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<p>The boundary of a wildlife reserve is quadrilateral \(KLMN\).</p>
<div class="graph-block" data-graph-config="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" data-width="420" data-height="260" style="width:420px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<p>\(KL=6.1\text{ km}\), \(LM=8.4\text{ km}\), \(MN=7.5\text{ km}\), \(\angle LMN=68^\circ\) and \(\angle KNL=36^\circ\).</p>
<p>The angle \(\angle LKN\) is acute. Give non-exact numerical answers correct to \(3\) significant figures.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q06-q-a" data-list-item-path="emath-aot-q06-q-a"><div class="list-item-paragraph">Find diagonal \(LN\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q06-q-b" data-list-item-path="emath-aot-q06-q-b"><div class="list-item-paragraph">Find \(\angle LKN\).<span class="inline-marks">[2]</span></div></li></ol>

The boundary of a wildlife reserve is quadrilateral \(KLMN\).

\(KL=6.1\text{ km}\), \(LM=8.4\text{ km}\), \(MN=7.5\text{ km}\), \(\angle LMN=68^\circ\) and \(\angle KNL=36^\circ\).

The angle \(\angle LKN\) is acute. Give non-exact numerical answers correct to \(3\) significant figures.

Sine Rule

Cosine Rule

Boundary of a wildlife reserve

Trigonometry

<ol class="abc-list"><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i1" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i1"><div class="list-item-paragraph">Find all prime numbers satisfying \(-5\leq2x-3\leq7\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i2" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i2"><div class="list-item-paragraph">Simplify \(\dfrac{18c^3d^3}{5f^3}\div\dfrac{3c^5d^2}{10f}\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i3" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i3"><div class="list-item-paragraph">Simplify \(\left(\dfrac{16y^8}{81x^{12}}\right)^{-\frac34}\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="sec4-em-2025-yckss-p2-q1-b1-i4" data-list-item-path="sec4-em-2025-yckss-p2-q1-b1-i4"><div class="list-item-paragraph">Write \(\dfrac{3x}{(2x-1)^2}-\dfrac2{2x-1}\) as a single fraction in its simplest form.<span class="inline-marks">[2]</span></div></li></ol>

Find all prime numbers satisfying \(-5\leq2x-3\leq7\).

Simplify \(\dfrac{18c^3d^3}{5f^3}\div\dfrac{3c^5d^2}{10f}\).

Simplify \(\left(\dfrac{16y^8}{81x^{12}}\right)^{-\frac34}\).

Write \(\dfrac{3x}{(2x-1)^2}-\dfrac2{2x-1}\) as a single fraction in its simplest form.

Multiplication and Division of Algebraic Fractions

Solving Simultaneous Linear Inequalities

Prime Numbers and Composite Numbers

Rational Indices

Addition and Subtraction of Algebraic Fractions

2025 YCKSS PRELIMS P2 Q1

Algebraic Expressions, Identities and Formulae

Indices

Linear Equations and Inequalities

Primes, HCF and LCM

James has 600 one-centimetre cubes. He arranges all the cubes into a cuboid whose base is a square. Find the smallest possible height of the cuboid.

Highest Common Factor and Its Applications

2025 NASS PRELIMS P1 Q5

<p>A vertical lookout tower has base \(B\) and top \(T\). Observation posts \(P\) and \(Q\) lie on the same straight, level path from \(B\).</p>
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<p>\(P\) is between \(B\) and \(Q\), \(PQ=35\text{ m}\), and the angles of elevation of \(T\) from \(P\) and \(Q\) are \(31^\circ\) and \(18^\circ\), respectively.</p>
<p>Give non-exact numerical answers correct to \(3\) significant figures.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q08-q-a" data-list-item-path="emath-aot-q08-q-a"><div class="list-item-paragraph">Find \(BP\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q08-q-b" data-list-item-path="emath-aot-q08-q-b"><div class="list-item-paragraph">Find the height \(BT\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q08-q-c" data-list-item-path="emath-aot-q08-q-c"><div class="list-item-paragraph">Find \(QT\).<span class="inline-marks">[2]</span></div></li></ol>

A vertical lookout tower has base \(B\) and top \(T\). Observation posts \(P\) and \(Q\) lie on the same straight, level path from \(B\).

