inline-marks">[1]</span></div></li></ol>

Aqil's father is $x^{2}$ years old while Aqil is $x$ years old now. In $5x$ years' time, Aqil's father is twice as old as him.

Form an equation in $x$ and show that it reduces to $x^{2}-7x=0$ .

Hence, find the age of Aqil's father now.

XMS 2EM WA2 2021 Q5

<p>An object was launched from a machine. Its height, $y$ metres above ground, $t$ seconds after launch, can be modelled by a quadratic function $y = f(t)$. Four seconds after launch, the object reaches its maximum height of $81$ m and then dropped to a height of $54$ m after a further $3$ seconds. </p>
<ol class="abc-list"><li data-list-item-id="item-1773454586595-nxo7pguih"><div class="list-item-paragraph">Find the initial height of the object when it was launched.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773454587503-2ajg6lta7"><div class="list-item-paragraph">Without solving a quadratic equation, explain and determine how long did the object stay above $54$ m ?<span class="inline-marks">[2]</span></div></li></ol>

An object was launched from a machine. Its height, $y$ metres above ground, $t$ seconds after launch, can be modelled by a quadratic function $y=f(t)$. Four seconds after launch, the object reached its maximum height of 81 m and then dropped to a height of 54 m after a further three seconds.

Find the initial height of the object when it was launched.

Without solving a quadratic equation, explain and determine how long did the object stay above 54 m?

2025 CGSS PRELIMS P2 Q1 

<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-paragraph">A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given by the formula $h = 7 + 20t - 5t^2$.<br>Find the maximum height attained by the ball and the time at which this occurs.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-content"><span class="list-item-text">Find the set of values of the constant $k$ for which the curve $y = x^2 + 6x + k$ lies entirely above the line $y = k(x - 1)$.</span><span class="inline-marks">[4]</span></div></li></ol>

A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given by the formula $h=7+20t-5t^2$. Find the maximum height attained by the ball and the time at which this occurs.

Find the set of values of the constant $k$ for which the curve $y=x^2+6x+k$ lies entirely above the line $y=k(x-1)$.

Quadratic Expressions Positive or Negative for All Real Values

2025 DSS PRELIMS P2 Q1

<p>A sport club purchased an equipment of \$$1500$ to be shared by a group of $x$ members. It was agreed that each member of the group would pay an equal share of the equipment fee.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the amount of money each member of the group had to pay.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>Three members of the group then quit the club. The club decided that it would contribute \$$140$ from its funds and that the balance of the fee was to be shared equally by the remaining members.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Write down an expression, in terms of $x$, for the amount of money each remaining member had to pay after the three members quit.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">As a result of the three members quitting, the amount each member had to pay was \$$5$ more than the initial amount.</div><div class="list-item-paragraph">Form an equation in $x$ and show that it reduces to $x^2 + 25x - 900 = 0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Solve the equation $x^2 + 25x - 900 = 0$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the amount of money that each member had to pay initially.<span class="inline-marks">[2]</span></div></li></ol>

A sport club purchased an equipment of \$$1500$ to be shared by a group of $x$ members. It was agreed that each member of the group would pay an equal share of the equipment fee.

Write down an expression, in terms of $x$, for the amount of money each member of the group had to pay.

Three members of the group then quit the club. The club decided that it would contribute \$$140$ from its funds and that the balance of the fee was to be shared equally by the remaining members.

Write down an expression, in terms of $x$, for the amount of money each remaining member had to pay after the three members quit.

IP2 Class Test (May 2025) Q11

<p>A golfer hits a ball such that the height, $h$ metres, of the ball above the ground $t$ seconds later is given by $h = 0.2t(40 - t)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">By expressing the function in the form $h = a(t - m)^2 + n$, where $a$, $m$ and $n$ are constants, explain whether the ball can reach a height of $100$ metres.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $t$ for which the ball is at most $75$ metres above the ground.</span><span class="inline-marks">[4]</span></div></li></ol>

A golfer hits a ball such that the height, $h$ metres, of the ball above the ground $t$ seconds later is given by $h = 0.2t(40 - t)$.

By expressing the function in the form $h = a(t - m)^2 + n$, where $a$, $m$ and $n$ are constants, explain whether the ball can reach a height of $100$ metres.

Find the range of values of $t$ for which the ball is at most $75$ metres above the ground.

