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PSLE

<p>$AB$, $AC$ and $AD$ are chords of a circle such that&nbsp;$AB = 4$ cm, $AC = 10$ cm and $AD = 11$ cm. Given that&nbsp;$AC$ bisects $\angle BAD$, find the area of the circle.</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770728619550-uu6wi9exw8l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3Mjg2MTk1NTAtdXU2d2k5ZXh3OGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY2NSwiZXhwIjoxNzkyNjYyOTY1fQ.k6hEnT1tjAIZmE3RwKJpO9zvvE5MmLtYWfcXl0biAc0" style="max-width: min(160px, 100%); height: auto; display: block;" alt="Chords in a circle" class="alignright custom-width" /></figure></div>

$AB$, $AC$ and $AD$ are chords of a circle such that $AB = 4$ cm, $AC = 10$ cm and $AD = 11$ cm. Given that $AC$ bisects $\angle BAD$, find the area of the circle.

Angles in the Same Segment

Equal Chords and Subtended Angles

Cosine Rule

Angles in a Semicircle and the Converse

Circumference and Area of Circles

IP2U14 HOT Q5

Properties of Circle

Trigonometry

Perimeter and Area of Plane Figures

<p>A cylinder is three times as high as it is wide.</p>
<p>The diagonal of the axial rectangular cross-section shown is $20\text{ cm}$; its width is the cylinder’s diameter.</p>
<p>Find the <strong>EXACT</strong> surface area of the top of the cylinder (as shaded in the image).</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 style="text-align: left">(Note: Your solution will have the symbol $\pi $ in it.)</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768919168947-al2ixqh65g.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg5MTkxNjg5NDctYWwyaXhxaDY1Zy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjYyNjY1LCJleHAiOjE3OTI2NjI5NjV9.ULW83GU4Uy3JTZJujxESwD4QUiGJkMC0mc0c01XQJvg" style="max-width: min(160px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A cylinder is three times as high as it is wide.

The diagonal of the axial rectangular cross-section shown is $20\text{ cm}$; its width is the cylinder’s diameter.

Find the **EXACT** surface area of the top of the cylinder (as shaded in the image).

Pythagoras’ Theorem in Three Dimensions

IP1U13 SWII (Pythagoras) Q14

Pythagoras’ Theorem

<p>A lawn mower has a blade of breadth $0.75$ m, and its two front wheels are identical of diameter $20$ cm, its two rear wheels are identical of diameter $40$ cm. A gardener uses this lawn mower to mow a square lawn of dimension of $18$ m by $18$ m. He alternates his mowing directions, as shown in the figure below, at a speed of $5$ m/min.</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/1767770230557-ep6ty5s4h1v.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzAyMzA1NTctZXA2dHk1czRoMXYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY2NSwiZXhwIjoxNzkyNjYyOTY1fQ.CZrLbw07N_MR3cdMbT8n1vT1ep7Hw7WYPFopdXSz8kI" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Assuming there is no overlapping of mowing areas, find</p>
<ol class="abc-list"><li data-list-item-id="item-1767770241397-x243aajbw" data-list-item-path="item-1767770241397-x243aajbw"><div class="list-item-paragraph">the number of times he needs to move from one side to the opposite side of the lawn to finish mowing the whole area of the lawn,</div></li><li data-list-item-id="item-1767770241933-2iwwf0hbj" data-list-item-path="item-1767770241933-2iwwf0hbj"><div class="list-item-paragraph">the number of rotations of the front and rear wheels, respectively, to finish mowing from one side to the opposite side, ignoring the length of the mower (giving your answers to the nearest integer),</div></li><li data-list-item-id="item-1767770242517-byd10cg6g" data-list-item-path="item-1767770242517-byd10cg6g"><div class="list-item-paragraph"><br>the least possible time needed for the gardener to fully mow the lawn.</div></li></ol>
<p>Clarification for (c): the intended model counts time along the straight mowing strips only, excluding the turns and movement between strips. The source gives no turning model, so it does not determine the actual minimum total journey time.</p>

A lawn mower has a blade of breadth $0.75$ m, and its two front wheels are identical of diameter $20$ cm, its two rear wheels are identical of diameter $40$ cm. A gardener uses this lawn mower to mow a square lawn of dimension of $18$ m by $18$ m. He alternates his mowing directions, as shown in the figure below, at a speed of $5$ m/min.

