A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>$5-2$ </p>
<p>$5-\left( -2 \right)$ </p>
<p>$-5-\left( -2 \right)$</p>

Addition and Subtraction involving Negative Numbers

Subtraction involving Negative Numbers

Real Numbers

Shading and Identifying Set Regions

Foundations of Set Theory: Mastering Union and Intersection Operations

Sets

Increasing Speed; Constant Speed; Decreasing Speed

Interpreting and Sketching Motion Graphs

Distance Travelled and Average Speed

Conversion of Speed-Time Graph to Distance-Time Graph

Functions and Graphs

<p>If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by </p>
<p style="text-align: center">$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$</p>
<p>This formula for solving quadratic equations is usually used when the <strong>quadratic expression cannot be factorised easily. </strong></p>
<p style="margin-top: 1.5rem"><strong>Tip: </strong></p>
<p>Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.</p>

If $a{{x}^{2}}+bx+c=0$ , where where $a,b$ and $c$ are real numbers and $a\ne 0$, then the solutions are given by 

$x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

This formula for solving quadratic equations is usually used when the **quadratic expression cannot be factorised easily. **

Always express the quadratic equation in the general form $a{{x}^{2}}+bx+c=0$ so that the correct value for $a,b$ and $c$ can be substituted into the formula.

Solving Quadratic Equations Using the Quadratic Formula

Solving Quadratic Equations by using Formula

Quadratic Functions, Equations and Inequalities

<p>The Number Line</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760777236067-bvb4a4kmsaf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA3NzcyMzYwNjctYnZiNGE0a21zYWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzNzE5OCwiZXhwIjoxNzkzNTM3NDk4fQ.LXPzfmKaqb-Od_vUGB8EHDcdvPB4z67dn9f8XvrLhVY" width="400" class="aligncenter" /></figure>
<p>Mark the following numbers on the number line:</p>
<p>$4$, $-6$, $-3$, $\frac{5}{7}$, $3\frac{1}{2}$, $-\frac{3}{4}$, $\pi$, $1.\dot{3}$</p>

Mark the following numbers on the number line:

$4$, $-6$, $-3$, $\frac{5}{7}$, $3\frac{1}{2}$, $-\frac{3}{4}$, $\pi$, $1.\dot{3}$

Ordering Real Numbers and the Number Line

Negative and Positive Numbers

<p>When subtracting two vectors $\overrightarrow{AB}$ and $\overrightarrow{CB}$:</p>
<p style="text-align: center">$\overrightarrow{AB} - \overrightarrow{CB} = \overrightarrow{AC}$</p>
<p>For any two column vectors, $\mathbf{a} = \begin{pmatrix} p \\ q \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} r \\ s \end{pmatrix}$, then</p>
<p style="margin-left: 12.5rem">$\!\!\!\!$ $\mathbf{a} - \mathbf{b} = \begin{pmatrix} p - r \\ q - s \end{pmatrix}$</p>

When subtracting two vectors $\overrightarrow{AB}$ and $\overrightarrow{CB}$:

$\overrightarrow{AB} - \overrightarrow{CB} = \overrightarrow{AC}$

For any two column vectors, $\mathbf{a} = \begin{pmatrix} p \\ q \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} r \\ s \end{pmatrix}$, then

$\!\!\!\!$ $\mathbf{a} - \mathbf{b} = \begin{pmatrix} p - r \\ q - s \end{pmatrix}$

Vector Subtraction

Vectors

<ol class="abc-list"><li data-list-item-id="tkss-p2-q11-q-0-content-items-0" data-list-item-path="tkss-p2-q11-q-0-content-items-0"><div class="list-item-paragraph">Prove $\dfrac{\cos 2\theta}{\tan \theta}+\sin 2\theta=\cot \theta$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="tkss-p2-q11-q-0-content-items-1" data-list-item-path="tkss-p2-q11-q-0-content-items-1"><div class="list-item-paragraph">Solve $3\sin x\tan x-\sec x=1$ for&nbsp;$-180^\circ\leq x\leq 180^\circ$.<span class="inline-marks">[6]</span></div></li></ol>

Prove $\dfrac{\cos 2\theta}{\tan \theta}+\sin 2\theta=\cot \theta$.

Solve $3\sin x\tan x-\sec x=1$ for $-180^\circ\leq x\leq 180^\circ$.