2023 Ahmad Ibrahim Sec AM S3 EOY Q4

<p>An arched underpass has the shape of a parabola as shown. In the diagram, the curved underpass can be modelled by the quadratic function $y = a(x - h)^2 + k$, where $a$, $h$ and $k$ are constants. In this model, $x$ m is the horizontal distance from $O$ and $y$ m is the height of the arch.</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/1768287539506-lozwia7s2nm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyODc1Mzk1MDYtbG96d2lhN3Mybm0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2NzkwOCwiZXhwIjoxNzkyODY4MjA4fQ.RSU8IOvIv-CUEmLBD7rTQYrRbVpmXdOi0Zq0d_f-4ik" style="max-width: min(220px, 100%); height: auto; display: block;" alt="Diagram of an arched underpass with a vehicle" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">A one-way road passing through the arch is $4.2$ m wide and the maximum height of the arch is $4.2$ m. State the coordinates of the maximum point of the quadratic function.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the exact value of $a$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Determine whether it is possible for a vehicle that is $2.8$ m wide and $2.4$ m tall to navigate through the underpass. Explain your reasoning by showing your workings clearly.</span><span class="inline-marks">[3]</span></div></li></ol>

An arched underpass has the shape of a parabola as shown. In the diagram, the curved underpass can be modelled by the quadratic function $y = a(x - h)^2 + k$, where $a$, $h$ and $k$ are constants. In this model, $x$ m is the horizontal distance from $O$ and $y$ m is the height of the arch.

A one-way road passing through the arch is $4.2$ m wide and the maximum height of the arch is $4.2$ m. State the coordinates of the maximum point of the quadratic function.

Determine whether it is possible for a vehicle that is $2.8$ m wide and $2.4$ m tall to navigate through the underpass. Explain your reasoning by showing your workings clearly.

2023 ACSBR AM S3 EOY Q4 

<p>A tank is firing rounds during a target practice.  The height, $h$ metres, of a projectile above the ground during flight is governed by the equation $h=5t-\frac{1}{5}{{t}^{2}}$ where $t$ is the time in seconds for the projectile in flight.</p>
<ol class="i-list"><li><div class="list-item-paragraph">What is the maximum height achieved by a projectile?</div></li><li><div class="list-item-paragraph">Sketch the curve $h=5t-\frac{1}{5}{{t}^{2}}$ for $h\ge 0$. </div></li><li><div class="list-item-paragraph">Find the range of values of $t$ for which a round is at least $10$ metres above the ground.</div></li></ol>

A tank is firing rounds during a target practice.  The height, $h$ metres, of a projectile above the ground during flight is governed by the equation $h=5t-\frac{1}{5}{{t}^{2}}$ where $t$ is the time in seconds for the projectile in flight.

<p>A piece of wire, $100$ cm in length is divided into two parts.  One part is bent to form a square of side $x$ cm and the other part a square of side $y$ cm.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $y$ in terms of $x$.</div></li><li><div class="list-item-paragraph">Show that the total area, $A$ cm$^{2}$, of the two squares is given by $A={{x}^{2}}+{{(25-x)}^{2}}$.</div></li><li><div class="list-item-paragraph">Calculate the value of $x$ for which $A$ has a stationary value. Determine whether this value of $x$ makes $A$ a maximum or a minimum.  Hence, find the value of $A$.</div></li></ol>

A piece of wire, $100$ cm in length is divided into two parts.  One part is bent to form a square of side $x$ cm and the other part a square of side $y$ cm.

Application to Real Life Problems

<p>The speed a boat in still water is $55$ km/h and that of the current is $x$ km/h.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find an expression for the speed of the boat</div><ol class="i-list"><li><div class="list-item-paragraph">against the current,</div></li><li><div class="list-item-paragraph">with the current.</div></li></ol></li><li><div class="list-item-paragraph">Find an expression for the time required to travel $120$ km</div><ol class="i-list"><li><div class="list-item-paragraph">against the current,</div></li><li><div class="list-item-paragraph">with the current.</div></li></ol></li><li><div class="list-item-paragraph">If the boat takes $40$ minutes longer to travel against the current than it takes to travel with the current, write down an equation in $x$ and show that it can be expressed as&nbsp;${{x}^{2}}+360x-3025=0$.</div></li><li><div class="list-item-paragraph">Solve the equation ${{x}^{2}}+360x-3025=0$, giving both answers correct to two decimal places.<br></div></li><li><div class="list-item-paragraph">Find the time taken to travel $80$ km with the current.</div></li></ol>

The speed a boat in still water is $55$ km/h and that of the current is $x$ km/h.