Assuming there is no overlapping of mowing areas, find

Clarification for (c): the intended model counts time along the straight mowing strips only, excluding the turns and movement between strips. The source gives no turning model, so it does not determine the actual minimum total journey time.

Speed, Distance and Time

Perimeter and Area of Rectangles and Squares

Perimeter and Area of Common Plane Figures

Arc Length and Area of Sector

Ratio, Rate and Speed

<p>A dart board made up of $3$ circles with $3$ different colours is shown in the diagram. Each circle has a radius of $2$ cm, $6$ cm and $8$ cm respectively. Assume that all darts thrown hit the board.</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;"><ol class="abc-list"><li data-list-item-id="item-1773731735577-c4upj30qs" data-list-item-path="item-1773731735577-c4upj30qs"><div class="list-item-paragraph">Show that the probability of a dart hitting the red area is $\frac{1}{16}$.</div></li><li data-list-item-id="item-1773731736094-cpv90njoh" data-list-item-path="item-1773731736094-cpv90njoh"><div class="list-item-paragraph">If&nbsp;$2$ darts are thrown, one at a time, find the probability of both darts hitting the orange area.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767699347486-1b2bgfgqzq3.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTkzNDc0ODYtMWIyYmdmZ3F6cTMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY2NSwiZXhwIjoxNzkyNjYyOTY1fQ.KiVkuE4ctsDhUoBrA26-y70DC7Lr9db_MDuwFkN2Geo" style="max-width: min(240px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>
<p>Assume each dart lands uniformly over the area of the board, and the two throws are independent.</p>

A dart board made up of $3$ circles with $3$ different colours is shown in the diagram. Each circle has a radius of $2$ cm, $6$ cm and $8$ cm respectively. Assume that all darts thrown hit the board.

Assume each dart lands uniformly over the area of the board, and the two throws are independent.

Independent Events

Probability of a Single Event

Geometric Probability on a Dartboard

Probability

<p>A factory produces circular iron discs out of rectangular iron boards of dimension of $32$ cm by $64$ cm. For Type A of circular iron disc, at most $8$ of which can be produced out of one rectangular iron board, with each disc being fitted exactly in the rectangular iron board as shown in the figure below.</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/1767691426530-q4wxphjjfn9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTE0MjY1MzAtcTR3eHBoampmbjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY2NSwiZXhwIjoxNzkyNjYyOTY1fQ.dtcnJX60G1FG2ISgui1CgGjh20tx2a6jovPzxVAmgig" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find, for Type A of circular iron disc,</p>
<ol class="abc-list"><li data-list-item-id="item-1767691435888-c7077f6j8" data-list-item-path="item-1767691435888-c7077f6j8"><div class="list-item-paragraph">the radius, perimeter and area, respectively,</div></li><li data-list-item-id="item-1767691436432-4mycuq65c" data-list-item-path="item-1767691436432-4mycuq65c"><div class="list-item-paragraph">the area of the shaded region,</div></li><li data-list-item-id="item-1767691437472-v1pcpoev6" data-list-item-path="item-1767691437472-v1pcpoev6"><div class="list-item-paragraph"><br>the utilisation efficiency of the rectangular iron board, denoted by the total area of circular iron discs as a percentage of the area of the rectangular iron board. Correct your answer to $1$ decimal place.</div></li></ol>
<p>Another two types of circular iron discs, Type B and Type C, have radii of $2$ cm and $4$ cm, respectively.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1767691481241-as0dy1n4x" data-list-item-path="item-1767691481241-as0dy1n4x"><div class="list-item-paragraph">Find the utilisation efficiency of the rectangular iron board for Type B and Type C, respectively. Correct your answer to $1$ decimal place.<br>Hence, state your finding(s) about the utilisation efficiency of Types A, B and C.</div></li></ol>
<p>For (d), arrange Types B and C in the same rectangular rows-and-columns pattern as Type A. The efficiency comparison concerns this packing pattern, not an unrestricted maximum-packing problem. In (b), count only the shaded border gaps; the three unshaded internal gaps are excluded.</p>

A factory produces circular iron discs out of rectangular iron boards of dimension of $32$ cm by $64$ cm. For Type A of circular iron disc, at most $8$ of which can be produced out of one rectangular iron board, with each disc being fitted exactly in the rectangular iron board as shown in the figure below.

