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Laws of Indices

Sec 3/4 E Math T03-Q03 - Solving an exponential equation

Indices

<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.

Applications of Trigonometry — General

Boundary of a wildlife reserve

Trigonometry

<p>Mei Lin deposits $ \$3000$ in an account that pays compound interest at $2\%$ per year.</p>
<ol class="abc-list"><li data-list-item-id="t03-q043-parts-1" data-list-item-path="t03-q043-parts-1"><div class="list-item-paragraph">Calculate the amount in the account after $4$ years.</div></li><li data-list-item-id="t03-q043-parts-2" data-list-item-path="t03-q043-parts-2"><div class="list-item-paragraph">Hence, calculate the total interest earned.</div></li></ol>

Mei Lin deposits $ \$3000$ in an account that pays compound interest at $2\%$ per year.

Calculate the amount in the account after $4$ years.

Hence, calculate the total interest earned.

Compound Interest

Sec 3/4 E Math T03-Q43 - Calculating compound interest

Standard Form and Compound Interest

<p>Solve the simultaneous inequalities $-4\le 3x+2&lt;17$.</p>

Solve the simultaneous inequalities $-4\le 3x+2<17$.

Solving Simultaneous Linear Inequalities

Sec 3/4 E Math T02-Q01 - Solving simultaneous linear inequalities

Linear Equations and Inequalities

<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.

Prime Numbers and Composite Numbers

Algebraic Fractions

2025 YCKSS PRELIMS P2 Q1

Algebraic Expressions, Identities and Formulae

Primes, HCF and LCM

<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>
<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>$P$ 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.

Hill lookout and observation posts

<p>A straight line passes through $P(-2,-1)$ and $Q(3,3)$. Find its equation in the form $ax+by=c$, where&nbsp;$a$, $b$ and $c$ are integers.</p>

A straight line passes through $P(-2,-1)$ and $Q(3,3)$. Find its equation in the form $ax+by=c$, where $a$, $b$ and $c$ are integers.

Equation of a Straight Line

Sec 3/4 E Math T04-Q02 - Finding the equation of a straight line

Coordinate Geometry

<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.

Surveying a coastal reserve

<p>The points $A(1,6)$, $B(4,16)$ and $C(7,p)$ are collinear. Find $p$.</p>

The points $A(1,6)$, $B(4,16)$ and $C(7,p)$ are collinear. Find $p$.

Gradient of a Straight Line

Sec 3/4 E Math T04-Q11 - Using collinearity to find an unknown coordinate

<p>Mei Lin deposits $ \$5000$ in an account that pays compound interest at $2\%$ per year.</p>
<ol class="abc-list"><li data-list-item-id="m02-q002-parts-1" data-list-item-path="m02-q002-parts-1"><div class="list-item-paragraph">Calculate the amount in the account after $4$ years.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="m02-q002-parts-2" data-list-item-path="m02-q002-parts-2"><div class="list-item-paragraph">Hence, calculate the total interest earned.<span class="inline-marks">[4]</span></div></li></ol>

Mei Lin deposits $ \$5000$ in an account that pays compound interest at $2\%$ per year.

Sec 3/4 E Math Mock 2-Q02 - Mock Paper 2: Calculating compound interest

Consider the expression $x^2-4x-3$.
Express $x^2-4x-3$ in the form $(x-a)^2-b$.
Hence, solve $x^2-4x-3=0$, giving your answers correct to 2 decimal places.

Express $x^2-4x-3$ in the form $(x-a)^2-b$.

Hence, solve $x^2-4x-3=0$, giving your answers correct to 2 decimal places.

Solving Quadratic Equation by Completing the Square

Sec 3/4 E Math T01-Q25 - Completing the square and solving a quadratic equation

Quadratic Functions, Equations and Inequalities

<p>A machine is bought for&nbsp;$\$45000$. Its value depreciates by $14\%$ each year. Calculate its value after $7$ years, giving your answer to the nearest dollar.</p>

A machine is bought for $\$45000$. Its value depreciates by $14\%$ each year. Calculate its value after $7$ years, giving your answer to the nearest dollar.

Sec 3/4 E Math T03-Q07 - Applying repeated percentage depreciation

<p>A sample contains $2.8\times10^{18}$ identical particles. The mass of each particle is $4.1\times10^{-22}$ grams. Calculate the total mass of the particles, giving your answer in standard form.<span class="inline-marks">[2]</span></p>

A sample contains $2.8\times10^{18}$ identical particles. The mass of each particle is $4.1\times10^{-22}$ grams. Calculate the total mass of the particles, giving your answer in standard form.

Standard Form

Sec 3/4 E Math Mock 1-Q04 - Mock Paper 1: Applying standard form to a scientific quantity

<p>Solve $\dfrac{3}{x-3}+\dfrac{2}{x+4}=1$, giving your answers correct to 2 decimal places where necessary.<span class="inline-marks">[2]</span></p>

Solve $\dfrac{3}{x-3}+\dfrac{2}{x+4}=1$, giving your answers correct to 2 decimal places where necessary.

Solving Equations involving Algebraic Fractions

Sec 3/4 E Math Mock 1-Q01 - Mock Paper 1: Solving an equation involving algebraic fractions

<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>
<div class="graph-block" data-graph-config="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" data-width="369" data-height="190" style="width:369px;height:190px;max-width:100%;display:block;margin:0 auto;"></div>
<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 $\cos\angle BDC$.

Cosine Rule

2025 SJI PRELIMS P1 Q3

<p>The first four terms in a sequence are $1, 4, 9$ and $16$.</p>
<ol class="abc-list"><li data-list-item-id="item-1785756533528-2co5fuelw" data-list-item-path="item-1785756533528-2co5fuelw"><div class="list-item-paragraph"><p>Write down the $6$th term of this sequence.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1785756534704-9z7mggxiy" data-list-item-path="item-1785756534704-9z7mggxiy"><div class="list-item-paragraph"><p>Find an expression, in terms of $n$, for the $n$ th, $T_n$, of the sequence.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1785756630861-b8kpl7gee" data-list-item-path="item-1785756630861-b8kpl7gee"><div class="list-item-paragraph"><p>Is the number $99$ part of the sequence above? Explain your answer.</p><span class="inline-marks">[1]</span></div></li></ol>

The first four terms in a sequence are $1, 4, 9$ and $16$.

2022 AMKSS EOY P1 Q3

Number Patterns

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

Factorise $x^2+x-20$ completely.
Hence, solve the equation $x^2+x-20=0$.

Hence, solve the equation $x^2+x-20=0$.

Solving Quadratic Equations

Sec 3/4 E Math T01-Q32 - Factorising and solving a quadratic equation

<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="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" 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.

E Math Question Bank