\(P\) is between \(B\) and \(Q\), \(PQ=35\text{ m}\), and the angles of elevation of \(T\) from \(P\) and \(Q\) are \(31^\circ\) and \(18^\circ\), respectively.

Give non-exact numerical answers correct to \(3\) significant figures.

Angles of Elevation and Depression

Hill lookout and observation posts

<p>Simplify \(5p-4-(3+2p)+p-6\).<span class="inline-marks">[2]</span></p>

Addition and Subtraction of Linear Algebraic Expressions

2025 YCKSS PRELIMS P1 Q1

A polygon has n sides. When the number of sides is doubled, its interior angle increases by 20. Find n.

A polygon has \(n\) sides. When the number of sides is doubled, its interior angle increases by \(20^\circ\). Find \(n\).

Polygons

2025 SJI PRELIMS P1 Q5

<p>Triangle \(ABC\) has&nbsp;\(AB=26\) cm, \(BC=10\) cm, and \(AC=24\) cm. A circle passes through \(A,B,C\).</p>
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<p>Explain, with reasons, why \(AB\) is the diameter of the circle.<span class="inline-marks">[3]</span></p>

Triangle \(ABC\) has \(AB=26\) cm, \(BC=10\) cm, and \(AC=24\) cm. A circle passes through \(A,B,C\).

Explain, with reasons, why \(AB\) is the diameter of the circle.

The Converse of Pythagoras Theorem

Angles in a Semicircle and the Converse

2025 YCKSS PRELIMS P1 Q12

Properties of Circle

Pythagoras’ Theorem

- Use prime factors to explain why 18 x 63 is not a perfect square. - The number 18p is a perfect cube. Find the smallest positive integer value of p.

Use prime factors to explain why \(18\times63\) is not a perfect square.

The number \(18p\) is a perfect cube. Find the smallest positive integer value of \(p\).

Prime Factorisation and Its Applications

Laws of Indices

2025 SST PRELIMS P1 Q2

<p>The diagram shows a coastal reserve bounded by four survey markers \(P\), \(Q\), \(R\) and \(S\) on level ground.</p>
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<p>The bearing of \(P\) from \(S\) is \(342^\circ\) and \(SP=176\text{ m}\). Marker \(Q\) is \(80\text{ m}\) due west of \(P\).</p>
<p>The angle \(PQR\) is \(117^\circ\) and the bearing of \(P\) from \(R\) is \(052^\circ\).</p>
<p>Give non-exact numerical answers correct to \(3\) significant figures unless a different degree of accuracy is stated.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q-a" data-list-item-path="emath-aot-q-a"><div class="list-item-paragraph">Find the bearing of \(R\) from \(P\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-b" data-list-item-path="emath-aot-q-b"><div class="list-item-paragraph">Find \(\angle QPR\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-c" data-list-item-path="emath-aot-q-c"><div class="list-item-paragraph">Show that the distance \(PR\) is \(168.66\text{ m}\), correct to two decimal places.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-d" data-list-item-path="emath-aot-q-d"><div class="list-item-paragraph">Find the area of triangle \(PQR\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q-e" data-list-item-path="emath-aot-q-e"><div class="list-item-paragraph">Find \(\angle RPS\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-f" data-list-item-path="emath-aot-q-f"><div class="list-item-paragraph">Hence, calculate the distance \(SR\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-g" data-list-item-path="emath-aot-q-g"><div class="list-item-paragraph">An observation tower, \(24\text{ m}\) high, stands vertically at \(P\).</div><div class="list-item-paragraph">Find the angle of elevation of the top of the tower from \(S\).<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a coastal reserve bounded by four survey markers \(P\), \(Q\), \(R\) and \(S\) on level ground.

The bearing of \(P\) from \(S\) is \(342^\circ\) and \(SP=176\text{ m}\). Marker \(Q\) is \(80\text{ m}\) due west of \(P\).

The angle \(PQR\) is \(117^\circ\) and the bearing of \(P\) from \(R\) is \(052^\circ\).

Give non-exact numerical answers correct to \(3\) significant figures unless a different degree of accuracy is stated.