Trigonometric Equations

Compound and Double-Angle Identities

Proving and Simplifying Trigonometric Identities

2025 TKSS PRELIMS P2 Q11

Trigonometry

It is given that $\sin A=-\frac{1}{\sqrt{5}}$ and $\tan B=-\frac{3}{4}$, where $A$ and $B$ are in the same quadrant. Without using a calculator, find the exact value of

Given that $\sin A=q$, where $A$ is an obtuse angle, express $\tan^2 A$ in terms of $q$.

Symmetry of Trigonometric Functions

Exact Trigonometric Values and the Unit Circle

2025 NASS PRELIMS P1 Q6

<p>Solve the equation $9 \sin x = -2 \sec x$ for&nbsp;$-\pi \le x \le 0$.<span class="inline-marks">[4]</span></p>

Solve the equation $9 \sin x = -2 \sec x$ for $-\pi \le x \le 0$.

2025 BVSS PRELIMS P1 Q2

<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

<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

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

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.

Basics of Pythagoras’ Theorem

2025 NASS PRELIMS P1 Q9

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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

Area of Triangle

2025 AISS PRELIMS P1 Q12

Perimeter and Area of Plane Figures

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Without using a calculator, solve the inequality $\frac{4}{x + 2} \geqslant \frac{x - 3}{x}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph">Hence, solve the inequality $\frac{4}{\vert{}x\vert{} + 2} \geqslant \frac{\vert{}x\vert{} - 3}{\vert{}x\vert{}}$.<span class="inline-marks">[2]</span></div></li></ol>

Without using a calculator, solve the inequality $\frac{4}{x + 2} \geqslant \frac{x - 3}{x}$.

Hence, solve the inequality $\frac{4}{\vert{}x\vert{} + 2} \geqslant \frac{\vert{}x\vert{} - 3}{\vert{}x\vert{}}$.

Inequalities involving Substitution

Inequalities involving Rational Functions

Inequalities involving Modulus Functions

N2024 P1 Q4

Inequalities

<p>Two vectors $\mathbf{a}$ and $\mathbf{b}$ are such that&nbsp;$\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3$. It is given that&nbsp;$\mathbf{a}$ is a unit vector and $\mathbf{b} \cdot \mathbf{b} = 9$.</p>
<p>Show that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are perpendicular.<span class="inline-marks">[3]</span></p>

Two vectors $\mathbf{a}$ and $\mathbf{b}$ are such that $\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3$. It is given that $\mathbf{a}$ is a unit vector and $\mathbf{b} \cdot \mathbf{b} = 9$.

Show that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

Vector (Cross) Product

Perpendicular Vectors Using the Scalar Product

Scalar (Dot) Product

N2024 P1 Q2

<p>A warehouse is $18\text{ m}$ long and $12\text{ m}$ wide, with vertical walls $6\text{ m}$ high.</p>
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<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.

Give non-exact numerical answers correct to $3$ significant figures.

Trigonometry in Two and Three Dimensions

Angles of Elevation and Depression

Warehouse crane beneath a roof

<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.

Hill lookout and observation posts

<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 term, $T_n$, 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 $16$.

General Term of a Sequence

Testing Sequence Membership

2022 AMKSS EOY P1 Q3

Number Patterns

<p>The curve $y=\tan x$ meets the line $y=-\dfrac{2\sqrt3}{\pi}x+\dfrac{2}{\sqrt3}$ at $A$, where the $x$-coordinate of $A$ is $\dfrac{\pi}{6}$. The line meets the positive $x$-axis at $B$.</p>
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<ol class="i-list"><li data-list-item-id="amarea-q21-parts-item-1" data-list-item-path="amarea-q21-parts-item-1"><div class="list-item-paragraph">Find the coordinates of $B$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amarea-q21-parts-item-2" data-list-item-path="amarea-q21-parts-item-2"><div class="list-item-paragraph">Find the exact area of the shaded region bounded by the curve, the line and the $x$-axis.<span class="inline-marks">[5]</span></div></li></ol>

The curve $y=\tan x$ meets the line $y=-\dfrac{2\sqrt3}{\pi}x+\dfrac{2}{\sqrt3}$ at $A$, where the $x$-coordinate of $A$ is $\dfrac{\pi}{6}$. The line meets the positive $x$-axis at $B$.

Find the exact area of the shaded region bounded by the curve, the line and the $x$-axis.

IP Math Question Bank