Boat Journey Problem Leading to a Quadratic Equation

<p>Jenny is an online seller, bought $x$ dresses for a total of $\$720$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down an expression for the cost price of each dress in terms of $x$.</div></li><li><div class="list-item-paragraph">She sold $16$ of them for&nbsp;$\$480$ and the rest at a loss of $\$2$ per dress due to minor defects.</div><ol class="i-list"><li><div class="list-item-paragraph">Write down, in terms of $x$, the number of dresses that were sold at a loss.</div></li><li><div class="list-item-paragraph">Write down in terms of $x$, the selling price of each dress that was sold at a loss.</div></li><li><div class="list-item-paragraph">Show that the total, in dollars, she received from the sale of all dresses can be written as&nbsp;$\$\left(1232-\frac{11520}{x}-2x\right)$.</div></li></ol></li><li><div class="list-item-paragraph">Given that Jenny made a profit of $\$120$ altogether, form an equation in $x$ and show that it reduces to ${{x}^{2}}-196x+5760=0$.</div></li><li><div class="list-item-paragraph">Solve the equation ${{x}^{2}}-196x+5760=0$.</div></li><li><div class="list-item-paragraph">Given that the cost price of each dress is more than $\$10$, find the cost of each dress.</div></li></ol>

Jenny is an online seller, bought $x$ dresses for a total of $\$720$.

Selling Dresses: Forming and Solving a Quadratic Equation

<p>Town $P$ is $200$ km due north of Town $Q$.  </p>
<p>A van travels from Town $P$ at a constant speed of $80$ km/h due west.  </p>
<p>At the same time, a lorry travels at a constant speed of $60$ km/h due south from Town $Q$.</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/1765509728944-0c2qte79vir5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU1MDk3Mjg5NDQtMGMycXRlNzl2aXI1LnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI4Njc5MDgsImV4cCI6MTc5Mjg2ODIwOH0.mWTcekfaQ8PkqnzmEdsQ1JfmN2fCuuY1vurUtDYXOLw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">If the van and lorry travelled for&nbsp;$t$ hours, write down in terms of $t$, the distance of</div><ol class="i-list"><li><div class="list-item-paragraph">the van from Town $P$ after $t$ hours,</div></li><li><div class="list-item-paragraph">the lorry from Town $P$ after $t$ hours.</div></li></ol></li><li><div class="list-item-paragraph">Given that after $t$ hours, the vehicles are $300$ kilometres apart, show that&nbsp;$5{{t}^{2}}+12t-25=0$.<br></div></li><li><div class="list-item-paragraph">Solve the equation $5{{t}^{2}}+12t-25=0$, giving your answers correct to $2$ decimal places.</div></li></ol>

Town $P$ is $200$ km due north of Town $Q$.  

A van travels from Town $P$ at a constant speed of $80$ km/h due west.  

At the same time, a lorry travels at a constant speed of $60$ km/h due south from Town $Q$.

Vehicles Travelling from Two Towns: A Quadratic Equation Problem

<p>A trailer travels its entire $80$ km journey at an average speed of $x$ km/h.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down, in terms of $x$, for the time taken, in hours, for the trailer to travel its journey.</div></li></ol>
<p>On a certain day, due to heavy traffic, the trailer was delayed and took $30$ minutes longer than scheduled to travel the first $20$ km.  </p>
<p>The remainder of the journey was completed at an average speed of $\left( x+20 \right)$km/h.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the time taken, in hours, to travel</div><ol class="i-list"><li><div class="list-item-paragraph">the first $20$ km,</div></li><li><div class="list-item-paragraph">the remaining journey.</div></li></ol></li><li><div class="list-item-paragraph">On the day of the delay, the trailer managed to arrive at its destination on time.</div><ol class="i-list"><li><div class="list-item-paragraph">Write down an equation in $x$, and show that it reduces to ${{x}^{2}}+20x-2400=0$.</div></li><li><div class="list-item-paragraph">Solve ${{x}^{2}}+20x-2400=0$.</div></li></ol></li><li><div class="list-item-paragraph">Find the time taken for the normal scheduled journey.</div></li></ol>

A trailer travels its entire $80$ km journey at an average speed of $x$ km/h.

On a certain day, due to heavy traffic, the trailer was delayed and took $30$ minutes longer than scheduled to travel the first $20$ km.  

The remainder of the journey was completed at an average speed of $\left( x+20 \right)$km/h.

Travel Time Problem Leading to a Quadratic Equation

<p>The price of petrol in July was $x$ dollars per litre and Mr Lee spent $\$72$ on petrol.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the amount of petrol he filled his car with in July.</div></li></ol>
<p>In August, the price of petrol increased by $15$ cents per litre.  </p>
<p>Mr Lee spent the same amount of money on petrol.  </p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the amount of petrol he filled his car with in August.</div></li></ol>
<p>Mr Lee noticed that he was getting $2$ litres of petrol less in August than in July.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Form an equation in $x$ and show that it reduced to $20{{x}^{2}}+3x-108=0$.</div></li><li><div class="list-item-paragraph">Solve the equation $20{{x}^{2}}+3x-108=0$.  </div></li><li><div class="list-item-paragraph">Hence, find the price of petrol in August.</div></li></ol>

The price of petrol in July was $x$ dollars per litre and Mr Lee spent $\$72$ on petrol.