Area of Triangle

Bearings

Surveying a coastal reserve

<p>A cylindrical solid has a hemispherical top of the same radius \(r\text{ cm}\). The cylinder has height \(2r\text{ cm}\), and the volume of the entire solid is \(576\pi\text{ cm}^3\).</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/1789189430874-y1vjytfj43.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODkxODk0MzA4NzQteTF2anl0Zmo0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkxODgzMTMwLCJleHAiOjE3OTE4ODM0MzB9.I-dD4AUPE-5nVClV4TruCAOJE60pcsj--v66qDwFo2c" style="max-width: min(150px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i1" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i1"><div class="list-item-paragraph">Show that&nbsp;\(r=6\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i2" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i2"><div class="list-item-paragraph">The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.<span class="inline-marks">[4]</span></div></li></ol>

A cylindrical solid has a hemispherical top of the same radius \(r\text{ cm}\). The cylinder has height \(2r\text{ cm}\), and the volume of the entire solid is \(576\pi\text{ cm}^3\).

The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.

Volume and Surface Area of Cylinders

Applications of Pythagoras’ Theorem

Volume and Surface Area of Spheres

Volume and Surface Area of Cones

Volume and Surface Area of Composite Solids

2025 MGS(S) PRELIMS P2 Q7

Volume and Surface Area of Prisms

Mensuration (Pyramid, Cones and Spheres)

F is directly proportional to v2. Find the percentage change in F when v is reduced by 20%.

\(F\) is directly proportional to \(v^2\). Find the percentage change in \(F\) when \(v\) is reduced by 20%.

Direct Proportion

Percentage Change and Reverse Percentage

2025 SJI PRELIMS P1 Q17

Percentages

Direct and Inverse Proportion

- Express x2-6x+8 in the form a(x+b)2+c. - The curve y=x2-6x+8 is drawn. Using your answer in (a), write down - the coordinates of the turning point, - the equation of the line of symmetry of the curve.

Express \(x^2-6x+8\) in the form \(a(x+b)^2+c\).

The curve \(y=x^2-6x+8\) is drawn. Using your answer in (a), write down

Completing the Square and Vertex Form

Applications of Graphs

2025 MGS(S) PRELIMS P1 Q12

Quadratic Functions, Equations and Inequalities

Functions and Graphs

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<p>\(X\), \(Y\) and \(Z\) are three points on horizontal ground. \(\angle ZXY=35^\circ\), \(\angle XYZ=90^\circ\) and \(XZ=20\text{ cm}\).</p>
<ol class="abc-list"><li data-list-item-id="sec4-em-2025-aiss-p1-q12-b3-i1" data-list-item-path="sec4-em-2025-aiss-p1-q12-b3-i1"><div class="list-item-paragraph">Calculate the distance \(YZ\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="sec4-em-2025-aiss-p1-q12-b3-i2" data-list-item-path="sec4-em-2025-aiss-p1-q12-b3-i2"><div class="list-item-paragraph">Calculate the area of triangle \(XYZ\).<span class="inline-marks">[2]</span></div></li></ol>

\(X\), \(Y\) and \(Z\) are three points on horizontal ground. \(\angle ZXY=35^\circ\), \(\angle XYZ=90^\circ\) and \(XZ=20\text{ cm}\).

Basics of Trigonometric Ratios

2025 AISS PRELIMS P1 Q12

Perimeter and Area of Plane Figures

<p>\(ABD\) is a triangle, \(ADC\) is a straight line, \(AB=8\text{ cm}\), \(AD=9\text{ cm}\), and \(BD=10\text{ cm}\). Without using a calculator, find the exact value of \(\cos\angle BDC\).</p>
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<p>Find the exact value of \(\cos\angle BDC\).<span class="inline-marks">[2]</span></p>

\(ABD\) is a triangle, \(ADC\) is a straight line, \(AB=8\text{ cm}\), \(AD=9\text{ cm}\), and \(BD=10\text{ cm}\). Without using a calculator, find the exact value of \(\cos\angle BDC\).

Find the exact value of \(\cos\angle BDC\).

2025 SJI PRELIMS P1 Q3

Given 2x-y=8 and xy=4, without solving for x and y, find 4x2+y2.

Given \(2x-y=8\) and \(xy=4\), without solving for \(x\) and \(y\), find \(4x^2+y^2\).

Evaluating Expressions using Special Identities

2025 PHS PRELIMS P1 Q17

Solve 8x-3y=25 and 12x+5y=9. You must show your working.

Solve \(8x-3y=25\) and \(12x+5y=9\). You must show your working.