Another two types of circular iron discs, Type B and Type C, have radii of $2$ cm and $4$ cm, respectively.

For (d), arrange Types B and C in the same rectangular rows-and-columns pattern as Type A. The efficiency comparison concerns this packing pattern, not an unrestricted maximum-packing problem. In (b), count only the shaded border gaps; the three unshaded internal gaps are excluded.

Perimeter and Area of Composite Figures

Expressing One Quantity as a Percentage of Another

Percentages

<p>A roller brush used for wall-painting has a circular cross section of diameter $8$ cm, and the breadth of the roller is $21$ cm. A painter uses this roller brush to paint a wall of breadth $4.2$ m and height $5$ m. He paints in vertical rectangular paint columns of breadth $21$ cm and in a speed of $10$ cm/s in the vertical direction. </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/1767690677788-jn9mox3202r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTA2Nzc3ODgtam45bW94MzIwMnIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY2NSwiZXhwIjoxNzkyNjYyOTY1fQ.MrZLxffL_rAIbuzUxlebNvA_h037BI18rVfFWr28lvI" style="max-width: min(340px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Assuming there are no overlapping paint columns, find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the number of paint columns needed to finish painting the whole area of the wall,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the number of rotations of the roller to finish painting one column, correcting your answer to the nearest integer,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the least possible time needed for the painter to fully paint the wall.</span></div></li></ol>

A roller brush used for wall-painting has a circular cross section of diameter $8$ cm, and the breadth of the roller is $21$ cm. A painter uses this roller brush to paint a wall of breadth $4.2$ m and height $5$ m. He paints in vertical rectangular paint columns of breadth $21$ cm and in a speed of $10$ cm/s in the vertical direction. 

Assuming there are no overlapping paint columns, find

the number of paint columns needed to finish painting the whole area of the wall,

the number of rotations of the roller to finish painting one column, correcting your answer to the nearest integer,

the least possible time needed for the painter to fully paint the wall.

<p>In this question, the average tangential speed along the circular path is being calculated by the following formula:\[\text{Average tangential speed} = \frac{\text{Total circular distance travelled}}{\text{Total time taken}}\]The Singapore Flyer is one of the world’s most famous Ferris wheels, which has a radius of $75$ m.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Given that it takes $30$ minutes for the Singapore Flyer to finish one full rotation, find the average tangential speed of the Singapore Flyer in one full rotation in m/s.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the average speed of ascending in vertical direction, for a capsule of the Singapore Flyer to travel from the lowest point to the highest point.</div></li></ol>
<p>The London Eye is another famous Ferris wheel in the world, which also finishes one full rotation in $30$ minutes.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Given that the ratio of the radius of the Singapore Flyer to the radius of the London Eye is $5 : 4$, find the average tangential speed of the London Eye in one full rotation in m/s.</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Using your answer in (iii), find the average speed of descending in vertical direction, for a capsule of the London Eye to travel from the highest point to the lowest point.</div></li></ol>
<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/1767674389740-uhh3znrfqv.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2NzQzODk3NDAtdWhoM3pucmZxdi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjYyNjY1LCJleHAiOjE3OTI2NjI5NjV9.WbJPNKa97lsrRfRofmgtD3YgoBXbM00sVDn8uVK03i8" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Use the stated dimensions as a mathematical model and assume both wheels rotate uniformly. Then a half-turn takes half the full-turn time.</p>

In this question, the average tangential speed along the circular path is being calculated by the following formula:\[\text{Average tangential speed} = \frac{\text{Total circular distance travelled}}{\text{Total time taken}}\]The Singapore Flyer is one of the world’s most famous Ferris wheels, which has a radius of $75$ m.