In August, the price of petrol increased by $15$ cents per litre.  

Mr Lee spent the same amount of money on petrol.  

Mr Lee noticed that he was getting $2$ litres of petrol less in August than in July.

Petrol Price Changes and Quadratic Equations

<p>The diagram represents a piece of rectangular paper $ABCD$ which has been folded along the line $EC$ so that&nbsp;$D$ has moved to $F$.  </p>
<p>The length of $AB$ is $\left( 4x+2 \right)$cm, $BC$ is $\left( 3x-1 \right)$ cm and $ED$ is $2$ cm.</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/1765508069246-9gltsy6i68j.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU1MDgwNjkyNDYtOWdsdHN5Nmk2OGoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2NzkwOCwiZXhwIjoxNzkyODY4MjA4fQ._s_X_mJ9Czg1yyHR3GlFrhviuMg6tac1oPgytB9q59w" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down, in terms of $x$, and expression for the area of rectangle $ABCD$.</div></li><li><div class="list-item-paragraph">Write down, in terms of $x$, an expression for the area of the triangle $CDE$.</div></li><li><div class="list-item-paragraph">Given that the area of the shaded region $ABCFE$ is $570$ cm$^{2}$, form an equation in $x$ and show that it simplifies to $2{{x}^{2}}-x-96=0$.</div></li><li><div class="list-item-paragraph">Solve the equation $2{{x}^{2}}-x-96=0$, giving your answers correct to three decimal places.</div></li><li><div class="list-item-paragraph">Hence, find the area of the folded paper $CEF$, giving your answer correct to $2$ decimal places.</div></li></ol>

The diagram represents a piece of rectangular paper $ABCD$ which has been folded along the line $EC$ so that $D$ has moved to $F$.  

The length of $AB$ is $\left( 4x+2 \right)$cm, $BC$ is $\left( 3x-1 \right)$ cm and $ED$ is $2$ cm.

Folding a Rectangle: Area and Quadratic Equation

<p>A small pipe fills a water tank in $x$ minutes, whereas a larger pipe fills the same tank in $(x-8)\mathrm{ }$ minutes.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down an expression, in terms of $x$, for the fraction of the tank filled with water in one minute using</div><ol class="i-list"><li><div class="list-item-paragraph">the small pipe,</div></li><li><div class="list-item-paragraph">the large pipe.</div></li></ol></li></ol>
<p>If both pipes are used together, then the tank can be filled in $7.5$ minutes.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Form an equation involving $x$ and show that it reduces to ${{x}^{2}}-23x+60=0$.</div></li><li><div class="list-item-paragraph">Solve the equation ${{x}^{2}}-23x+60=0$, giving your answers correct to $2$ decimal places.</div></li><li><div class="list-item-paragraph">Hence, find the time taken to fill the tank using only the large pipe.</div></li></ol>

A small pipe fills a water tank in $x$ minutes, whereas a larger pipe fills the same tank in $(x-8)\mathrm{ }$ minutes.

If both pipes are used together, then the tank can be filled in $7.5$ minutes.

Pipes and Time: Forming and Solving a Quadratic Equation

<p>In the diagram given below, the circles having centres at $P,Q$and $R$ touch each other externally.</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/1770889732276-5l1rvwq9cj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4ODk3MzIyNzYtNWwxcnZ3cTljai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODY3OTA4LCJleHAiOjE3OTI4NjgyMDh9.UlH0P4t3L4k-tBqKUGq6q73cv1PGtCXQshWlIaUpTlk" style="max-width: min(270px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that the radius of the three circles are $3$ cm, $x$ cm and $\left( 3x+2 \right)$ cm respectively, and $\angle PQR=90{}^\circ $, form an equation in $x$ and show that it reduces to $2{{x}^{2}}-2x-3=0$.</div></li><li><div class="list-item-paragraph">Solve the equation $2{{x}^{2}}-2x-3=0$, giving your answers correct to $2$ decimal places.</div></li><li><div class="list-item-paragraph">Hence, calculate the perimeter of the triangle $PQR$.</div></li></ol>

In the diagram given below, the circles having centres at $P,Q$and $R$ touch each other externally.

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