Solving Simultaneous Linear Equations using Elimination Method

2025 YCKSS PRELIMS P1 Q13

Simultaneous Equations

<p>\(\mathcal E=\{x\in\mathbb Z:3&lt;x&lt;18\}\), \(P=\{\text{multiples of}\hspace{0.5em}5\}\), and \(Q=\{\text{perfect squares}\hspace{0.15em}\}\).</p>
<ol class="abc-list"><li data-list-item-id="sec4-em-2025-sst-p1-q1-b2-i1" data-list-item-path="sec4-em-2025-sst-p1-q1-b2-i1"><div class="list-item-paragraph">List the elements in \((P\cup Q)'\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="sec4-em-2025-sst-p1-q1-b2-i2" data-list-item-path="sec4-em-2025-sst-p1-q1-b2-i2"><div class="list-item-paragraph">On the Venn diagram, shade the region representing \((P\cap Q)\cup(P\cap Q')\).<span class="inline-marks">[1]</span></div><div class="list-item-after-content"><div class="graph-block" data-graph-config="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" data-width="420" data-height="260" style="width:420px;height:260px;max-width:100%;display:block;margin:0 auto;"></div></div></li></ol>

\(\mathcal E=\{x\in\mathbb Z:3<x<18\}\), \(P=\{\text{multiples of }5\}\), and \(Q=\{\text{perfect squares}\}\).

On the Venn diagram, shade the region representing \((P\cap Q)\cup(P\cap Q')\).

Venn Diagrams, Universal Sets and Complement of a Set

Combined Operations on Sets

2025 SST PRELIMS P1 Q1

Sets

Write 12-x- 3x+2x2-7x+10 as a single fraction in its simplest form.

Write \(\dfrac1{2-x}-\dfrac{3x+2}{x^2-7x+10}\) as a single fraction in its simplest form.