Given that it takes $30$ minutes for the Singapore Flyer to finish one full rotation, find the average tangential speed of the Singapore Flyer in one full rotation in m/s.

Find the average speed of ascending in vertical direction, for a capsule of the Singapore Flyer to travel from the lowest point to the highest point.

The London Eye is another famous Ferris wheel in the world, which also finishes one full rotation in $30$ minutes.

Given that the ratio of the radius of the Singapore Flyer to the radius of the London Eye is $5 : 4$, find the average tangential speed of the London Eye in one full rotation in m/s.

Using your answer in (iii), find the average speed of descending in vertical direction, for a capsule of the London Eye to travel from the highest point to the lowest point.

Use the stated dimensions as a mathematical model and assume both wheels rotate uniformly. Then a half-turn takes half the full-turn time.

Dividing Quantities in a Given Ratio

Speed

<p>The circumference of a circle is $15$ cm. Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the radius,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the area,</span></div></li></ol>
<p>of the circle, leaving your answers in $4$ significant figures.</p>

The circumference of a circle is $15$ cm. Find

of the circle, leaving your answers in $4$ significant figures.

Significant Figures

<p>The points $P$ and $Q$ lie on the circumference of a circle with centre $O$.<br>The area of the circle is $121\pi$ cm${}^2$.<br>The length of the major arc $PQ$ is $16.5\pi$ cm.<br>Calculate</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;"><ol class="i-list"><li><div class="list-item-paragraph">the radius of the circle,</div></li><li><div class="list-item-paragraph">the length of the minor arc $PQ$, giving your answer in terms of $\pi$,</div></li><li><div class="list-item-paragraph">the angle $POQ$ subtended by the minor arc $PQ$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767516450426-gwybhs41mge.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1MTY0NTA0MjYtZ3d5YmhzNDFtZ2UucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY2NSwiZXhwIjoxNzkyNjYyOTY1fQ.OnO3HboCc85F2M_-IY8FSFACzwyvNasmOC_u_lwWQMI" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The points $P$ and $Q$ lie on the circumference of a circle with centre $O$.
The area of the circle is $121\pi$ cm${}^2$.
The length of the major arc $PQ$ is $16.5\pi$ cm.
Calculate

Arc Length

Circular Measure: Arc Lengths and Sector Properties

<p>The diameter of a circular animal cell is $23.7$ micrometres.   ($1$ micro =$10^{-6}$)</p>
<ol class="i-list"><li><div class="list-item-paragraph">Express $23.7$ micrometres in metres, giving your answer in standard form.  </div></li><li><div class="list-item-paragraph">Find the circumference of the circular animal cell in metres, giving your answer in the standard form.</div></li></ol>

The diameter of a circular animal cell is $23.7$ micrometres.   ($1$ micro =$10^{-6}$)

Converting Metric Prefixes Using Powers of Ten

Calculations in Standard Form

Standard Form in Biological Measurements

Standard Form and Compound Interest

<p>The diameter of a circular organism is $7$ micrometres. ($1$ micro $={{10}^{-6}}$)</p>
<ol class="i-list"><li><div class="list-item-paragraph">Express $7$ micrometres in metres.</div></li><li><div class="list-item-paragraph">Find the area, in square metres, of the circular organism, giving your answer in standard form.</div></li></ol>

The diameter of a circular organism is $7$ micrometres. ($1$ micro $={{10}^{-6}}$)

Common Prefixes in our Daily Life

<p>The radius of a small circle is $(p-2q)$cm and the radius of a big circle is $(5p-12q)$cm.</p>
<p>The radius of the big circle is twice the radius of the small circle.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down and simplify a linear equation connecting $p$ and $q$.</div></li><li><div class="list-item-paragraph">If the sum of the circumference of the two circle is $16\pi $ cm.  Write down and simplify 	another linear equation connecting $p$ and $q$.</div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Find the values of $p$ and $q$.</div></li><li><div class="list-item-paragraph">Hence, find the radius of the small circle.</div></li></ol></li></ol>

The radius of a small circle is $(p-2q)$cm and the radius of a big circle is $(5p-12q)$cm.

The radius of the big circle is twice the radius of the small circle.

Math Question Bank