2025 SJI PRELIMS P1 Q10

<p>A warehouse is \(18\text{ m}\) long and \(12\text{ m}\) wide, with vertical walls \(6\text{ m}\) high.</p>
<div class="graph-block" data-graph-config="eyJheGlzIjpmYWxzZSwiZ3JpZCI6ZmFsc2UsIndpZHRoIjo1MDAsImhlaWdodCI6MzMwLCJlbGVtZW50cyI6W3siaWQiOiJxMTEtYWIiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAsMF0sInBvaW50MiI6WzcsMF0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtYmMiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzcsMF0sInBvaW50MiI6WzEwLDJdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoicTExLWNkIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlsxMCwyXSwicG9pbnQyIjpbMywyXSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6InExMS1kYSIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMywyXSwicG9pbnQyIjpbMCwwXSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6InExMS1hZSIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMCwwXSwicG9pbnQyIjpbMCwzLjNdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoicTExLWJmIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOls3LDBdLCJwb2ludDIiOls3LDMuM10sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtY2ciLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzEwLDJdLCJwb2ludDIiOlsxMCw1LjNdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoicTExLWRoIiwiZGFzaCI6MiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlszLDJdLCJwb2ludDIiOlszLDUuM10sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtZWYiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAsMy4zXSwicG9pbnQyIjpbNywzLjNdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoicTExLWZnIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOls3LDMuM10sInBvaW50MiI6WzEwLDUuM10sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtZ2giLCJkYXNoIjoyLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzEwLDUuM10sInBvaW50MiI6WzMsNS4zXSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6InExMS1oZSIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMyw1LjNdLCJwb2ludDIiOlswLDMuM10sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtZWkiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAsMy4zXSwicG9pbnQyIjpbMy41LDYuNV0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtZmkiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzcsMy4zXSwicG9pbnQyIjpbMy41LDYuNV0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtaGoiLCJkYXNoIjoyLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzMsNS4zXSwicG9pbnQyIjpbNi41LDguMl0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtZ2oiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzEwLDUuM10sInBvaW50MiI6WzYuNSw4LjJdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoicTExLWlqIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlszLjUsNi41XSwicG9pbnQyIjpbNi41LDguMl0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJxMTEtYWciLCJkYXNoIjoyLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAsMF0sInBvaW50MiI6WzEwLDUuM10sInN0cm9rZVdpZHRoIjoxLjJ9LHsiaWQiOiJxMTEtYWoiLCJkYXNoIjoyLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzAsMF0sInBvaW50MiI6WzYuNSw4LjJdLCJzdHJva2VXaWR0aCI6MS4yfSx7IngiOjAsInkiOjAsImlkIjoicTExLWEiLCJ0eXBlIjoicG9pbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludFNpemUiOjEuMywicG9pbnRTdHlsZSI6ImNpcmNsZSJ9LHsieCI6MTAsInkiOjUuMywiaWQiOiJxMTEtZyIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50U2l6ZSI6MS4zLCJwb2ludFN0eWxlIjoiY2lyY2xlIn0seyJ4IjozLjUsInkiOjYuNSwiaWQiOiJxMTEtaSIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50U2l6ZSI6MS4zLCJwb2ludFN0eWxlIjoiY2lyY2xlIn0seyJ4Ijo2LjUsInkiOjguMiwiaWQiOiJxMTEtaiIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50U2l6ZSI6MS4zLCJwb2ludFN0eWxlIjoiY2lyY2xlIn0seyJ4IjotMC4zNSwieSI6LTAuMiwiaWQiOiJxMTEtbGEiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoQVxcKSJ9LHsieCI6MTAuMTIsInkiOjUuMywiaWQiOiJxMTEtbGciLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoR1xcKSJ9LHsieCI6My4yNSwieSI6Ni43MywiaWQiOiJxMTEtbGkiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoSVxcKSJ9LHsieCI6NS45NiwieSI6OC4yNCwiaWQiOiJxMTEtbGoiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoSlxcKSJ9LHsieCI6My4yLCJ5IjotMC4zNSwiaWQiOiJxMTEtMTIiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoMTJcXHRleHR7IG19XFwpIn0seyJ4Ijo4LjY1LCJ5IjowLjc1LCJpZCI6InExMS0xOCIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCgxOFxcdGV4dHsgbX1cXCkifSx7IngiOjcuMTUsInkiOjEuNTUsImlkIjoicTExLTYiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoNlxcdGV4dHsgbX1cXCkifSx7IngiOjEuMSwieSI6NS4zNSwiaWQiOiJxMTEtNzUiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoNy41XFx0ZXh0eyBtfVxcKSJ9XSwieEF4aXNMYWJlbCI6IiIsInlBeGlzTGFiZWwiOiIiLCJib3VuZGluZ0JveCI6Wy0xLDguNSwxMS4zLC0xLjJdLCJvcmlnaW5MYWJlbCI6IiIsInNob3dBeGlzTGFiZWxzIjpmYWxzZSwic2hvd05hdmlnYXRpb24iOmZhbHNlLCJrZWVwQXNwZWN0UmF0aW8iOmZhbHNlfQ==" data-width="500" data-height="330" style="width:500px;height:330px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Its symmetric pitched roof has sloping edges \(7.5\text{ m}\) long in each end cross-section. The ridge is directly above the centre line of the warehouse.</p>
<p>Give non-exact numerical answers correct to \(3\) significant figures.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q11-q-a" data-list-item-path="emath-aot-q11-q-a"><div class="list-item-paragraph">Find the vertical height of the roof ridge above the ground.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q11-q-b" data-list-item-path="emath-aot-q11-q-b"><div class="list-item-paragraph">Find the distance from a bottom corner to the opposite top corner of the vertical walls.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q11-q-c" data-list-item-path="emath-aot-q11-q-c"><div class="list-item-paragraph">From the same bottom corner, find the angle of elevation of the ridge point at the opposite end.<span class="inline-marks">[3]</span></div></li></ol>

A warehouse is \(18\text{ m}\) long and \(12\text{ m}\) wide, with vertical walls \(6\text{ m}\) high.

Its symmetric pitched roof has sloping edges \(7.5\text{ m}\) long in each end cross-section. The ridge is directly above the centre line of the warehouse.

Trigonometry in Two and Three Dimensions

Warehouse crane beneath a roof

Quadrilateral ABCD is made from triangles ABC and BCD. AB=3 cm, AC=4 cm, BD=2 cm, CD= 3 cm, and angle ABC=90. - Find BC. - Is triangle BCD right-angled? Show your working clearly.

Quadrilateral \(ABCD\) is made from triangles \(ABC\) and \(BCD\). \(AB=3\text{ cm}\), \(AC=4\text{ cm}\), \(BD=2\text{ cm}\), \(CD=\sqrt3\text{ cm}\), and angle \(ABC=90^\circ\).

Is triangle \(BCD\) right-angled? Show your working clearly.

E Math Question Bank